#statistical thermodynamics

[secondwheel]

Now it is our turn to study statistical mechanics...

[oliviabutsmart]

Physics Friday #7: It's getting hot in here! - An explanation of Temperature, Entropy, and Heat

THE PEOPLE HAVE SPOKEN. This one was decided by poll. The E = mc^2 post will happen sometime later in the future

Preamble: Thermodynamic Systems

Education level: High School (Y11/12)

Topic: Statistical Mechanics, Thermodynamics (Physics)

You'll hear the word system being used a lot ... what does that mean? Basically a thermodynamic system is a collection of things that we think affect each other.

A container of gas is a system as the particles of gas all interact with each other.

The planet is a system because we, and all life/particles on earth all interact together.

Often, when dealing with thermodynamic systems we differentiate between open, closed, and isolated systems.

An open system is where both the particles and energy contained inside the system interact outside the system. Closed systems only allow energy to enter and exit the system (usually via a "reservoir").

We will focus mainly on isolated systems, where nothing enters or exits the system at all. Unless if we physically change what counts as the "system".

Now imagine we have a system, say, a container of gas. This container will have a temperature, pressure, volume, density, etc.

Let's make an identical copy of this container and then combine it with it's duplicate.

What happens to the temperature? Well it stays the same. Whenever you combine two objects of the same temperature they stay the same. If you pour a pot of 20 C water into another pot of 20 C water, no temperature change occurs.

The same occurs with pressure and density. While there are physically more particles in the system, the volume has also increased.

This makes things like Temperature, Pressure, and Density intensive properties of a system - they don't change when you combine systems with copies of itself. They act more like averages.

However, duplicating a system and combining it with itself causes the volume to double, it also doubles the amount of 'stuff' inside the system.

Thus things like volume are called intensive, as they appear to be altered by count and size, they act more like proportional values.

This is important in both understanding heat and temperature. The energy of a system is an intensive property, whereas temperature is intensive.

Temperature appears to be a sort of average of thermal energy, which is how we can analyse it - but this only works in the case of gasses, which isn't universal. It's useful to use a more abstract definition of temperature.

Heat, on the other hand, is much more real. It is concerned with the amount of energy transferred cased by a change in temperature. This change is driven by the second law of thermodynamics, which requires a maximisation of entropy.

But instead of tackling this from a high-end perspective, let's jump into the nitty-gritty ...

Microstates and Macrostates

The best way we can demonstrate the concept of Entropy is via the analogy of a messy room:

We can create a macrostate of the room by describing the room:

There's a shirt on the floor

The bed is unmade

There is a heater in the centre

The heater is broken

Note how in this macrostate, we can have several possible arrangements that describe the same macrostate. For example, the shirt on the floor could be red, blue, black, green.

A microstate of the room is a specific arrangement of items, to the maximum specificity we require.

For example the shirt must be green, the right heater leg is broken. If we care about even more specificity we could say a microstate of the system is:

Atom 1 is Hydrogen in position (0, 0, 0)

Atom 2 is Carbon in position (1, 0, 0)

etc.

Effectively, a macrostate is a more general description of a system, while a microstate is a specific iteration of a system.

A microstate is considered attached to a macrostate if the conditions of the macrostate are required. "Dave is wearing a shirt" and "Dave is wearing a red shirt" can both be true, but it's clear that if Dave is wearing a red shirt, he is also wearing a shirt.

What Entropy Actually is (by Definition)

The multiplicity of a microstate is the total amount of microstates attached to it. It is basically a "count" of the total permutations given the restrictions.

We give the multiplicity an algebraic number Ω.

What we define entropy as is the natural logarithm of Ω.

S = k ln Ω

Where k is Boltzmann's constant, to give the entropy units. The reason why we take the logarithm is:

Combinatorics often involve factorials and exponents a.k.a. big numbers, so we use the log function to cut it down to size

When you combine a system with a macrostate of Ω=X with a macrostate of Ω=Y, the total multiplicity is X × Y. So this logarithm makes Entropy extensive

So that's what Entropy is, a measure of the amount of possible rearrangements of a system given a set of preconditions.

Order and Chaos

So how do we end up with the popular notion that Entropy is a measure of chaos, well, consider a sand castle,

Image Credit: Wall Street Journal

A sand castle, compared the surrounding beach, is a very specific structure. It requires specific arrangements of sand particles in order to form a proper structure.

This is opposed to the beach, where any loose arrangement of sand particles can be considered a 'beach'.

In this scenario, the castle has a very low multiplicity, because the macrostate of 'sandcastle' requires a very restrictive set of microstates. Whereas a sand dune has a very large set of possible microstates.

In this way, we can see how the 'order' and 'structure' of the sand castle results in a low-entropy system.

However this doesn't explain how we can get such complex systems if energy is always meant to increase. Like how can life exist if the universe intends to make things a mess of particles, AND the universe started as a mess of particles.

The consideration to make, as we'll see, is that chaos is not the full picture, large amounts of energy can be turned into making entropy lower.

Energy Macrostates

There's still a problem with our definition. Consider two macrostates:

The room exists

Atom 1 is Hydrogen in position (0, 0, 0), Atom 2 is Carbon in position (1, 0, 0), etc.

Macrostate one has a multiplicity so large it might as well be infinite, and is so general it encapsulates all possible states of the system.

Macrostate two is so specific that it only has a multiplicity of one.

Clearly we need some standard to set macrostates to.

What we do is that we define a macrostate by one single parameter: the amount of thermal energy in the system. We can also include things like volume or the amount of particles etc. But for now, a macrostate corresponds to a specific energy of the system.

This means that the multiplicity becomes a function of thermal energy, U.

S(U) = k ln Ω(U)

The Second Law of Thermodynamics

Let's consider a system which is determined by a bunch of flipped coins, say, 10 flipped coins.

H T H T H H H T T H

This may seem like a weird example, but there is a genuine usefulness to this. For example, consider atoms aligned in a magnetic field.

We can define the energy of the system as being a function of the amount of heads we see. Thus an energy macrostate would be "X coins are heads".

Let's now say that every minute, we reset the system. i.e. we flip the coins again and use the new result.

Consider the probability of ending up a certain macrostate every minute. We can use the binomial theorem to calculate this probability:

Here, the column notation gives the choose function, which accounts for duplicates, as we are not concerned with the order in which we get any two tails or heads.

The p to the power of k is the probability of landing a head (50%) to the power of the number of heads we get (k). n-k becomes the number of tails obtained.

The choose function is symmetric, so we end up with equal probabilities with k heads as with k tails.

Let's come up with some scenarios:

There are an equal amount of heads and tails flipped

There is exactly two more heads than tails (i.e. 6-4)

All coins are heads except for one

All coins are heads

And let's see these probabilities:

Clearly, it is more likely that we find an equal amount of coins, but all coins being heads is not too unlikely. Also notice that the entropy correlates with probability here. A large entropy is more likely to occur.

Let's now increase the number of coin flips to 1000:

Well, now we can see this probability difference much more clearly. The "all coins are heads" microstate is vanishingly unlikely, and microstates close to maximum entropy are very likely comparatively.

If we keep expanding the amount of flips, we end up pushing the limits of this relationship. And thus we get the tendency for this system to maximise the entropy, simply because it's most likely.

In real life, systems don't suddenly reset and restart. Instead we want to consider a system where every minute, one random coin is selected and then flipped.

Consider a state in maximum entropy undergoing this change. It's going to take an incredibly long amount of time to perform something incredibly unlikely.

But for a state in minimum entropy, any deviation from the norm brings the entropy higher.

Thus the system has the tendency to end up being "trapped" in the higher entropy states.

This is the second law of thermodynamics. It doesn't actually make a statement about a particularly small system. But for small systems we deal with statistics differently. For large systems, we end up ALWAYS seeing a global increase in entropy.

How we get temperature

Temperature is usually defined as the capacity to transfer thermal energy. It sort of acts like a "heat potential" in the same way we have a "gravitational potential" or "electrical potential".

Temperature and Thermal Energy

What is the mathematical meaning of temperature?

Temperature is defined as a rate of change, specifically:

(Apologies for the fucked image formatting)

(Reminder this assumes Entropy is exclusively a function of energy)

The reason we define it as a derivative of entropy as entropy is an emergent property of a system. But thermal energy is something we can personally change.

What this rule means is that a system with a very low temperature will react greatly to minor inputs in energy. A low temperature system thus is really good at drawing energy from outside.

Alternatively, a system with very high temperature will react very slightly to minor inputs in energy.

At the extremes, absolute zero is where any change to the internal energy of the system will cause the system to escape absolute zero. According to the third law of thermodynamics this is where entropy reaches a constant value, because it can't be changed any more after being changed by an infinite amount.

Infinite temperature is where it's effectively impossible to pump more heat into the system, because the system is so resistant to accepting new energy.

Negative Temperature???

Considering our coin-flipping example, let's try and define some numbers. Let's say that the thermal energy of the system is equal to the amount of heads flipped.

This gives us an entropy of:

S ≈ U ln[n/U - 1] + n ln[1 - U/n]

(hopefully this is correct)

The derivative of this is:

dS/dU = ln[n/U - 1] - 2n/(n-U)

Note how this value becomes negative if n is large enough. Implying a negative temperature!

But how is this possible? What does this mean?

Negative temperatures are nothing out of the ordinary actually, they just mean that inputting energy decreases entropy and removing energy increases entropy.

What this means is that a system at a negative temperature, when put in contact with another system, will spontaneously try to dump energy out of itself as it aims to increase entropy.

This actually means that negative temperature is beyond infinite temperature. A fully expanded temperature scale looks like this:

[The Planck temperature is the largest possible temperature that won't create a black hole at 1.4 × 10³² Kelvin]

[Note that 0 K = -273.15 C = -459.67 F and 273.15 K = 0 C = 32 F]

This implies that -0 is sort of the 'absolute hot'. A temperature so hot that the system will desperately try to bleed energy much like the absolute zero system tries to suck in energy.

Heat and the First Law of Thermodynamics

So, what do we do with this information then? How do we actually convert this into something meaningful?

Here, we start to pull in the first law of thermodynamics, which originates with the thermodynamic identity:

dU = T dS - P dV

Note that these dx parts just mean a 'tiny change' in that variable. Here, U is expanded to include situations where the thermal energy of the system has to include things like compression/expansion work done to the system.

This identity gives us the first law:

dU = Q - W

Where Q, the heat energy of the system is re-defined as being dS/dQ = 1/T.

And W, the work (mechanical) energy of the system is defined as being dU/dV.

Both heat and work describe the changes to the energy of the system. Heating a system means you are inputting an energy Q.

If no energy is entering or exiting the system, then we know that any work that is being applied must be matched by an equal change in heat energy.

Since we managed to phrase temperature as a function of thermal energy, we can now develop what's known as an equation of state of the system.

For an ideal gas made of only hydrogen, we have an energy:

U = 3/2 NkT

Where N is the number of particles and k is the boltzmann constant.

We can also define another equation of state for an ideal gas:

PV = NkT

Which is the ideal gas law.

So what is heat then?

Q is the change of heat energy in a system, whereas U is the total thermal energy.

From the ideal gas equation of state, the thermal energy is proportional to temperature. In most cases, we can often express thermal energy as being temperature scaled to size.

Thermal energy is strange. Unlike other classical forms of energy, it can be difficult to classify it as potential or kinetic.

It's a form of kinetic energy as U relies on the movement of particles and packets within the system.

It's a form of potential energy, as it has the potential to transfer it's energy to others.

Generally, the idea that thermal energy is related to temperature is related to the 'speed' at which particles move is not too far off. In fact, we often relate:

1/2 m v^2 = 3/2 N k T

When performing calculations, as it's generally useful.

Of course, temperature, as aforementioned, is a sort of potential describing the potential energy (that can be transferred) per particle.

Heat, then effectively, is the transfer of thermal energy caused by differences in temperature. Generally, we quantify the ability for a system to give off heat using it's heat capacity - note that this is different from temperature.

Temperature is about how much the system wants to give off energy, whereas heat capacity is how well it's able to do that. Heat aims to account for both.

Conclusion

This post honestly was a much nicer write-up. And I'd say the same about E = mc^2. The main reason why is because I already know about this stuff, I was taught about it all 1-4 years ago in high school or university.

My computer is busted, so I'm using a different device to work on this. And because of that I do not have access to my notes. So I don't actually know what I have planned for next week. I think I might decide to override it anyways with something else. Maybe the E = mc^2 one.

As always, feedback is very welcome. Especially because this topic is one I should be knowledgeable about at this point.

Don't forget to SMASH that subscribe button so I can continue to LARP as a youtuber. It's actually strangely fun saying "smash the like button" - but I digress. It doesn't ultimately matter if you wanna follow idc, some people just don't like this stuff weekly.

[forgottenbones]

[fingons-rad-harp]

maybe the real eldritch horror was the math we did along the way

[thatoneluckybee]

What does this mean,

[teet-swea]

excuse me wh a t???????? im sorry i need to know more desperately

Update on this!! It was not quantum and I just misremembered! It was statistical mechanics, which is thermodynamics! Not too far off but still horrendous!

[factsandstuffbyd]

[wtdore]

continually thinking about how people misunderstand Tenet as a time-travel film, when of course it's really a thermodynamics film.

And as we all know, its impossible to understand an introductory course in thermodynamics.

[otaku553]

Flames (Part 1)

(NEXT) ->

(Spade Pirate Sabo AU Masterpost)

This one took AGES i hope you all enjoy!! It's probably a 2-parter but pending how my finals week is looking I might take a hiatus for next week. As I'm writing this post I've just finished my stat mech final which is great! (I wish unspeakable terrors on the person who decided that stat mech was required for the neurobiology-focused major. i get that it's useful for some bayesian statistics but like,,,,, come on i dont need to know thermodynamics)

I did originally have 2 more pages to post with this but scrapped them unfortunately, but I do hope the revised version, which will come out with the next part, will be enjoyable. I was also waffling briefly on whether to color this comic since it turned out so long but there were some panels that I envisioned so clearly in color that I just decided I might as well color everything. I'm especially proud of how the fruit turned out, I want to eat it

Feel free to send in asks if you want annotations of different things I decided on and details I added in each page :) I'm quite proud of this one hehe

[thymesignature]

Honestly I'm gonna really miss this stat mech class. It was a ton of work but also so much fun.

[subskz]

ʚïɞ butterfly bandage - 01

note: this is part 1 of a series (part 2, part 3, part 4, part 5)

content: bang chan/reader, university au, strangers to friends to lovers, themes of soulmates, reader is female and referred to with she/her pronouns, mentions of past unhealthy relationships, a bit of pining, kissing, slight suggestiveness but sfw (eventually nsfw)

summary: after the past three years you’ve had, whether or not you make it through the fourth all comes down to a single thread. fortunately, you find that thread, with chan on the other end. now, it’s just a matter of who needs it more—you, or him.

word count: 15.7k

By the time the spring semester of your senior year rolled around, you were coming apart at the seams.

It was subtle, not something anyone else would notice—you wouldn’t let them. Angling and maneuvering yourself so that it could never be visible to others was a skill that came all too naturally.

Still, you knew it wasn’t a question of if those seams would ever come completely loose, it was a question of when.

The past three years had been a near-constant fight to keep yourself afloat, with each one lining up to present a brand new, life-altering event tailored just for you. Two of which seemed like the end of the world, and one that truly was.

A heartbreak of your own volition. The loss of someone irreplaceable. Another heartbreak for good measure, also of your own volition. With the number of lessons the universe had packed in for you, you were certain that you’d be able to pass on to your next life without any problems.

Third time's the charm.

That was how the saying went, but for your own sake, you had to enter your final year of university stubbornly clinging to the hope that surely, fourth time would be the charm instead.

Incidentally, charm did come, in the form of Bang Christopher Chan.

It had begun with the most trivial of interactions. On the first day of your PHYS 408: Thermodynamics and Statistical Mechanics course, out of breath and—despite the cool February air—nearly working up a sweat from racing around the physics building like some kind of bright-eyed and bushy-tailed freshman, you’d made the very innocuous decision to take the first empty seat you could find. Near the back of the class, close to the door; the perfect spot for someone looking to get their credits in and clock out.

What you’d failed to notice until after you’d sat down, however, was the brooding statue of a boy occupying the chair right next to yours, resting his cheek on his hand and staring at the whiteboard with a look so fierce you would think it had personally wronged him somehow.

Seo Changbin. You’d seen him around more than once, having shared a handful of classes with him over the years, but never daring to approach him. You weren’t exactly someone you’d describe as faint of heart, but something about his muscular build and intense gaze, always made darker by the shadow of a cap, had you wary enough to keep a distance.

Not that it was difficult to steer clear of him, anyway, when he was the last to arrive and the first to leave as soon as each lecture hit its designated time limit—and that was if he’d even shown up to begin with.

You still remembered the first thing you’d noticed after settling down next to him, that being, that he was surprisingly much shorter than you’d initially thought. All those times you’d spotted him from afar, tapping along to the beat of his music or killing time in the activity center between classes, had given you the impression that he was as gifted in height as he was in muscle.

That didn’t change the fact that his intimidating presence more than made up for it, and you had taken great care to not veer into his personal space when you slipped your notebook and pencils out of your bag to prepare for what was sure to be a grueling learning experience.

The second thing you’d noticed about Changbin, was that he himself didn’t have a bag—or any kind of work materials, for that matter. There he sat on the first day of class, with nothing but a caseless Samsung S23 Ultra, a pair of headphones, and a ridiculously large bottle of what you’d assumed to be some kind of energy drink. It was almost impressive, in a way, how he hadn’t even tried to fool himself into thinking he’d be productive this semester.

You’d heard horror stories from your upperclassmen about this Thermodynamics professor. His strict grading criteria and endless list of hyper-specific rules were enough to make anyone with your degree plan dread taking his course; the most notable of said rules being that he prohibited any and all forms of technology in his classroom. It hadn’t taken long for him to single out every student who had dared to present even the tiniest flash of fiberglass around him, and Changbin was no exception.

In retrospect, it should’ve been inevitable to you that twenty minutes into the introductory lecture, he’d lean over and awkwardly ask you if he could borrow a pencil.

Wordlessly, you’d nodded and passed him a complimentary sheet of paper along with your pencil bag, allowing him to choose for himself. To your astonishment, he’d reached for your pink, Sanrio-themed mechanical pencil without a single moment of hesitation, whispering his thanks.

You’d never thought a smirk could be described as shy before you saw his. It was unexpected, coming from someone who looked like he bent iron bars for fun, but a welcome surprise regardless.

What had been even more surprising, was that this strange affinity for cuteness wasn’t a one time thing for him—not even close. With every passing Tuesday and Thursday morning you spent in his company, you soon came to discover that the Seo Changbin you’d created in your mind and the Seo Changbin existing before you were two very, very different people.

“You’re here!” he piped, loud enough to turn a few heads in his direction. “I saved you a seat.”

The flimsy, neglected notebook occupying your chair as some kind of placeholder was such a pitiful sight that you couldn’t help but snort.

“The seat I’ve sat in every day since our first class?” you hummed. “Thanks, Bin.”

“You’d better mean that,” he complained. “This place is lawless, someone might get bold one day and take your spot.”

“They’d beg me to take it back after five minutes of your nagging.” You passed his notebook back to him with a grin. It was hardly used and horribly undersized for a course as rigorous as this one, but you still considered it an improvement over the sorry state he’d been in when you first met.

You slipped into the familiar spot, unzipping your bag and preparing your study materials. “Shouldn’t I be the one surprised that you’re here, anyway?” you pointed out. “To what do we owe the honor of Seo Changbin having perfect attendance in an 8:00 a.m. class?”

“You know exactly what,” Changbin shuddered. Beneath the visor of his cap, you saw his eyes dart towards the podium, landing briefly on your demon of a professor. “Besides, senior year and all. It’d be pretty sad to take an extra semester just ‘cause I slacked off.”

You made a small noise of agreement. “So, fear and pressure,” you dropped your pencil bag dramatically on the table. “Now you sound like a real college student.”

Changbin perked up as he spotted the coveted flash of pink amidst your sea of pens and highlighters. “There she is,” he breathed a sigh of relief. “Thought I’d have to make it through this quiz without my lucky charm for a sec.”

“You keep calling it that,” you mused, fishing the pencil in question out from your pouch. “What makes it so special?”

Solemnly, he took it from your hand, curling his fingers around the pink plastic with all the grace and delicacy in the world. He gestured for you to lean in closer, as if preparing to share some deep, profound secret with you.

“It never runs out of lead.”

You nodded, putting on your best fascinated face. You didn’t have the heart to tell him that you’d been the one refilling it.

“Plus, I’ve aced every quiz I’ve taken with it so far.” Changbin’s eyes gleamed as he continued. “It’ll get me through midterms for sure.”

You reached out mischievously, threatening to swipe it from his hands. “In that case, I might just use it for myself.”

“Don’t even joke about that!”

Though your mood was light, it still soured the slightest bit at the mention of grades. Of the three quizzes you’d taken so far this semester, Changbin had scored better than you on two of them. It was a silly thing to be bothered by. You knew by now that he wasn’t lacking in intelligence by any means, but you also knew that intelligence alone wasn’t enough when it came to this course—or astrophysics in general. Certain levels of discipline and hard work were just as essential to your success, and it was difficult to ignore the question of what you seemed to be missing in those departments, especially when Changbin came across as so carefree about his studies.

With the way everything else had been crumbling around you since you’d begun university, the last hope you could cling to was at least maintaining your GPA until graduation. It had been the one constant in your life, an oddly comforting escape that you could pour your focus into when all else failed. You couldn’t afford to slip up—to be anything less than exceptional—for even a moment, not when your field of study was so fiercely competitive.

“You’ve definitely been doing well for yourself,” you commented. “It can’t all be thanks to Cinnamoroll, can it?”

“Oh?” the corner of his mouth curved up into a smirk. “Is that your way of complimenting me?”

You rolled your eyes, immediately accepting that you wouldn’t get anywhere without buttering him up first.

“I just think it’s unfair to give my pencil all the credit instead of that genius mind of yours, that’s all.”

Your tone was far too sweet to be natural, and you were sure that Changbin could see right through it. Even if he did, he played along anyway, lifting his chin proudly and letting out a satisfied hum.

“It’s true, it’s true,” he boasted. “Keep going.”

“Beauty, brawn, and brains,” you marveled, throwing a hand over your heart to really sell the idea. “You’re living proof that a guy can have it all.”

It was hard to describe the strange, high-pitched sound he made in response. Whatever it was, it helped your efforts feel just a bit more justified. Changbin scrunched up his nose, suddenly at a loss for words, and you were once again reminded of how utterly laughable it was that just two months ago, you’d found him intimidating.

“Ah, seriously,” he cleared his throat, trying to recover from the momentary lapse in bravado. “Alright, I’ll be honest. I get a lot of help from my friend.”

Your interest piqued, and you inched a bit closer. “Your friend?”

He crossed his arms, looking contemplative, and for a second, you thought he might demand more compliments before going into any further detail.

“He’s a couple years older than us, but still studying. He used to be on the astrophysics track before switching to music composition senior year.”

Your eyes widened a bit, half-perplexed, half-impressed. Astrophysics to music. It was a bold change to say the least, not one you could ever imagine yourself making, especially if it’d been close enough to his graduation that he had to take extra semesters.

A lightbulb flickered to life in your head, effectively cutting off whatever you’d planned to say next. “Wait a minute, music composition? Don’t tell me—?”

Changbin clicked his tongue, that same, sheepish expression creeping its way right back onto his face.

“Yes.”

“The same guy you—?”

“Yes,” he repeated. “Chan. The same guy I make music with.”

No matter how hard you tried, you could never suppress your amusement when you remembered the deeply unserious name Changbin and his friends had chosen for themselves.

“So, he’s one third of the famed 3RACHA,” you said it with a bit too much glee, your smile only widening when he shushed you as if the word were some kind of bad omen.

“Why are you embarrassed? The stuff you’ve shown me is really good.”

“I know.” A genuine compliment amidst your teasing only seemed to fluster him further, and he averted his eyes with a grumble. “Ah, forget it. Can’t believe I was gonna be nice and ask if you wanted to study with us.”

You paused. It was easy to forget sometimes that Changbin could be more observant than he let on. Still, you wondered if your earlier shift in demeanor had really been that obvious.

A part of you, the more prideful part, wanted to dismiss his offer right away. It would be like admitting that you were struggling with the course—which, realistically, you knew was ridiculous to care about when every one of your peers was going through the same thing. If the average class scores that your professor so proudly made known were any indication, it’d be a miracle if you weren’t struggling.

Before you could open your mouth to respond, however, the dull, lifeless voice of Dr. Choi rang out through the room, signaling the beginning of the lecture. You put away your study materials begrudgingly, cursing yourself for becoming too immersed in your chat with Changbin to get any last-minute cramming in.

Changbin, on the other hand, looked relaxed as ever, tapping your pencil lazily against the tabletop while the quizzes were passed out. You braced yourself, mind racing with all the knowledge you’d accumulated over the past weeks as a copy of the deceptively short quiz was slid over to you. It was a mere three questions long, but you’d be lucky if you finished them all in the time given to you.

Your eyes landed on the first Gaussian Probability Distribution word problem, and your head went blank. That was all it took for you to lean over to Changbin and whisper.

“I might have to take you up on that.”

。⋆。˚ ʚïɞ ˚。⋆。⋆。˚ ʚïɞ ˚。⋆。⋆。˚ ʚïɞ ˚。⋆。

Noon couldn’t come fast enough.

Your Thermodynamics quiz, not to mention the lecture that followed, had effectively drained your energy before the clock had even struck 9:00, with a full day of classes and assignments still lined up on the horizon.

As anticipated, you’d barely managed to complete the set of problems, even with all your preparation and practice. It could’ve gone much worse, but it was still enough to solidify your decision to join in on the study sessions Changbin had proposed.

He’d eagerly sorted out the details with you after class, planning to meet later this week at his and Chan’s apartment. It hadn't dawned on you until that moment that the latter of the two would probably be expecting some kind of payment for his tutoring services. After all, him helping Changbin out was one thing, but you were a complete stranger.

Changbin, however, had shut the possibility down as quickly as you’d brought it up. According to him, not only would Chan not ask you for any compensation, he’d outright refuse to accept it, even if you tried.

“The only thing Chan loves more than meeting people is helping them,” he’d told you, sounding so sure of himself that you were inclined to believe it.

Even so, it was a bit odd. A former astrophysics major, making a degree switch as drastic as music composition, and still being willing to revisit the same, headache-inducing subjects he’d so narrowly escaped, for free? The more you learned about this Chan character, the more you began to question what kind of person he really was.

Your stomach grumbled, reminding you that you were, in fact, ravenous.

You picked up your pace, drawn in by the welcoming aromas wafting from the campus food court. The feeling of your cell phone vibrating against your thigh made your steps falter a bit, and before you even slipped it out of your pocket, you already had a good idea of who the caller might be.

“Hi, Iseul.”

“Where are you?” she sounded expectant and slightly annoyed, sending your brain on an urgent mission to recall if you’d somehow lost track of plans with her.

“In the student union?” you answered cautiously. “Why?”

You were met with a dramatic huff crackling through the phone speaker.

“I’m outside your place,” she said, as if it were obvious. “Please tell me you didn’t have lunch already. I picked some up for us.”

You blinked, thoroughly confused for what was neither the first nor the last time as to what this girl’s thought process could possibly look like. After two years of friendship, you could confidently say that you had no idea.

“Sorry, did you text me or something?” You pulled your phone away from your ear to open your messages.

“No,” came her reply, tinged with the slightest hint of defensiveness. “But is it so crazy for me to expect you to actually be at your apartment? Y’know, the place where you live?”

“At noon on a Tuesday? A little,” you said plainly. You chose not to bring up the fact that she had to be well aware of your schedule to organize this meeting the very instant your lunch break started.

Another huff. “Well, are you coming or not? There’s a million things I need to talk to you about and I don't know how much longer I can wait here before that security lady accuses me of loitering again.”

You checked the time. It was only a short, ten minute walk to your apartment complex, you could definitely make it before your next lecture.

“Alright, alright. I'm on my way.”

“You’re the best,” her tone changed so abruptly that you almost laughed out loud. “See you soon!”

The call ended before you could get your own goodbyes in. With how quickly she’d hung up, you’d think she had something else to do besides stand around waiting for you to arrive.

Regardless, you hardly felt irritated, well-acquainted with Iseul’s behavior by now.

Your friendship with her had blossomed by pure accident, even with some reluctance on your part. One too many times sophomore year, you’d encountered her in the computer lab at the same ungodly hour as you, battling an army of technical issues with no one around to solve them considering that even the lab assistants had long taken their leave for the night. The first two instances you’d spotted her, slamming her mouse against the desk and cursing violently at her monitor, you’d kept to yourself—albeit with a tinge of guilt—and focused on your own approaching deadlines. After the third time, however, you’d figured the universe was trying to tell you something, and decided to help her out before she rendered every piece of equipment in the lab unusable in her academia-induced fits of rage.

From there, she’d latched on to you in a heartbeat. After all, someone who could help with tasks as incomprehensible to her as troubleshooting Microsoft Excel was sure to be reliable in other areas. On top of that, her newfound interest in you had only doubled when she’d found out that you happened to be living in the newest phase of apartments on campus. Suddenly, she had made the executive decision that you were the best of friends, and that every waking moment of your free time should be spent together at your place.

You might have been offended by her comically transparent motives if you hadn’t discovered soon after that your floorplan was just a few square feet bigger than hers. What she probably wanted most, you’d figured, was a friend.

Your initial misgivings aside, you were grateful to have Iseul in your life. She was someone who could be kept at a safe distance. Not physically, (her constant barging into your space would never allow that) but emotionally. A bit too preoccupied with herself to ever delve into personal matters that you’d rather keep to yourself, but still considerate enough to care about you. At least, in the bare minimum of ways, which was really all you needed from her. She was convenient and comfortable, and you’d long found your rhythm with her despite many labeling her a pain to get along with.

As you began making your way out of the dining hall to meet her, the sight of someone entering from the far side of the building made your heart drop to your stomach.

You froze, suddenly rooted in your place, feet heavy as cinderblocks. It shouldn’t have come as such a shock to you. You were bound to see him again, eventually, whether on campus or through some other unfortunate crossing of paths later down the line. You’d known this and braced yourself for it, too.

Still, no amount of time would’ve ever been long enough.

A very specific type of dread crept up on you, one you hadn’t felt so intensely for almost a year now. But the way it filled up your chest and spread through your skin was all too familiar, like it had never left your system to begin with. Like the kind of person you were before was still inside you, lying dormant.

Resentment and remorse fought for their place in your mind. Somehow, they both felt unjustified. He didn’t deserve to be the target of those emotions, and you didn’t deserve to have them. He hadn’t done anything—that was exactly it: he hadn’t done anything.

You told yourself that you had no right to feel this way. But it didn’t change the fact that he embodied everything you wanted to forget about the past three years.

He hadn’t noticed you yet; at least, you hoped desperately that he hadn’t. You weren’t going to stick around until he did, either. You shook your head, as if to forcibly expel the thoughts before they took root in your brain, and spun on your heels, making your way towards the exit located as far away from him as possible.

In that moment, you were more grateful for Iseul’s impulsive tendencies than ever.

。⋆。˚ ʚïɞ ˚。⋆。⋆。˚ ʚïɞ ˚。⋆。⋆。˚ ʚïɞ ˚。⋆。

You rubbed your hands together in an attempt to warm them up, praying that the clouds would hold out until you and Changbin made it to his apartment. It was an unusually cold day by April’s standards, and the sharp winds and ominous gray sky promised a rain that was sure to be bone-chilling for whoever got caught in it.

“Right there,” Changbin pointed at the building you were approaching, finger landing in the direction of a balcony on its third floor. There was a soccer jersey for a team you didn’t recognize hanging off the railing, flapping in the wind so wildly that you were concerned it may fly away altogether. “See, the walk isn’t so bad, right?”

It had been nearly half an hour. Granted, the journey home took longer than expected thanks to Changbin, despite having lived in this complex for two years, still managing to lose his way somehow.

“I’m starting to understand why getting to class on time is so hard for you.”

“I told you, I’ve never taken this route before!” he objected. “I’m just not used to coming from the east side of campus.”

You relented, deciding you’d teased him enough along the way. “It’s alright, it was a bonding experience,” you gave him a playful smile. “I just hope Chan won’t mind that we’re late.”

Changbin waved his hand dismissively. “Nah, don’t worry. He’s probably holed up in his room working right now. Doubt he even knows what time it is.”

It sounded like a dig at the older boy, but there was no hint of scorn in Changbin’s voice, just honest affection.

A strange feeling had been periodically bubbling up inside you all week, and at the mention of Chan, it made its presence known yet again. Whether curiosity or anxiety was at the root of it, you weren’t quite sure, but it grew stronger and stronger with each step you took up to their apartment. By the time you reached the third floor, you found it hard to focus on anything else.

Changbin fumbled with his keys for a moment before unlocking the door and swinging it open. You made note of the plated number on the wall next to you as he did. 8-325.

“Well, we made it in one piece,” he stepped to the side, inviting you in. You accepted with an appreciative nod, and as you slipped off your shoes, your eyes scanned over the living room and kitchen areas in front of you. They were surprisingly neat, with just a few stray socks and water bottles scattered here and there. Even the state of the kitchen sink wasn’t all that bad. No rotting food, no mountain of dishes, no overflowing trashcan.

“Wow,” you murmured, impressed. “It’s clean.”

Changbin snickered at that, as if he’d anticipated your exact reaction. “Minho raises hell if we let it get any worse than this.”

Minho. You’d almost forgotten about their other roommate. Like in the case of Chan, you hadn’t met him, but you’d heard a few things here and there from Changbin. He was a year older than you—a Computer Science major, if you remembered right—but still an undergraduate due to him taking a gap year after high school to work. You wondered if Changbin was some kind of magnet for these people, with his unique balance of childish antics and emotional maturity giving any upperclassmen he came into contact with no choice but to take him under their wing, even sticking around until he graduated like true, responsible older brothers.

“Chan!” Changbin’s voice rang out through the apartment, louder than you thought was probably necessary. “Chan! We’re here!”

There was no response for a minute or so, and just as you shrank back in preparation for another ear-splitting shout from Changbin, you registered the faint sound of a door opening down the hall.

“Coming!”

For some reason, you held your breath.

Shrouded in a mass of black, from his hoodie, to his pants, to the beanie on his head, out shuffled Chan.

He was just an inch or two taller than Changbin, but similarly to him, he had a strong presence. Maybe it was the way his clothes made him look like a walking void, or maybe it was the way he appeared so friendly in contrast to them. His eyes were gentle and his face was weary, but kind. He looked like someone who smiled a lot.

“Sorry,” he pulled his headphones down, letting them rest around his neck. “I lost track of time.”

Changbin gave you a knowing look, as if to remind you that he’d told you so. “It’s okay, I figured.” He conveniently left out the fact that you and him had arrived beyond schedule.

Chan turned to you, tired eyes finding you for the first time. You introduced yourself with a quick dip of your head, and he did the same. You thought it would end at that, but to your surprise, he reached out his hand, wiggling it around slightly to push back the oversized sleeve that had been covering his palm.

“Nice to meet you!” he chirped.

You took his hand, unable to stop yourself from flinching the instant your skin brushed against his.

He was warm. Unnaturally so.

It set off every last one of your nerve-endings, seared through your veins. You might’ve attributed it to his clothing, but all three of you were dressed in thicker attire given the weather. Surely, he had to be cooking up a ridiculous level of heat in that hoodie for his skin to be burning the way it was. On top of that, he didn’t look sweaty or flushed in the slightest. There was just a natural, rosy complexion to his cheeks (which, upon second look, you noted were quite soft in comparison to the rest of his masculine features).

You blinked, realizing with a start how long you’d gone without returning Chan’s greeting.

Changbin bumped his shoulder against yours, and you cringed inwardly. That had to be some kind of record for how fast a first impression could crumble.

“Nice to meet you, too.” you tried to quell the awkwardness, but the way you pulled back all too quickly only seemed to make things worse.

Chan eyed you for a split second longer, his stare flickering down to your hand so briefly that you weren’t sure if you’d imagined it. He flexed his fingers once, then the look of concern on his face morphed into a polite smile.

It was an unfortunate moment for you to notice that he had dimples.

“Is it too toasty in here?” He angled his head towards the thermostat. “I can change it if you’re uncomfortable!”

Just a minute ago, you would’ve told him that you were fine. You’d been perfectly content with your body temperature up until you’d come into contact with the human furnace that was Bang Chan.

You had half a mind to question if he was the uncomfortable one, with all the heat that was practically radiating off of him, but Changbin spoke up first.

“Have you been outside today?” He shivered. “Trust me, this is perfect.”

At that, he strolled over to the kitchen table and plopped down his belongings, looking more prepared to learn than you had ever seen him in class. Chan's smile didn’t waver despite the fact that he obviously hadn’t been asking for Changbin’s opinion, and he exchanged a glance with you, as if you were old pals rolling your eyes over a mutual friend.

You smiled back at him, determined to let this guy believe that you were, in fact, capable of understanding social cues.

“I'm gonna grab my old notes,” he informed you. “Make yourself at home!”

You thanked him quietly, making your way over to the table and joining Changbin in the seat closest to him. As soon as Chan was out of earshot, he nudged you curiously.

“What was that?”

You put on your best neutral front. “What?”

Changbin squinted, eyeing you up and down. “You were acting weird.”

You considered playing dumb, but quickly decided against it. Knowing him, he wouldn’t stop pestering you until you gave him the answer he wanted.

“He was hot,” you shrugged.

“He was what!?”

You tensed up. “No, no, not like that. I mean he was hot, like, physically.”

His mouth hung open, and you weren’t sure what to be more annoyed with: your abysmal choice in words, or his seemingly deliberate misunderstanding of you.

“He felt hot,” you clarified. “Like, his skin. That's all.”

The explanation only seemed to tickle Changbin further, and you elbowed his side irritably, trying to shush his delighted cackles.

“Okay, so, you weren’t acting weird. You just are weird.”

“I'm serious!” you protested.

“He's not better looking than me, is he?” he continued dramatically. “You didn’t do anything like that when we first met.”

You exhaled, composing yourself before you grew defensive over something so ridiculous. “Because your hand didn’t feel like the surface of the sun.”

Changbin nodded solemnly as if he understood, but the look on his face was still completely unconvinced. “Yeah, yeah,” he clicked his tongue. “Just don’t go falling in love with him, alright?”

You snorted, not bothering to dignify him with a response.

That was the last thing you needed—the last thing you wanted, even. To spend another few years building something that you could already predict the demise of. Another few years constructing a tower that you would never even get to see completed, let alone make a home in. Because it was sure to crumble; that was the only thing it could do when its foundation was never fit to support anything to begin with.

The sound of Chan’s approaching footsteps snapped you out of your unpleasant thoughts. He'd taken longer to return than you’d expected, and you could only pray that he hadn’t overheard your conversation with Changbin. He did seem like the type, after all. To pretend like he was still in the other room so that you could be spared the embarrassment of getting caught in the middle of a conversation about him.

“Alright, let’s get started.”

Two notebooks, a laptop, and the colossal textbook required for PHYS 408: Thermodynamics and Statistical Mechanics—co-written by Dr. Choi himself, of course—were all dropped on the table before you. You felt a glimmer of hope. Chan seemed to be serious about helping out, so much that you wondered if this arrangement truly could be the extra boost you needed to finish the semester with an A.

He settled into the chair opposite you and Changbin. “So, next up is the midterm, yeah? I guess we should start from the beginning.”

“Inexact differentials, please,” Changbin requested. “I still don’t get them.”

Chan raised an eyebrow, lips twitching in amusement. “Since when are you so ready to study?”

“Since we got our new recruit,” he leaned back in his chair. “There’s less pressure on me now that your wrath is split between us.”

You let a soft chuckle slip at that, trying to imagine what it might take to anger someone who appeared as good-natured as Chan. Said boy cleared his throat, looking a bit embarrassed.

“I swear, I’m not that harsh.”

You nodded, fully aware of Changbin’s talent for exaggeration. “I don’t think anything can scare me after Dr. Choi, anyway.”

“That’s true,” he giggled. For how charming it was, it didn’t last nearly long enough.

You pulled your eyes away before landing yourself in another incriminating situation.

“Alright, inexact differentials it is.”

。⋆。˚ ʚïɞ ˚。⋆。⋆。˚ ʚïɞ ˚。⋆。⋆。˚ ʚïɞ ˚。⋆。

Forty-five minutes into your first study session, you’d come to make two very important realizations about Bang Chan.

The first, being, that he wasn’t exactly the best at explaining things.

He’d typically start on the right track, but it wasn’t long before he’d veer off on tangent upon tangent, tacking on more and more information until it became a full-blown ramble, all loosely connected with a series of “um”s and “y’know”s before being clumsily wrapped up with a final “so…uh, yeah!”

You didn’t hold it against him. He was clearly a smart guy, and you knew firsthand what a nightmare these topics could be to teach to other people, especially taking into account that it had been two years since he’d learned them. Even with his less than articulate methods, you still found yourself grasping concepts exponentially better than you ever did in your thermodynamics lecture, and that was because Chan seemed to be gifted with what you could only assume was an endless supply of patience. He’d repeat himself as many times as deemed necessary, perfectly content with rereading his notes, checking the textbook, and even searching things up online until he was certain that both you and Changbin had understood.

The second realization you’d come to, was that your concerns about whether or not you might get to hear more of his laughter had quickly been put to rest.

He giggled at everything. At you, at Changbin, at himself. Sometimes, he giggled at nothing at all, just to fill the silence. It was admittedly fascinating to see the way his face would change, from the stern expression he wore when offering guidance, to the sheepish smile that’d appear when he stumbled over his words.

After hearing his laughter for the better part of an hour, infectious and melodic and, occasionally, ending with the faintest squeak, you still hadn’t gotten sick of it. Though, you did find yourself thinking that he had to be either an extremely self-conscious person, or an extremely giddy one for giggling to come as naturally to him as breathing.

“Does that make sense?” Chan tilted his head. “Let me know if you wanna go over it again!”

“I think I got it,” you smiled.

In truth, you didn’t, but it was a matter of dignity at this point. Enthalpy was one of the most basic properties you needed to know in order to build on concepts infinitely more complicated than it, and if you held up the review any longer to focus on something so mundane, you may not be able to show your face around this guy ever again.

It didn’t help that somewhere along the line, the looming clouds outside had broken at last, bringing about the downpour that you’d anticipated all day. Each explosive clap of thunder chipped away at your focus more and more, making you prone to stupid, easily avoidable mistakes that frustrated you to no end.

You thought your answer had been convincing, even making sure to look him in the eye when you’d said it, but Chan still didn’t let up.

“Are you sure?” he pressed.

“Oh my God,” Changbin’s voice turned up in a whine, his earlier enthusiasm nowhere to be found. “If you explain this one more time I’m seriously gonna go crazy.”

Before Chan could respond, the sound of keys jingling amidst the steady patter of rain caught everyone’s attention. You turned your head just in time to see the door creak open, letting in a violent gust of wind, and, with it, the lean figure of a stranger.

He was soaked. Dark hair plastered to his forehead, clothes hanging off of him like heavy drapes, and sneakers forming small puddles on the wooden floor.

“It’s raining,” he announced.

Changbin broke out into a fit of laughter, and you bit your lip to prevent yourself from doing the same. Chan, though clearly on the verge of losing it as well, still rose from his chair like a reflex and grabbed a hand towel from the kitchen. He tossed it to the other boy, who you could only guess was Minho.

“I was starting to wonder where you were,” Chan remarked, voice shaking with barely contained glee.

“I got sick of waiting for the rain to stop, so I made a run for it.” Minho dumped the water out of his shoes and shut the door in disgust. “Then I remembered why I don’t run.”

The small towel didn’t do much for his drenched state, and after a few moments of shaking it haphazardly in his hair, he gave up and let it rest around his neck instead.

“You should shower and dry off,” Chan told him. “You’ll catch another cold.”

Minho grunted in acknowledgement, but rather than following through, he strolled over to the kitchen. As he did, his gaze landed on you for the first time, giving you a clear view of his face.

Every striking feature of his was balanced out with a soft counterpart. Sharp, intense eyes with puffy bags underneath, a sharp, prominent nose between full cheeks, and sharp, catlike lips above a round chin. It was a delicate combination that not only made him attractive, but interesting to look at, as well.

He studied you for a moment too long, just enough to spark a sense of unease inside you.

“That’s no good, Changbin,” he clicked his tongue at last. “Don’t tell me you’re such a hopeless case that Chan had to find you a second tutor.”

“It’s a study group!” Changbin cried indignantly. “And what the hell kind of introduction is that? Say hi!”

The corner of Minho’s mouth curved into a smirk, like it was made to do exactly that. Similar to Changbin’s, it wasn’t sultry, but unlike Changbin’s, it wasn't shy. It was mischievous and playful, like that of a child’s cheeky grin.

His attention shifted back to you, and he gave you a proper greeting. It was surprisingly polite, all things considered, even ending with a short bow.

He popped open the refrigerator door, leaning forward in a way that had to be uncomfortably cold given that he was still dripping wet.

“I had a few pudding cups left in here. At least two,” he called out.

“Wasn’t me,” Chan piped with the speed of someone who was accustomed to being the first suspect.

Minho pulled his head out from behind the door, accusatory glare locking right on Changbin.

The boy shifted guiltily next to you, unable to hold eye contact with Minho for longer than a few seconds.

“Sorry,” he muttered.

Minho shut the fridge with a hum. “That’s alright.” His voice was breezy and sweet, a complete contrast to what came out of his mouth next. “Just sleep with your door locked until you buy me new ones.”

“Hey,” Changbin whined. “That’s scary.”

He tugged at your arm as if expecting you to rush to his defense, and you settled for giving him a comforting pat on the back, not nearly familiar enough with Minho to joke around with him like that. Given how Chan was watching in amusement, you figured this was a regular occurrence for them, anyway.

Following Minho’s arrival, your review session more or less fell apart. The idle chit chat eventually led into a full on conversation, and when Changbin shut his textbook with a luxurious stretch, you knew there was no chance of getting him to open it again.

You didn’t mind, really. The three of you had covered a lot of ground in the time you’d spent studying, and you were already worlds more confident about the upcoming exam. Your main concern, now, was how you were going to get home. It was well past sunset, and the thick sheet of clouds had darkened the night more so than usual, not allowing even a single drop of moonlight to break through. That, coupled with the fact that it was still very much pouring outside, complicated your plans a bit.

Sitting there as the odd one out among the group of friends, you couldn’t help but feel like you were overstaying your welcome, but any attempts you made at suggesting that you brave the storm and head home were emphatically shut down.

“It’s okay,” you tried to convince them. “I really should get back and have dinner.”

“Have dinner with us!” Changbin didn’t miss a beat.

You hesitated, uncertain as to whether it would be more rude to accept or decline.

“It doesn’t look like the rain’s gonna stop anytime soon,” Chan reasoned. “Why don’t we eat first?”

Minho, in vengeance of his fallen pudding cups, loudly declared that he wouldn’t be cooking dinner for anyone. It became clear to you in that moment that he was probably the only thing standing between his roommates and malnutrition, because their go-to second option (if not their only other option) was instant ramyeon.

So, there the four of you sat, crammed together on their living room couch, watching some obscure superhero movie that Changbin seemed to know every line of, and slurping away at your noodles.

They had turned out tasty enough, with the extra spices and sauces you’d added to make the flavor a bit more appealing, but with the way Chan scarfed down his share, you might’ve thought it was the best meal he’d ever had. He was all satisfied noises and delighted fist shakes, looking happier eating instant cup noodles than you’d seen some people look their entire lives.

He was cute, you decided.

Though the movie lessened some of the pressure you felt to socialize, a faint air of awkwardness still lingered around you, only ever really ebbing when you and Changbin would interact in between his passionate lore discussions with Chan and his bickering with Minho.

Chan seemed to sense early on that you weren’t fully relaxed with the atmosphere; at least, you assumed as much judging by his periodic efforts to pull you back into the conversation.

“Everything good?” he’d asked at one point, leaning over so you could hear his whisper above the movie.

Even with Changbin serving as a buffer between you two, his persistent warmth still found you.

“Oh, yeah.”

Not your most eloquent response. To be fair, you hadn’t anticipated his question. It didn’t seem to have convinced him, but he’d given you a smile, anyway.

“Alright. Just know that you’re more than welcome here, yeah?”

You were grateful for his kindness, but at the same time, it had caught you off guard. It wasn’t a regular thing for you, being read with such ease by someone you hardly knew, and you couldn’t decide if you were just being uncharacteristically transparent that day, or if Chan was too perceptive for his own good.

Changbin was Changbin. That in itself helped you loosen up a bit, as well. He behaved in virtually the exact same way around the older boys as he did with you—albeit, leaning more into his childish side—and it filled your chest with a pleasant sort of relief. He considered you a friend; close enough to treat you with the same intimacy that he treated people he’d known for years.

Minho, on the other hand, was more of an enigma. Not rude by any means, but not overly accommodating, either. The one thing you were certain of was that he was incredibly funny. Witty, too. He didn’t speak as much as Chan or Changbin, but when he did, it was always something memorable. His voice had a playful lilt to it that never seemed to go away, like nothing he said was meant to be taken too seriously.

As the night continued and the four of you had all eaten your fill—or, several fills in Chan’s case—your reservations slowly but surely melted away. You spoke more naturally, joked with Changbin the way you always did when you were together, and even found yourself comfortable enough to make a few snarky comments about the film’s ridiculous plot and cringeworthy special effects, to which Changbin took great offense and Minho had let out a few laughs.

As for Chan’s laughter, another few hours of it still hadn't made it any less endearing. In fact, the more you heard it, the more hooked on it you became.

By the time the storm had passed and you could finally head home safely, you found yourself a bit wistful that your impromptu gathering had come to an end.

。⋆。˚ ʚïɞ ˚。⋆。⋆。˚ ʚïɞ ˚。⋆。⋆。˚ ʚïɞ ˚。⋆。

“So,” Iseul’s eyes twinkled. “What’s the verdict? Is he cute?”

Straight to the point. It was something you liked about her, usually, but in this scenario, you almost wished she’d never asked.

Ever since that day, you’d felt an inexplicable sense of…well, you didn’t quite know what it was. Discomfort, unease, foreboding; they were all too extreme to describe the feeling. All you knew was that something peculiar stirred inside you whenever you thought back to Chan. Maybe it was because of your clumsy first interaction, or maybe it was because of that nagging, uncanny belief that he could see right through you from the very first moment you met.

It was unfair, in a way, because you knew for a fact that he’d been nothing but friendly every time you’d hung out with him—a delight to be around, really. You could easily see why he was the social butterfly that Changbin made him out to be.

“Hello?” Iseul complained. “I'm not gonna stop asking, even if you ignore me.”

In retrospect, telling her about your new study routine with Changbin and his mystery friend—however offhanded it had seemed at the time—probably wasn’t your smartest move.

“Yeah. Really cute, actually.”

You may as well have told her that he’d asked for her hand in marriage with the squeal she let out. “I knew it, I knew it! Tell me everything.” She nearly knocked her drink over in her rush to scoot closer to you.

It was hard to keep a straight face. Even when you knew it was short-lived, her enthusiasm over the simplest of things was contagious.

“What’s there to tell?” you feigned nonchalance in a way that was sure to annoy her. “I go to him and Bin’s place, we study, I leave.”

“Come on,” Iseul pouted. “There has to be more to it than that. What’s he like? Do you have a picture?”

“A picture?” you echoed incredulously. “You take a commemorative selfie every time you study thermo?”

“Like, his Instagram or something!”

“He has three posts, and none are of his face.”

Iseul deflated at that, and you broke out into proud chuckles. You were being difficult, sure, but the part about his profile was at least true. A picture of his hand holding up a peace sign at the beach, a picture of what you assumed to be his dog back home, and a surprisingly clear shot of the moon; those were the three precious images Bang Chan had felt compelled to share with the world, with the most recent one being from almost two years ago.

“He’s got a nice smile,” you offered.

Iseul took the bait instantly, perking back up. “Yeah?”

You nodded. “Dimples, too.”

“Cute.” She clasped her hands together, looking lost in a dream. “That’s it, I have to see him.”

“What’s got you so interested, anyway?” you mused. “Aren’t you talking to someone?”

With the way her face dropped right back into a grimace, you knew you’d touched on a sore subject. “No,” she said curtly. “I mean, am I? Does it count as talking when you’re lucky to get a reply every six hours?”

“You’re just clingy,” you teased, already bracing yourself for when her hand flew out to swipe at you.

“I’m totally low maintenance!” she cried. “Anyway, I don't even want Chan for me. This is about you.”

You shifted in your spot, that same, strange feeling twisting in your stomach, stronger this time.

“Me? What do you mean?”

Iseul put her chopsticks to the side, giving you a look that was far too serious given the topic.

“I’m finding you a boytoy.”

You nearly laughed out loud, only stopping yourself in the nick of time when you caught that she wasn’t joking in the slightest. 

“No, you’re not.”

“I am!” she insisted, bravely holding her ground in the face of your disbelief. “What are you gonna do when I settle down and don’t have time for you anymore? I gotta make sure you have someone to entertain yourself with!”

Your amusement wavered just a bit. You knew she meant well, but when it came to Iseul—or anyone, for that matter—trying to do things for your sake, you’d long accepted that you’d prefer if they didn’t even bother. 

“There’s no rush,” you pointed out. “You have to actually get a text back before you can settle down, right?”

“Oh my God! I'm trying to help you and this is the thanks I get?”

“Thanks, Iseul.” You reached out to give her an apologetic pat. “But I don’t need any help with that.”

Suddenly, her lips curved into a devious smirk, and you had a sneaking suspicion that she’d misunderstood what you meant.

“Oh, I know you don’t,” she drawled. “Never forgetting that dreamboat you had following you around like a lost puppy all sophomore year. What was his name again—?”

“Doesn’t matter,” you cut her off a bit too harshly, regretting it as soon as you did.

Iseul frowned. “It was just a question.”

“You’re right, sorry.”

“What ever happened to him, anyways?” she continued, apparently not taking the hint. “Things ended so suddenly with you two.”

You tried not to bristle. After your near-encounter in the dining hall the other week, he’d been occupying your thoughts far too often for your liking. That, coupled with those peculiar feelings that had sparked within you upon meeting Chan, had you unreasonably on edge ever since. 

“I told you,” you tried to sound casual. “It just wasn’t a good match. I don’t think he really liked me all that much.”

Iseul scoffed, not buying it for a second. “Please, he was obsessed with you.”

The urge to tell her everything right then and there was more tempting than ever. To unload all the bitterness, the guilt that had been building up and weighing you down for the better part of two years now. You knew you couldn’t, though, not when it meant having to break the very same news to her that had led to the end of your relationship. The chances of her reacting the same way that he had were slim, but even the smallest possibility was more than enough reason for you to stay quiet. You’d kept it tucked away for far too long now, anyway. She’d only get upset if she found out now.

“Obsession isn’t the same as love.”

Iseul grew quiet for a moment.

“I guess,” she mumbled.

She turned her attention back to her soda, as if the conversation had suddenly become too heavy for her tastes.

You didn’t blame her, but it further solidified your decision to leave what you’d wanted to say buried in your heart.

。⋆。˚ ʚïɞ ˚。⋆。⋆。˚ ʚïɞ ˚。⋆。⋆。˚ ʚïɞ ˚。⋆。

Several doses of caffeine were in order.

Anyone who happened to witness the unfortunate sight of you and Changbin stumbling out of Room 118 of the physics building, spiritually battered and bruised and barely able to process your surroundings, might’ve thought you’d just gone to war.

It wasn’t much of a stretch, considering the exam you’d just taken. You felt ridiculous for ever thinking the two hour time slot was overkill; in actuality, it had been a rare display of mercy from Dr. Choi.

“I’m dropping out,” Changbin declared.

“Absolutely not.”

“I’ll do it,” he insisted. “Before I lose my mind for real.”

He slumped heavily against you, and it took all your strength to support his muscular body so that the both of you wouldn’t be sent toppling to the floor.

“After everything Chan’s done for you? You might just break his heart.”

Changbin seemed to take your joke a bit too seriously, a horrified look crossing his face. “Can you imagine how that would’ve gone without his help?”

“Don’t even wanna think about it,” you shuddered.

For how excruciating the thermodynamics midterm had been, it was more because of the psychological torture aspect than the difficulty of the content itself—though, its difficulty was nothing to sneeze at, either. The one positive that had come from this hellish experience was confirmation that choosing to study with Chan had undoubtedly been the right choice for you. Every topic you’d managed to review over the few meetings you’d had so far stayed fresh in your mind during the exam, so vividly that you could even recall the inflections in Chan’s voice whenever he’d sing his sentences at random. You weren’t sure if it was intentional, or if it was even something he was aware that he did, but you’d caught on to it right away.

Because his melodies helped you remember better, of course, not because you found it endearing.

“We really need to thank him,” Changbin bumped his head against yours. “Let’s bake him a cake.”

“You can’t even crack an egg.”

“Who told you that!?” he bolted upright, miraculously regaining his energy.

You kept your lips sealed, but it didn’t take long for him to narrow down the suspects.

“Minho…” he muttered. “Who the hell shares that story with someone they just met?”

“I agree that we should do something for Chan, though,” you tried to stay on topic before Changbin could get riled up about Minho. He was already sour on him after he’d bought replacement pudding cups as threatened, only to smugly be told that they were the wrong brand.

“I’ll think of something when my brain isn't fried.” Changbin shoved his hands in his pockets, looking contemplative for a second. “You never answered my question, y’know.”

“Hm?”

“About him being better looking than me.”

His words caught you so off guard that you actually stopped in your tracks, turning to give him a look of pure disbelief.

“Seo Changbin,” you said plainly. “You can’t be serious.”

“What?” his tone grew defensive. “This is important stuff! You’re supposed to be around the same level of attractiveness as your friends. It’s scientifically proven.”

You so badly wanted to hold your unimpressed stare, but it was impossible when the man in front of you was speaking without an ounce of shame.

“First of all,” you began. “I've told you a million times that it had nothing to do with his appearance.”

It was Changbin's turn to look unimpressed, but he waited for you to finish.

“Second of all, you’re a very handsome guy, Binnie,” you tacked on the nickname for maximum effectiveness. “So if I were to fall in love with anyone, it’d obviously be you.”

You truly meant the compliment, but a little extra flattery never hurt when it came to him. A wide, embarrassed smile spread across his face like clockwork, and he reached out to smack your shoulder, giggling at an unnaturally high pitch.

“Geez, don’t say it like that,” he complained. “I wasn't ready.”

You shook your head. “You’re so simple.”

For both Changbin’s peace of mind and your own, you hoped now that the issue would be dropped. You had enough confusing feelings about Chan already without Iseul and Changbin blowing things completely out of proportion.

“Wanna get some coffee?” you suggested. “There’s a really good kiosk on the first floor of the library.”

“I think I’m gonna head home and nap, actually. I’ve got another exam tonight.”

You let out a sympathetic hum. “That’s rough. Good luck, Bin.”

“Thanks,” he sighed dramatically. “Treat me for all my hard work once midterms are over.”

“Sure, I’ll even save up so I can afford your rich kid tastes.”

Changbin grinned at that. “On second thought,” he pulled his hand out of his coat pocket to reveal your pencil; his lucky charm. “You’ve given me more than enough.”

He attempted to pass it back to you, but you nudged his hand away gently.

“Keep it. Maybe it’ll help with your next exam.”

From there, you and Changbin said your goodbyes for the day. You decided to head to the coffee shop on your own, in desperate need of some kind of energy boost so you wouldn’t crash the instant you returned to your apartment.

As you made your way over to the campus library, your mind drifted back to Chan. It seemed to do that a lot, recently.

You wanted to do something to express your gratitude to him, but it was difficult to decide on what when you knew so little about the guy. Changbin could always help in that department, of course, but then there was the issue of actually getting Chan to accept it.

Despite not having walked nearly long enough to work up a sweat, you felt strangely heated when you approached the library entrance. Not only that, your hands were clammy, and you had to wipe your palm on your clothes before reaching out for the door handle. The warm, addictive scent of coffee flooded your senses as you entered the building. You almost connected your sudden rise in temperature to its cozy atmosphere—that was, until your eyes zeroed in on a figure seated at the table directly across from where you stood.

He was hunched over his laptop, consumed by his dark clothes so that he was hardly visible to anyone passing by, but you’d already reached a point where you could’ve recognized that side profile anywhere. A distinctive nose peeked out from behind the hood pulled over his head, thumb brushing over his lips as he concentrated on the screen before him.

Driven by an urge you couldn’t quite place, your feet drew you in his direction, and you had to force yourself to come to a sudden halt. He looked busy—exhausted, too—it was probably best to leave him alone.

Just as you turned to continue over to the coffee stand, dark eyes flickered up to find you, as if on cue. Recognition flooded his face, lighting up with a smile.

You gave him a small wave, and to your surprise, he gestured enthusiastically for you to come over to him. You adjusted the strap of your bag, feeling unusually self-conscious, like you’d given too much away with just your stare. Still, you steeled yourself and padded over to his table.

“Hey!” Chan removed his headphones, hood slipping off along with them. “I was just thinking of you.”

You blinked. “You were?”

“Yeah, you and Bin had your exam today, didn’t you?”

“Oh, right. He just headed home, actually.”

He pulled out the chair next to his, inviting you to take it. You hesitated for a moment before accepting, giving him a grateful nod.

As you settled in next to him, it dawned on you that this was the first time you’d ever seen him without some kind of hat or beanie on his head. You hadn't even known that his hair was curly. It felt akin to a crime to have been robbed of the sight; soft, brown ringlets falling just above his eyes and swooping out at his nape, almost like the tail of a duck.

“How’d it go?” He tilted his head curiously. “Alright, I hope?”

“Well, let’s just say I understand why you switched majors.”

Chan’s laughter filled your ears, a blissful compensation for the past two hours you’d just had. He reached out to tap your shoulder lightly as he giggled, and you weren’t sure why it made your heartbeat pick up.

“That bad, huh?”

“It would’ve gone a lot worse without your help,” you confessed. “Thanks again for studying with us, you’re a lifesaver.”

“Yeah, of course!” Chan chirped. “I’m glad to help.”

“Are you sure there’s really nothing I can do in return? I hope you’re not holding back just ‘cause I’m Changbin’s friend.”

You were careful to ask a second time after your failed attempt at convincing him to accept some kind of payment—favor, anything—during your first study session. Just as Changbin had predicted, he’d brushed you off with a polite smile, insisting that it was the least he could do. Despite your best efforts, you’d ultimately stopped pressing the issue to avoid coming off as too pushy.

Chan waved his hand, dismissive, yet again. “Nah, you don’t have to worry about that. It’s no trouble at all!”

“How about I buy you a coffee?” You motioned in the direction of the kiosk. “Just one cup, and I’ll stop nagging.”

“Ah.” He scrunched up his nose in distaste. “Sorry, I don’t really drink it.”

You stared, waiting for some kind of indication that he was just messing with you, but it never came. Suddenly, his perpetually worn-out state made perfect sense.

“A college student who doesn’t drink coffee? They should study you.”

He grinned, looking a bit embarrassed. “If you need me as the subject for your research next semester, just let me know.”

“Thanks,” you giggled. “But then I’d owe you double.”

He seemed to be lost in thought for a moment, and you tried not to focus on the way his thumb came to run over his full lips again. You’d never seen lips shaped like his before; when you looked closely enough, they resembled a soft-edged heart.

“How about this? Give me your number and we’ll call it even.”

Your mouth nearly fell open. You hadn’t pegged him as the type.

“That way, we can say we’re officially friends,” he continued, completely oblivious to your shock. “And helping out a friend is normal, yeah?”

Friends. It was odd to hear him say that. You weren’t really sure if you could consider someone you’d spent just a handful of hours with your friend, but for what it was worth, he seemed to be speaking sincerely.

Your brief moment of panic melted away. Another case of unintentional flirtatiousness on his part, after all. It was relieving, in a way, because you could only imagine the effect someone like him might have on people if only he knew how to utilize his charm.

“Alright, you win. Just a warning, though, I’m not the best texter.”

“Me neither,” he admitted. “But if you ever need anything or wanna chat, I’ll be there!”

As you exchanged phone numbers, every one of your instincts called for you to be suspicious of Chan, to believe that, surely, he must have some kind of ulterior motive behind his eagerness to befriend you. But you knew what ill-intent looked like by now,—you’d be a fool if you didn’t—and there was none behind his eyes, just an honest desire to help in any way that he could.

It was almost foreign to you, something you’d never really seen in any other person but one.

“There! You’re debt-free.” Chan handed your phone back to you. He’d taken it upon himself to add a wolf emoji next to his contact name, and you shot him an amused look.

“My friends say it reminds them of me,” his voice turned a bit sheepish, as if realizing how silly it felt to say out loud.

You softened. “That’s cute.”

“You think so?” He reached up to fiddle with his piercing, and you noticed for the first time how red the tip of his ear had become. Probably a side effect of his concerning levels of body heat. “What should I put next to yours?”

“A flame?” you joked. “So you can remember me as the girl who sucks at thermo.”

Chan flexed his fingers. “I like it,” he giggled.

You stole a glance at his laptop as he edited your contact, met with a sea of sound waves, audio files, and incomprehensible icons taking up his screen.

“So, were you working on something?”

He perked up. “Oh, yeah! Just messing around with some sounds, really.”

You leaned in a bit closer despite not understanding much of what you were looking at. Even with your lack of expertise, you could see that whatever he was doing was more than just messing around.

“Is it for a class?” your interest piqued. “Or for 3RACHA?”

Chan’s breath hitched, loud enough for you to hear, and you wondered for a moment if you’d said something wrong.

“You know about that?”

“Bin’s shown me a few songs! You guys are really good.”

He ducked his head, the flush on his ears creeping up to paint his cheeks the same shade. Oh. He really had been flustered the entire time. It excited you more than it probably should have.

“Ah, thank you,” he chuckled breathlessly. “Sorry, I’m just a little caught off guard, I think.”

You considered changing the subject for the sake of his comfort. What he said next, however, quickly quelled any concerns you had. “Which one did you like the most?”

He lifted his gaze shyly, looking so hungry for approval that you made a mental note to ask him more about his music in the future.

“Zone!” you didn’t miss a beat. “I especially love the lines in Māori.”

His face broke out into a grin so wide that his eyes almost squeezed shut from sheer happiness. “I sing that part,” he beamed. 

Of course he did. You tried to imagine it—the bubbly, unassuming boy in front of you delivering lines with such power and confidence. It intrigued you, just like everything else about him. From the first day Changbin had described him to you, he was like a puzzle that you were determined to collect all the pieces of, to bring your understanding of him to completion.

Your original goal in coming to the library now long forgotten, the two of you stayed at his table for at least another hour, chatting about all sorts of things. You learned that while all three members of 3RACHA had a hand in composing and songwriting (a fact that you made note of for future, Changbin-teasing purposes) Chan played the biggest role when it came to arrangement. With a bit of prompting on your part, he gave in and showed you a snippet of what he’d been working on before you arrived.

Placebo was the working title. It had a hopeful, upbeat melody that made you feel light and strangely nostalgic. There were no lyrics yet—Chan was still waiting on Jisung, the final third of the boys, to finish up his parts. As it turned out, he was the wide-eyed, messy-haired junior you’d spotted hanging around Changbin all those instances over the years, and one of the first people that Chan had befriended upon moving from Australia. How they’d come to meet when Chan was three years older than him, you had no idea, but you figured this guy could become best friends with his prison guard if he really wanted to, so it didn’t seem worth questioning.

Even with its half-finished instrumental and lack of lyrics, you could already sense a potential new favorite in Placebo. Though, if you were being honest, given the expression on Chan’s face as he played it for you—earnest and giddy and biting his fist in anticipation—you might've said the same regardless of which song it was.

“Do you really like it?” He kept his eyes on the screen, but you could see the glee plastered on his face.

“I do! It makes me happy.” You slipped his headphones off and passed them back to him. “You have to show me when it’s finished, okay?”

It didn’t seem possible, but his smile grew, cheeks rising and dimples flashing. “Okay, promise.”

He held out his pinky to seal the deal. You hesitated, wincing inwardly when you remembered what had happened the last time your skin touched his. Even so, you were determined not to fumble another interaction with him, and you braced yourself before hooking your fingers together.

The heat was still very much there, though not quite as drastic as before. It didn’t jolt through your nerve-endings like it had when you’d first met; instead, it kindled at your point of contact and spread steadily along your skin, from your pinky to your palm until it warmed your entire body. Gentle and intense, all at once.

Chan looked like he had something to say, but just as he opened his mouth to speak, your phone buzzed to life on the table. Reluctantly, you unlaced your pinky from his and reached for the device, unsurprised when you saw Iseul’s name in glowing white letters.

“Sorry, one sec,” you excused yourself, knowing that if you didn’t take her call now, many more were to come.

“Hello?” your voice came out winded, and you swallowed hard to steady it.

“Are you busy?”

Your eyes darted to Chan. He’d turned his attention back to his laptop, humming quietly to himself.

“Kinda, is everything alright?”

“Oh,” she paused. “What’s up?”

“Just in the library,” you left out the fact that you were with Chan, not keen on fueling her newfound desire for matchmaking.

“I need help planning my schedule for next semester,” she sounded stressed, but you knew by now that even the most easily-solved of problems could be the end of the world in her eyes. “Literally none of these marketing sections work for me and I need this credit to graduate. I’m going fucking crazy trying to move my other classes around.”

There was no excuse for you to say no, other than the fact that academics were the last thing you wanted to think about after the midterm you’d just had. That, and, you were enjoying your time with Chan more than you’d like to admit.

“Alright, I can help you figure it out. I’ll just need some time to get to your place.”

"You’re the best,” she breathed a sigh of relief. “Hurry, please.”

At that, she hung up, probably to get right back to abusing her laptop’s trackpad with furious clicks. You slipped your phone into your pocket, and when you began gathering up your belongings, Chan’s gaze shifted back to you.

“Heading out?”

“Yeah,” you wished you didn’t feel so wistful about it. “My friend needs help with her fall schedule, she’s kinda freaking out.”

A knowing look crossed his face, lip twitching with the faintest hint of amusement. It wasn’t lost on you, and you raised an eyebrow at him.

“What is it?”

“Nothing,” he hummed. “Just think I understand now why you wanted to repay me so bad.”

You had half a mind to be taken aback, but it felt strangely expected of him, like you’d known that such a minor detail would be enough for him to catch on. That tendency you’d noticed from the first day you’d met him, making itself known more and more each time you crossed paths. 

“Think you’re the only one who can do people favors?” 

“Course not,” his smile mirrored yours. “I hope things work out with your friend.”

“Thanks.” You rose from your spot, wondering briefly if you should say what was on your mind before parting ways with him. “It was nice talking with you.”

“You, too.” He held up his phone, wiggling it around as a reminder. “We’ll talk more soon!”

In the end, you left the library without a single drop of caffeine in your system, yet somehow, you felt more energized than ever.

。⋆。˚ ʚïɞ ˚。⋆。⋆。˚ ʚïɞ ˚。⋆。⋆。˚ ʚïɞ ˚。⋆。

Six weeks and several study sessions later, you had come to make two more very important realizations about Bang Christopher Chan.

The first being, that he hadn’t been exaggerating when he’d warned you about his texting habits. You’d always thought you were bad at responding in a timely manner, hell, you’d thought Changbin was bad; but when Chan said he wasn’t the best texter, he didn’t just mean that he could be dry or slow or forgetful, he meant that most of the time, he simply didn’t text at all.

Instead, he liked to call.

You didn’t really mind—you tended to prefer talking to people over texting, anyway, but you’d admittedly been stunned when, after a week of radio silence following your encounter in the library, he’d contacted you out of the blue for no reason other than to “catch up”. No warning, no opening text, just an unexpected call that ended up stretching into a thirty minute conversation before you had to hang up and head to your next class. Another short period of no contact, and then, it had happened again. This time, just a few days following your first chat.

His calls, you’d also noticed as time went on, sometimes came at the most ungodly hours of the night. Once or twice, you’d woken up in the morning to find a missed call notification followed by an apologetic text a few hours later.

chan 🐺 (5:23 a.m.) sorry haha, didn’t realize how late it was

It left you perplexed as to when this man ever got a wink of sleep.

Even with your conversations being so sporadic, you found yourself looking forward to them regardless. He always had something interesting to share with you, from stories about people he’d met and the places he’d been, to music discussions and recommendations, to a vast array of space knowledge that he seemed to have neatly filed away in his brain. He talked about space a lot, like it was his friend. The moon, especially. It was undoubtedly your favorite topic of conversation, not only because it was a shared interest, but because the pure wonder and adoration with which he spoke of it stirred a warmth inside you like no other.

On top of all that, he always made an effort to check in with things on your end as well—in fact, it was always the first thing he asked about the moment you’d pick up, which might have been the most confusing detail of all. He was simultaneously the most absent and the most attentive communicator you’d ever met.

Once it had been made apparent to you that this routine may very well become commonplace with Chan, your curiosity had piqued enough for you to finally question him about it. His explanation, however, almost had you wishing you’d never asked, because nothing could’ve prepared you for his simple, sincere, “It’s just nice to hear your voice, y’know?”

That led into your second, more troubling realization. Somewhere along the line, you seemed to have developed a bit of a soft spot for Chan.

It had dawned on you some weeks ago, when the two of you had visited a new ice cream shop near campus that you’d mentioned was your favorite. When you’d recommended the place to him, you’d never once considered that he would take it as a suggestion for you to accompany him in trying it out. In the end, he’d ordered not one, not two, but all three of the signature flavors you told him you liked the most, detailing his thoughts about each one, with plenty of delighted hums and vocalizations in the process. Much to your horror, you’d listened to him chat passionately away with the most hopelessly endeared, involuntary smile on your face, knowing right then and there that your fate was sealed.

For that reason, your limited interaction with him was more like a blessing in disguise to you. The moment you’d discovered just how often your thoughts seemed to be preoccupied with him, your first instinct had been to distance yourself, to cut off all unnecessary contact until the pesky, ever-present daydream of his melodic laughter was forcibly expelled from your brain. Your regular meetings with him and Changbin, however, had made your efforts increasingly difficult, and you couldn’t shake the fear that, with how naturally Chan seemed to tune in to your emotions, it was only a matter of time before he noticed you behaving differently around him.

Today brought with it another moment of reckoning, another test of your resolve in the form of a two hour study session. You’d managed to get by the last few without any major slip-ups, making you especially grateful that Changbin was around to ensure you behaved more like your usual self.

bin 😑 (5:36 p.m.) oh, i forgot to tell you i can’t make it today

You stared down at your phone in disbelief, nearly coming to a halt in the middle of the road.

You’d texted Changbin this morning to double check that you were still on for studying this evening, even making sure to reach out hours in advance so he could reply before it was too late. Clearly, you’d have to give him at least a day’s notice from now on, because you were just a minute away from his complex when he’d decided to graciously inform you that he wouldn’t be coming.

you (5:36 p.m.) are u serious??? i’m almost at your place

bin 😑 (5:38 p.m.) sorry sorry it’s game night w/ minho and jisung lol. but chan’s home dw

you (5:38 p.m.) game night...you do realize this is for the final right? why isn’t chan with you guys?

bin 😑 (5:39 p.m.) relax mom i’ll come to the next one ;;; and he said he’s fine studying w/ you instead

A sense of dread twisted in your stomach. Regardless of how kind-hearted Chan was, you knew there was absolutely no chance in hell he would’ve preferred to stay home on a Friday night, tutoring you on the most demonic subjects known to man, while his friends hung out without him.

bin 😑 (5:40 p.m.) are you mad ㅜ

you (5:41 p.m.) ur a bad kid

bin 😑 (5:41 p.m.) huuuu ㅜㅜ

you (5:42 p.m.) i’m just gonna head home and tell chan we should reschedule

bin 😑 (5:42 p.m.) noooo don’t do that chan doesn’t care i promise lol

bin 😑 (5:43 p.m.) he probably prefers it this way tbh

You paused, hand resting uncertainly on the stairway railing.

you (5:44 p.m.) what do you mean?

A minute passed, then another, and still no response. You huffed, assuming you’d reached your Changbin text quota for the day, and you locked your phone irritably. If Chan was expecting you, you supposed you had no choice.

It’s not a big deal, probably. You told yourself as you trudged up the stairs. Still, it felt like one. The prospect of being alone with him stressed you out as much as it excited you. No long-distance advantage of a phone call, no Changbin serving as a bridge between the two of you; just you versus Chan and his accidental charm for the next two hours.

Taking a deep breath, you knocked on the door of unit 8-325. You wondered briefly if he’d even heard, considering his headphones were virtually glued to his ears most of the time, but you didn’t get the chance to worry much about it before the door swung open, much sooner than you’d expected.

“Hi!” he greeted cheerfully. “How’ve you been?”

No hoodie on today. It made sense, given how much the weather had warmed up, but you personally felt that the muscle tank he had on instead wasn’t really necessary. His curls were out, too.

So, it was safe to say you weren’t doing well.

“Powering through the end of the semester,” you flashed a quick smile, shuffling inside and slipping off your shoes. “You?”

Chan shut the door with a noise of sympathy. “Same here.”

Your eyes scanned over the apartment. It felt undeniably empty without Changbin’s steady, familiar presence next to you or without Minho slinking back and forth between his room and the kitchen, making sure to cause as many distractions as possible each time he did.

“I’m so sorry,” you blurted out. “Changbin just told me that he wasn’t coming. If you wanna do this another night and go hang out with the others, that’s totally fine.”

He looked surprised for a moment, turning to look at you properly. “It's all good! They’ve been obsessed with that game for weeks, I got kinda sick of it, anyway.”

“Oh,” you frowned.

Chan sensed that you were still unconvinced—of course he did—and he gave you a reassuring smile. “I’m here because I wanna be.”

You knew it wasn’t his intention, but your heart still skipped a beat.

“That’s what I’m so confused about, I guess.”

He simply chuckled in response, as if that were enough to explain himself. Despite your lingering concerns, you decided not to press the issue any further, and you made your way over to the kitchen table as usual to set down your bag. You realized a moment too late that you had chosen the chair right next to where his laptop was placed. Just as you were debating whether or not you could get away with switching before he noticed, he slipped into the spot next to you, blissfully unaware of the impact it’d have on your psyche for the rest of the hour.

“I’m glad you came,” he commented, setting up his own study materials. “Feels like it’s been a while since we’ve talked.”

You wondered if that was his way of letting you know that he felt you’d been avoiding him. Well, avoiding was a bit of a stretch. More like limiting your exposure, taking him in moderation so you wouldn’t get addicted.

“It does,” you agreed. “And not just ‘cause you disappear off the face of the earth when I don’t see you in person.”

“Hey, hey!” It was defensive, but good-natured as ever. “I’m just not much of a phone guy.”

“Right, you’re more of a laptop guy.”

He grinned. “Exactly.”

“Speaking of,” you gestured to the device in question. “Have you made any progress on Placebo?”

He perked up, visibly brightening at your mention of the song. “A bit,” he chirped. “Actually, I rearranged some parts of it.”

“Oh?”

Chan’s eyes twinkled, and you got the feeling that something mischievous was brewing in his mind. “Not gonna show you yet, though.”

“And break our promise?” you feigned hurt.

“Our promise was for me to show you when it’s finished, yeah?” his grin was far too proud, like he’d been waiting for his chance to pull something like this. It was a newer side of him you hadn’t quite gotten used to yet—playful, cheeky.

“The fine print, huh?” you clicked your tongue in defeat. “Alright, you win.”

“That’s two for me, so far.”

With the way he giggled, it felt more like a win for you.

A good half hour had passed before the two of you began any actual studying, and it wouldn’t have bothered you—not in the slightest—if you weren’t already concerned about taking up too much of Chan’s evening. It didn’t help that he seemed to be a bit unfocused today as well, prone to veering off topic even more so than usual and leaving his attempts at explaining the material harder to follow than ever.

He pressed his lips together into an uncertain line, squinting at his laptop screen as he tried to make sense of the application of Sommerfeld expansion. Absent-mindedly, he crossed an arm over his chest to cup his neck, biceps bulging in the process. You’d learned from your talks with him that he was a swimmer, but you hadn’t quite expected him to look like that beneath the oversized jackets and hoodies that he wore so religiously. It was hard not to stare, to admire every toned curve and vein that protruded ever so slightly when he flexed his muscles. 

You wondered what it’d be like to touch them; if they were as firm and powerful as they looked, or if they were surprisingly much softer, just like his demeanor. You also wondered how they might look beneath you, held down by your grasp.

“Sorry,” he sighed at last, bringing you back to your senses. “I’m not really sure about this one.”

You tore your eyes away from his arms, face heating up despite not being caught. “No worries.” You put your pen down. “Do you wanna take a break? I feel like we’re both kinda out of it tonight.”

“Yeah, that’s a good idea.” He looked relieved, and a bit self-conscious. “To be honest, I barely even understood this stuff when I was an astrophysics major.”

It was an offhand comment, but it caught your attention. You’d admittedly begun to assume as much after your second or third study group under his guidance, given the way consulted outside sources so often, but to have it confirmed brought about a whole new level of respect for Chan. And, maybe something else.

“Have you been learning thermo all over again just for me and Bin?”

His gaze fell, as if realizing in alarm that he’d inadvertently exposed himself to you.  “You could say that,” he chuckled awkwardly. “I actually think I’m studying more now than I ever did when I took this class.”

A part of you wasn’t sure whether or not to be bothered that you’d been tutored by someone who wasn’t exactly qualified for the past month and a half. But no matter how badly his act of selflessness could have ended up for all three of you, you couldn’t bring yourself to feel anything but affection for the boy. Well, that, and a bit of guilt for even putting him in this position in the first place. He’d gone out of his way to re-teach himself concepts that were by no means easy to grasp, solely for the sake of helping you and Changbin out. And he had. You knew for a fact that you’d not only seen improvement in your scores since meeting him, but in your confidence in the subject as a whole.

“You’re seriously too nice for your own good,” you murmured.

He reached up for his ear, tugging at his piercing. “It’s nothing, really.”

“It’s not,” you said firmly. “Not many people would do that, especially for a stranger. So, thank you.”

“Of course,” his voice was light. “We’re friends, after all.”

“Right.”

Friends. The first time he’d said it, you’d been doubtful—both in regards to whether or not you could actually call yourselves friends, and in his intentions in doing so. Hearing it now, you felt just as strange about it, but not for the same reasons. You could safely say you were friends, that wasn’t the problem.

The problem was, you wanted to be more.

“Did you like astrophysics?” you asked the question before you had the chance to say something else, something far more stupid.

“I did,” he sounded genuine, but tense. “Well, for the most part. It just felt like the most…practical thing I could do, y’know?”

“Can I ask why you changed majors?”

It was a detail that had been nagging away at the back of your mind since Changbin had first mentioned it to you. You weren’t sure why it felt so important to know, like an essential piece of the puzzle.

Chan paused, an uncomfortable look crossing his face. It barely lasted a second, but it instantly had you wishing you’d curbed your curiosity and said nothing at all.

“It’s kinda a long story,” he said slowly. You could tell he was trying to sound casual about it. His body language, however, was more than enough for you to see that he wanted to change the subject as quickly as possible. “I guess it was just something I needed to do at the time.”

“I understand,” you decided to drop it, for his sake. “No need to get into it, if you don’t want to.”

He gave you a grateful smile. “Some other time, yeah? Can’t be telling you my life story when I’m supposed to be helping you prepare for finals.”

You hummed softly in agreement, and just like that, the atmosphere was relaxed again.

Still, the question lingered in your mind.

。⋆。˚ ʚïɞ ˚。⋆。⋆。˚ ʚïɞ ˚。⋆。⋆。˚ ʚïɞ ˚。⋆。

It was inevitable to you, at this point, that any and all sense of time would be lost whenever you and Chan got to talking. What you’d intended to be just a short break from studying to recharge, eventually morphed into another hour and a half of you two chatting away—with a few failed attempts to get back to work here and there. That was why, when the clock struck 9:00 p.m. to mark your third hour with him, you were hardly surprised.

“Why don’t I walk you home? It’s late.”

You tried to ignore the way his offer made your stomach flip.

“Oh, no you don’t have to.” The words were out of your mouth like an instinct. It was tempting, so, so tempting, but you knew that any more exposure to Chan was sure to make your soft spot for him develop into something much more troublesome. “It's a pretty far walk.”

He tilted his head, confused as to why the distance was even worth mentioning.

“Ohh, I see,” his voice took on that same, unfamiliar quality from before. “You don’t wanna spend any more time with me, is that it?”

You blinked, scanning his face for some sign of hurt or offense. Instead, all you found was a playful smile, eyes crinkling and dimples flashing.

He was teasing you.

“You got me,” you played along, throwing your bag dramatically over your shoulder. “I only spend my Friday nights studying thermo with people I can’t stand.”

Chan giggled. It was shy and cute; the giggle of someone completely unaware of how enamored with him you really were.

“In that case, making me walk there and back shouldn’t be a problem, right? Since you hate me so much.”

You relented. It was a losing battle from the start, anyway.

The air had grown a bit chillier after sunset, which, much to your relief, meant Chan had thrown on a jacket and covered up his criminally distracting arms. You felt a strange sense of peace as the two of you strolled along the sidewalk out of his apartment phase, stealing glances at him as often as the streetlights would allow. He had his hands in his pockets, swinging them with each step he took and swaying his head along with the breeze that brushed through his curls.

It was hopeless. You were so hopelessly taken by him.

“There she is,” you remarked, slowing your pace to gaze upwards. “That moon you love so much.”

It reflected a pure, white light among the sea of stars, owning the sky in all its Waning Gibbous glory.

“Beautiful,” you heard Chan murmur.

You looked over at him, hoping to catch a glimpse of his eyes illuminated by the moon as he stared up in awe. Instead, you found him staring right at you.

He seemed taken aback for a moment. Even so, for once, he didn’t look away. He simply smiled.

Warmth spread through your chest, and you knew this time you couldn’t blame it on his body heat.

“I think you have us both beat,” you said softly.

At that, he broke eye contact. He ducked his head with a shy puff of laughter, pressing his cheek into his shoulder to hide his face. You rode the high of it for the rest of your walk home together.

The two of you were mostly quiet as you neared your apartment complex, letting the silence hang comfortably around you. Despite the long walk, neither of you were in any particular hurry, and when you approached the front gate of your building, you couldn’t help but feel that the time had slipped away from you all too quickly.

“Thanks again for walking me home,” you murmured. “You really didn’t have to.”

“I wanted to,” he grinned.

Your hand rested tentatively on the handle, not yet wanting this moment to end.

“Not gonna try to return the favor, are you?” His eyes sparkled in the low light. Even when he was messing with you, he still sounded seconds away from becoming flustered himself.

You smiled. “I’ve got something in mind.”

Before he could say anything else, and before you could second guess yourself, you leaned forward and pressed a kiss to his cheek. It was quick and innocent, but it made his breath catch in his throat all the same. 

When you pulled back, Chan’s fingers came to hover over the spot your lips had been moments ago. You wished the lighting in the hallway was stronger, so that you could fully see the furious blush that you knew was spreading across his face.

His eyes flickered down to your lips. He didn’t say a word, he didn’t have to for you to hear him willing you to do it, begging you to do it.

So, you leaned in again and kissed him.

The heat that surged through you was different this time. It didn’t make you flinch or jolt back in alarm; it drew you in. However soft you’d imagined Chan’s lips to be—plush and heart-shaped and irresistible—the reality was infinitely softer.

Your hands reached up to cup his face. His warmth fed into yours, and vice versa, and somewhere in the back of your mind, it became clear that the fire had been coming from both of you this entire time. He sighed sweetly into the kiss, tilting his head forward, trying somehow to deepen it even further, like he wouldn’t satisfied until you were completely melded together.

The two of you might have stayed that way if your lungs hadn’t begun to cry for air. Reluctantly, you pulled away, leaving you both breathless and longing for each other’s warmth again. All the efforts you’d made to hold yourself back around him seemed so laughable now. You didn’t want him in moderation, you wanted all of him.

Chan’s eyes fluttered open, dark and dazed. The sight made you want to pull him inside with you, to take him apart bit by bit and put him back together again, over and over until you knew him inside and out.

Instead, you brushed your thumb over his burning cheek, touch harboring a gentleness that masked the ache inside you.

“Get home safe, Channie,” you whispered.

[hexagr]

There's a well-known correlation between depression, suicide, and the seasons of spring and summer, often attributed to seasonal allergies. The most plausible explanation is that pollen and allergies trigger inflammation, which can impair self-control and increase impulsivity. However, from a broader perspective, the underlying theme here is not merely allergies but thermodynamics. I say this because it's heat that triggers plants' release of increased amounts of pollen, exacerbating allergies. A related theory is the "heat hypothesis," which posits that higher temperatures correlate with increased aggressive behavior. Field experiments found evidence to support this hypothesis, while other lab experiments returned mixed results. But an empirical analysis would eventually lend more credence to the theory. For instance, researchers published a paper in 1997 in which they performed a statistical analysis looking at averages in temperature in comparison to US crime rates across the years 1950–1995. The analysis revealed a compelling pattern and suggested that higher temperatures are, in fact, linked to increases in crimes such as assault and murder. Through the lens of cognitive and physiological stress: since heat may impair cognitive performance and reduce self-control, it may also potentially make individuals more prone to aggression. 

[materialsscienceandengineering]

Thermodynamic Potentials

Specific state functions that depend on so-called natural variables (such as temperature, entropy, pressure, etc.) are known as thermodynamic potentials. The set of relevant natural variables is unique for each potential, though all rely on the number of particles and potential energy. These potentials can be thought of as similar to potential energy, which is the capacity to do work:

Internal energy is the capacity to do work as well as release heat, with variables including entropy and volume

Helmholtz free energy relates to mechanical and non-mechanical work, with variables including temperature and volume

Gibbs free energy is specific to non-mechanical work, with variables including temperature and pressure

And enthalpy is the capacity to do non-mechanical work and release heat, with variables including entropy and pressure

A fifth potential is also considered common, though less well known: the grand potential.

Sources/Further Reading: (Image source - Statistical Physics) (Wikipedia) (University of Frankfurt) (Hyperphysics) (LibreTexts)

[neet-aspirant]

27–01–2024, Saturday

definitely not showing off my new Fire-Boltt fire pods melody enc 501 that i bought today, in the pics. nope definitely not. I'd never.

psych

phy thermodynamics

maths statistics

bio

english project

[aurae-rori]

Oh yeah, I forgot about my actual thought when I got distracted by comedic potential!

I’ve brought up Nous no longer calculating futures before, and since I’ve been studying philosophy for the past month, it gave me this fun new insight into the whole thing when put next to my cursory knowledge of non-relativistic and informational/statistical physics! Don’t come for me if I get it wrong people, I despise any mathematics that isn’t statistics

Nous’s whole deal is determinism, something I’ve also mentioned before. Theoretically, if we could track every possible variable in the entire universe, then yes, reality is probably predetermined. And Nous is clearly capable of that.

But here’s another definition:

Entropy is the measure of chaos in a closed system.

In thermodynamics, isolated systems of high entropy tend towards thermodynamic equilibrium (hello HooH) where things don’t wanna change.

In other applications, it refers to energy dispersal. Basically, as time goes on, things get watered down, energy gets less efficient, heat death of the universe blah blah blah (Hello Nanook? Hello Qlipoth even???)

It’s a lot more complicated obviously, but this general principle, limited possibilities trending in a largely predictable way, is the so called “determinism.” Entropy levels cannot change, energy can’t really be destroyed, and the systems want to stabilize because of it. They want to be in a certain configuration.

But what happened if, suddenly, it’s not a closed system?

What if there’s suddenly enough outside interference that the initial conditions no longer apply? Think those multi-pendulums. Welt Yang, Acheron, the Trailblazer, and God knows who else. There’s a decent argument to be made that someone, something, multiples of people and things, threw off the whole system.

Energy cannot be created or destroyed. But it can be introduced. The Stellaron Hunters seem to be tracking these outside interferences on a macro level, collecting stellarons specifically. Is it any coincidence that this is what the Trailblazer has been put in possession of? Are stellarons some form of, for lack of a better word, outside data poisoning?

I’m just rambling wildly at this point, but I wasn’t allowed to start taking about spacetime theory in relation to philosophical determinism in class so I’ve gotta let off steam somehow.

- 🦋

this is so intelligent that my brain has been scrambling for words that are coherent enough to respond to this as neatly as youve placed it for me, but now i just dont care and youre gonna get my unhinged ramblings.

i love the theory. i fucking love it so much and i enjoy the idea of the stellaron hunters actively fucking with shit by bringing in unknown variables. open that closed system. fuck with that all knowing machine, nous. nous cant calculate due to the system being opened, i think is what you're proposing? and i fuck with that. like yeah "fuck your closed system im gonna open it forcefully with the power of this trash raccoon and this old ass man from hi3"

i do enjoy philosophy and i think i understand what you're getting at, but my brain was so fucking scrambled im so sorry butterfly anon i always love your asks sm KSDHKSLJGHLKjh <3

[minmin-vs-physics]

Min vs FA24

Now that I'm officially a college senior, I thought a post of what I will be up to is in order. (Especially since I was absconding last week) Gonna take some hard hitters for classes this semester, pray for me.

Intro to General Relativity: FINALLY. I've been waiting for this since before I was a physics major. I know it's gonna be good since my QM prof from last sem is teaching it. (Lowkey wanna switch to the grad version because my QM prof from last sem is teaching it)

Relativistic Quantum Field Theory: Another scary class but still highly anticipated! I've basically been doing QFT all summer, but the class is scarier because formalism. Of course, it will unlock some doors in particle theory.

Statistical Thermodynamics: lowkey im most nervous about this one. another beast of a topic in physics and i rlly want to learn it but idk we don't talk abt it much??? (except abt how much we're dreading it) the whole cohort will come together for this one.

Intro to Sociocultural Anthropology: always gotta throw one curveball in the schedule. not much to say bc im just taking it for a gen ed req.

Computational Physics: I should drop this bc taking four physics classes in grad apps season is kinda overkill. i wanted the lightest sem i could make but still ended up w this kraken. but no math class! (i had to pry out topology) this is the first and only semester i won't have a math class. in addition to courseload i also have

TAing for a CS class: ik my way around it so its not a problem but its still a time sink

TAing for a QM class: this is smth i def just do for the love of it, so another time sink basically but i look forward to it

Research: gotta work on that thesis y'all. i wanna make smth good out of it in time.

Physics GRE: broccoli on my plate

Grad Apps: waking nightmare. but it'll be fine i can drop out and become a finance bro.

but i also wanna make memories with all the other seniors because what? how are we seniors? (im writing this after going stargazing with my friends on a school night.)