Part 2

Brownian Motion, “Brownian Entropy” and “Entropo-dynamics”

The “Arrow of Time”

Entropy as “Disorder”

A Gedanken Experiment

Thermodynamic Equilibrium

“The Arrow of Time”

Physics is currently in the midst of a passionate affair with the concept of the “arrow of time”. It shows up everywhere, and it is very popular among non-scientists, too. Physicists point out that we don’t see all the molecules of air in a room move to one half of the room, just as we don’t see shattered cups reforming themselves into whole cups.

Actually, some scientists even refer to several “arrows of time”. For example, Stephen Hawking refers to three arrows of time, the “thermodynamic”, the “psychological”, and the “cosmological”. [S. Hawking, A Brief History of Time, p. 145.]

There are many synergizing... oversights here, too.

We will start by looking again at the “Brownian entropy” model we started looking at above.

Here are some plots of some state trajectories in some ergodic processes that give a very crude idea (and probably misleading idea if we e.g. try to study the distributions of changes thinking they are of a thermodynamic nature) of what entropy fluctuations might look like in an isolated system. As might be expected, the functions spend an “overwhelming” amount of time at-near the maximum “entropy”.

Figures 2a-e. Example trajectories of various ergodic processes.

If we examine a plot of an ergodic entropy function with time, allowing time to extend so that we see at least a few examples of the lowest entropy states being re-attained from the highest, we see... that’s right, we see entropy reversing over what might be a “longish period of time”. We will be able to find an arbitrary “starting” state with a high entropy and a companion “end” state, one with a low entropy, in the direction of increasing time, and when we draw the “arrow of time” we will get it pointing in the opposite direction to the one we are used to.

Because it also decreases (i.e. ergodicly and Brownianly)
entropy does not give an “arrow of time”
in the way that physicists have modeled it as so doing.
This derives from... oversights in thermodynamics
(which ostensibly gives us our “arrow of time”).

In fact, we didn’t need to introduce the concept of “Brownian entropy” to find this “arrow of time” pointing in the opposite direction, since just the ergodic hypothesis is sufficient to do that. But it helps to introduce it early, because the idea needs getting used to, and because there is much more to the story that concerns it.

Additionally (and this goes for purely ergodic analyses as well), in the plots above (and, without the careful study that is needed, we don’t know if this is thermodynamically misleading) we notice that the “down sides of the valleys” look like probabilistic mirror images of the “up sides of the valleys”. If we plotted increasing time from right to left instead of our usual left to right, we wouldn’t be able to tell the difference just by looking at the plot. I.e., if we pick a point on the curve where entropy is increasing, and then we “reverse time” and find entropy decreasing (which is roughly how the “arrow of time” got gedankened in the first place), we will see entropy decrease for perhaps only a short while, but it will eventually increase, perhaps a lot; and further, if we happen to plunk down our point where entropy just happens to be decreasing...

In any case, if we keep looking, we will eventually lose our “arrow of time” to the statistics of the fluctuations.

Just because we have entropy spending most of its time where it is most likely to be (which doesn’t seem to depend on the direction of time, a seemingly obvious point that is crucial to future studies) doesn’t mean that we can find an “arrow of time” in its activities.

It is very likely that when we (successfully) model the distributions of both short term (femtoseconds to days) and long term (to say, 10³⁰ years) Brownian “entropodynamic” fluctuations, we will find them to be symmetrical in time. If we find them to be asymmetrical, then we may again have a candidate for an “arrow of time”. Relatedly, it may turn out to be easy to prove mathematically that any “Brownian process” is an ergodic process, although it is easy to exhibit ergodic processes that are not Brownian. (A “Brownian process” might want to allow the systemically distributed Brownian “particle” to be immediately kicked back in the opposite “direction”, in the sense of having a non-zero probability of re-achieving the previous state — which state therefore cannot be the usual micro-state of momentum, etc. — as opposed to re-achieving merely the previous value of the state function.)

There is another issue worth raising, a paradoxical one. Pick any of the plots above and gedanken a horizontal line anywhere in the plot. Notice that the number of times it crosses the entropy curve when entropy is decreasing is within 1 of the number of times that it crosses it when entropy is increasing. Notice also that when we select our sample in this manner, we find that decreasing entropy is equally as likely as increasing entropy! I.e. the sum of the increases and decreases is necessarily a strict 0! The mathematics of this, which most of us are familiar with, tells us that it could hardly be any other way.

(Is there a standard resolution to this paradox? Yes, of course. The above graphs-plots do not tell us what would happen if we started in every possible state, let’s say at a given level of entropy. To do so, of course, tends to distinguish every possible micro-state instead of lumping them statistically in a macro-state. But what we get will — “probably” — be that almost all of these states will exhibit increasing entropy. But when we follow one particular trajectory of an ergodic system, we must per force get numbers of decreases and increase that differ at most by 1.)

In any case, all this should remind us of that ultimate truth:

The Zeroth Law of Statistics
   “There are lies;
     there are damn lies;
     and then there are... statistics.”

SECTIONS

Brownian Motion, “Brownian Entropy” and “Entropo-dynamics”

The “Arrow of Time”

Entropy as “Disorder”

A Gedanken Experiment

Thermodynamic Equilibrium

Entropy as “Disorder”

Our intent is to open the “cosmos” of worms of “time reversal” (a Gedanken Convenience Concept mistaken for something else). But first we will take a relevant detour through the cosmos of... oversights of the concept that entropy and disorder are really the same (effectively, or more formally, that “entropy measures disorder”). We will find that “order” is a completely “subjective” concept, since “order” as an “objective” concept has rather serious... inconsistencies.

THE STANDARD ENTROPY-DISORDER “RELATIONSHIP” CONCEPT

We will first look at the standardly accepted scientific concept of the relationship between entropy and order. Take a look at Figure(s) 3:

3a 3b 3c
Figure 3abc.

Assume that the black and white areas represent volumes of 2 different kinds of gas particles (e.g. molecules-atoms of an ideal gas) held apart by a barrier which can be removed without effecting either the entropy or the order. (Or we could assume that one of the areas represents a vacuum.) At the instant of the removal of the barrier, Figure 3a corresponds to a low entropy-high order (-low disorder) arrangement (sometimes called a “macro-state”) of those particles. A low entropy-high order arrangement is standardly considered to correspond to an “arrangement” (we could also say “macro-arrangement”) which has a low probability in the sense that it is 1 of a class of relatively few other “micro-arrangements” with which it is considered to be equivalent. (“Arrangements” because it has a different meaning; notice that things start to get confusing; thermodynamics gets very complicated.) I.e. if we only had two particles and both were in the upper half of the volume (and implicitly, if all micro-arrangements or micro-states with both of them in either half were considered equivalent), then such an arrangement would be less probable than one with 1 particle in the upper box and 1 in the lower. There is “1” way (“1” is a very strange number) to put them both in the upper box, “2” ways to put them 1 to each box.

In this way, Figure 3b corresponds to a medium entropy-medium order arrangement (as the particles mix), and Figure 3c corresponds to the highest entropy-lowest order arrangement of the 3 (with the mixing at “equilibrium”), and with certain obvious modest assumptions and limits, also corresponds to the highest possible entropy arrangement in this situation, with the highest possible disorder. If the particles were atoms of a gas, arranged as in Figure 3b, they would reasonably quickly, by human standards, rush to an arrangement like that in Figure 3c (especially if e.g. the white space actually represented a vacuum instead of a different kind of gas particle), and not that of Figure 3a.

This concept of order is standardly held to correspond (completely) to thermodynamic entropy (i.e. “thermodynamic entropy” is said to be a “measure” of “disorder”), which is held by the 2nd law of thermodynamics to increase monotonically in any isolated system (i.e. it does not decrease although it may increase only very slowly). Thus, increasing thermodynamic entropy is standardly held to mean increasing disorder, whereas wider variance in distribution (as in Figure 3a) is held to mean higher order since it has fewer arrangements that give rise to it. Our physicists (e.g. Richard Feynman) say things like “entropy measures disorder” and “the cosmos always goes from ‘order’ to ‘disorder’, so entropy always increases.”

That is the standard scientific concept of the relationship (scientifically considered necessary) between entropy and disorder in a very tiny nutshell. We will look a bit further at our concept of “disorder”.

OTHER VIEWS OF ENTROPY-DISORDER

Figure 4abc.
In Figure(s) 4 we see 3 arrangements of vertical lines. Which of the 3 arrangements is the “most ordered”? Which is the “least ordered”?

A “common sense”, “subjective” classification would have 4a (leftmost) as the “most disordered” (notice how it looks “chaotic”, as in chaos theory), and 4b (center) as the “most ordered” (least “chaotic”). But science’s current standard concept of “order” as corresponding to “lower entropy” (in the accepted sense of “fewer micro-arrangements giving rise to the macro-arrangement”) says that 4a is the “most ordered”, 4c (rightmost) is “more disordered” than 4a but “more ordered” than 4b. The arrangement 4b is even “maximally disordered” as there is no way for it to become “more disordered”.

If you find yourself saying that arrangement Figure 4a is really only 1 of many “disordered” arrangements that must all be considered together, and these arrangements when taken together are greater in number, and therefore both more highly probable and therefore more highly entropic not to mention more dis-ordered, than 4b, you should look at 4a again, carefully, very very carefully.

If you looked at 4a again very carefully, you will have noticed that it has a certain “order” that suggests that it not be classified as “disordered”. Hint: look at length of the smallest lines, and calculate the length of the other lines using that as the unit length. Especially note how our sense of how the arrangement was generated affects our judgment of whether it is “ordered” and “disordered”. Would it make any difference if the line lengths went something like: 2, 8, 2, 8, 1, 8, 2, 8, 4, 5, 9, 0...?

Because a particular macro-arrangement like Figure 4a doesn’t mean anything to us, we lump it together with all other such macro-arrangements that don’t mean anything to us, thus getting a large number of them (since there isn’t much that does mean anything to us). Things that don’t mean anything to us seem to be “disordered”. The number of micro-arrangements that make up this huge lump of unmeaningful macro-arrangements is quite a bit larger than the number that make up a similar lump of more meaningful macro-arrangements like Figure 4b, and is thus “disorder” is more likely, statistically, just as entropy is. Thus we find statistical mechanical support for the idea that increasing entropy and increasing disorder are equivalent... and inevitable. But this lumping is inherently subjective in nature, and we are reminded yet again that “there are lies, there are damn lies, and then there are... statistics.”

The Emperor’s New Clothes: “order” and “disorder” (and “chaos”) can now be seen to be subjective concepts, and distinctly not scientific ones. Now we can understand better the seeming paradox of Figure 4abc. If we think of arrangement 4a as temperatures or as water levels, they come to equilibrium at the level in 4b with both increasing thermodynamic entropy and increasing combinatoric entropy. But which looks more ordered?! “Common sense” says that 4b looks more ordered than 4a, just as the calm ocean looks more ordered than the chaotic stormy one. Whenever we have our own particular order, any “inappropriate” change in it seems to produce dis-order, whether there is increasing or decreasing entropy, complexity, or what have you.

I personally think that the entropy-disorder fallacy arose because, emotionally, we saw entropy as casting away stones that we had carefully gathered together and laid up as treasure, thus “destroying order”, and because we were reacting emotionally, we ignored that entropy also restored order, like dissipating the storm on the ocean. (Some people refer to “emergence” and such terms.) We have such terrifyingly powerful emotions about “order” and “chaos” (just as we do about “God” and “freedom”). But the nightmare isn’t over yet. What we will find by the time we finish analyzing entropy-disorder is The Emperor’s New Fall Fashion Line!

Note that scientifically we tend to confuse micro-arrangements with arrangements, micro-states with macro-states. We (scientifically) think “disorder” or “entropy” is a property of the micro-state (i.e. the actual state of the system), when it is actually only (and only perhaps) a property of (our context sensitive evaluation of) the macro-arrangement-state, e.g. of the implicit selection process that went into the choice of which micro-arrangements-states fall in the class and-or classification of that macro-arrangement-state. We (scientifically) think that the selection process is “objective” instead of “subjective” because... well, because we refer to physical attributes, we use numbers, even... statistics, we... all because we fail to notice the inherently subjective nature of our selection process. What universe do we select, to then select a subset, from which we select a further subset... which we subject to... statistical analysis? And how far do we try to extrapolate the results?!

When we extrapolate using the results of a statistical analysis, we rarely notice that we not only “go out” of the selection on which the statistics were based, we often go out of the selection from which they were selected, often even out of the originally selected universe. (Time tends to take us out of all these simultaneously in a “never the same river twice” sort of way, but again, as scientists we fail to notice this.) Each “going out” increases the error terms, often exponentially, almost always invisibly to our science. (Einstein’s relativity uses the concept of “manifold” where a similar process of extrapolation from “infinitesimal” “locally Euclidean regions” to “global levels” of the manifold also yields potentially exponentially increasing error terms, again scientifically unnoticed.)

“Selection” is almost a one-word tour of the original sin of... statistics.

All through this situation with the conjoined-twin concepts of “disorder” and “entropy” run the conjoined-twin concepts of “probability” and... “statistics”.

Whenever you think of probabilities, remember that probabilities are universally calculated by that process of nether-regional extraction known as... statistical analysis.

The probabilistic nature of the micro-arrangements or micro-states, the subjective nature of the selection (e.g. the criteria there for) of which micro-states are considered to make up a macro-state, all these and more should remind one, when contemplating “disorder” and “entropy”, of (repeated, for more “emphasis”):

The Zeroth Law of Statistics
“There are lies;
there are damn lies;
and then there are... statistics.”

“ORDER” IS A COMPLETELY “SUBJECTIVE” CONCEPT

Relevant Digression: If you are wondering why all the quotes for “subjective” and “objective”, it is because both these concepts are themselves “subjective”, even though science holds them to be “objective”. By this time this last should make sense.
(All other quotes probably indicate a similar attempt to remind the reader of the ill-defined nature of the beasts thus caged.)

We cannot escape it:

Our concepts of “order” and “disorder” are actually completely “subjective”.

But, it won’t hurt to give another example:

In mathematics, a random variable can have any possible distribution, even a constant one. It is not usually noted, so think of this: by our current scientific concepts of “entropy” and “order”, a constant (non-random?!) random variable has the “lowest order”, “highest disorder”, and of course the “highest possible entropy” of any random variable.

“Mom, this isn’t a Disaster Area; it’s a Pinnacle of the Highest Physical Order!”
“The more chaotic the storm makes the ocean, the more ordered the ocean becomes?! But this means that Chaos... and Order... ?!” Yes...

Questions that help put it all in perspective:

Is “complexity” more “orderly” or more “disorderly” than “simplicity”?! more “chaotic”?!
Is “order” more “complex” or more “simple” than “disorder”?! than “chaos”?!
Are “complexity” and “simplicity” completely subjective concepts just as “order” and “disorder” are?!
(We will eventually find that concepts such as “adequate complexity for emergence” are with regard to implicit subjectively chosen behaviors we wish to create or observe. I.e. the complexity of a system may be co-qualitatively-quantitatively adequate for 1 class of behaviors to emerge, but not others, even if the latter need no more complexity in a crude sense then the former.)

Enough for now. To reiterate the obvious:

Our concepts of “order” and “disorder” can not be held to be scientifically “objective”; they are in fact completely “subjective”.
If nothing else, the selection process that determines which micro- arrangements-states will be associated with which class(es) of macro-arrangements-states is itself always completely “subjective”.

and repeating the ever more obvious:

The Zeroth Law of Statistics
“There are lies;
there are damn lies;
and then there are... statistics.”

If you try to offer a different way of looking at the above situations that gives a contradictory evaluation, you will merely reinforce the demonstration of the “subjective” nature of the ostensibly “objective” and “scientific” concepts of “order” and “disorder”. For example, if you tried to refer to the “objective” nature of the probabilities of certain arrangements as a composite of the probabilities of the micro-arrangements, one response could-should be (as above) that the categorization of micro-arrangements into macro-arrangements is itself “subjective”: e.g. just as in the study of mechanics, we draw our sub-system boundaries in an arbitrary fashion, i.e. arbitrarily assigning micro-arrangements-states to macro-arrangements-states. E.g., it is “subjective” whether 2718281828459... and 31415926535... are examples of “disorder” — highly probable (when taken as a class; selection-statistics again) and interchangeable (within the class; more selection-statistics) waves on a stormy ocean — or “order” — unique and non-interchangeable initial sequences of digits in 2 fundamental transcendental constants (if we take such constants as non-interchangeable; more selection and statistics again).

The foundational nature of the confusion of entropy and disorder makes it desirable to objectify the subjective nature of the confusion. A gedanken experiment is in order.

SECTIONS

Brownian Motion, “Brownian Entropy” and “Entropo-dynamics”

The “Arrow of Time”

Entropy as “Disorder”

A Gedanken Experiment

Thermodynamic Equilibrium

A Gedanken Experiment

A simple gedanken experiment is offered below to show the (absolute lack of) correlation between thermodynamic entropy and disorder, but first:

QUICK SUMMARY AND EXPANSION

It is currently held as tautological that increasing thermodynamic entropy means increasing disorder. Our physicists (e.g. Richard Feynman) say things like “entropy measures disorder” and “the cosmos always goes from ‘order’ to ‘disorder’, so entropy always increases.” We hold “order”, “disorder”, and relatedly, the need to use energy (power and force) to restore and maintain “order” from “disorder” to be objective, scientifically based concepts. We hold that entropy always increases in the cosmos, or rather, that it increases in any isolated system (a system with no matter or energy transport into or out of it), and the cosmos, being all there is, is by definition an isolated system (standard physics).

Among other things, as we saw above, scientists use this concept of (strictly) monotonically increasing entropy to give us an “arrow of time” since we notice things breaking and mixing (or whatever) and when we reverse the movie we have made of them mixing, it shows an un-breaking and un-mixing (or un-whatever) that we do not see — at least not often — in real life. This is also the “reversibility-irreversibility of time” question-concept that is so popular now, and the answer currently given to it.

A SIMPLE BUT CRUCIAL GEDANKEN EXPERIMENT

There is a simple gedanken experiment that shows that “disorder” and “entropy” cannot relate in the way that science holds that they do:

The Entropy-Disorder Gedanken Experiment

In an isolated system, place one small (Earth-sized) planet, and, on its surface, place a pint bottle, make that a quart bottle, of that good English un-homogenized whole milk. (Sorry, James, shaken, not stirred. The entropy-disorder fallacy should now be immediately obvious. It is obvious that the gravitational field of the Earth will do work, but it will be doing it in the context of an isolated system that comes to “equilibrium”, a Brownian-ergodic equilibrium to be sure. Why has thermodynamics ignored this almost completely?! After all, even Boltzmann pointed out that fields are always affecting real world thermodynamic systems.)

In another isolated system, place a similar planet and another quart bottle divided by a zero-entropically removable barrier which separates the top and bottom halves in which are placed two differently color-dyed waters (let’s make them gedanken equidensity; again, the entropy-disorder fallacy should be immediately obvious).

Remove the barrier. As time increases, in the first system the cream and the milk separate while thermodynamic entropy increases, and simultaneously, in the second system the two colored waters merge (they may not merge “all the way” depending on actual densities) while thermodynamic entropy increases.

I.e. in the one system “order” increases with increasing thermodynamic entropy while in the other, “disorder” increases with increasing entropy... and this happens:
regardless of which direction you pick for “increasing” order or disorder,
and, we might add:
regardless of which direction you pick for time.

Some might still object that in the first isolated system the gravitational field is doing work which reverses the entropy. This argument has its validity in open systems where the gravitational field is outside of the system in question (the milk bottle), but transferring energy and entropy away from it. This argument also neglects that the gravitational field is also doing work in the second isolated system. The traditional view is to look at what happens when “the batteries run down” and the systems finally attain “equilibrium”. But in our case the final equilibrium gives us just what was found above. (The concept of “equilibrium” needs a closer look, which it will get in the next section.)

It seems to be an extremely absurd set of... oversights, but the science-physics of thermodynamics has almost completely ignored the effect of fields on the thermodynamics of systems. (A counter-example from classical thermodynamics is-are the attractive van der Waals forces between molecules, which have been studied for some time.) Scientists will deny this and point at Boltzmann’s distribution (of large numbers of small particles per unit volume of an ideal gas as a function of field derived potential energies of the particles, absolute temperature, etc.; it ignores the problem of the initial distribution of the particles and the possibility of potential energy wells that the particles would require e.g. tunneling to get in-out of), but they certainly left such knowledge-wisdom out of their reasoning about entropy and disorder.

They also left them out of another picture. This ignoring of field effects shows itself again in the seemingly paradoxical (and also seemingly... oversighted) case that a temperature gradient can exist at equilibrium without a classical thermodynamic net heat flow along it, paradoxical since classical-standard thermodynamics still holds it as an impossibility (despite e.g. Boltzmann’s distribution). But such a non-zero temperature gradient does occur in the atmosphere of a planet (even more noticeably in the gedanken atmosphere of a gedanken planet), along with both a mass-density gradient and number-of-molecules-density gradient, where the field of gravitational potential (easiest to think of in Newtonian terms) reduces the kinetic energies — and numbers, due to Maxwell’s kinetic energy distribution considerations — of particles that move “up” away from the planet, and increases the kinetic energies of those that move “down” toward the planet (and thus the average kinetic energies in both cases), but without classical net heat flow while doing so. (We should start looking for non-classical heat flow, taking into account the gravitational potential.)

This is without a doubt one or more synergistically interacting Emperor’s New Clothes situations. Once we start actually looking, much of the fallacy of our concept of entropy-disorder correlation-equivalence quickly become obvious. We did not sufficiently separate variables, as happens so often in situations involving inconsistency. We associated order with a physical arrangement that only associated with entropy in some situations, and failed to notice that those some were distinctly not all situations. Once we notice the Emperor’s New Clothes, we start finding examples all over the place. When beads of water or mercury join of themselves to form a larger bead in an isolated system, they do so with increasing thermodynamic entropy, but decreasing combinatoric entropy. But do they always do so with increasing “disorder”? decreasing?!

A common fallacy is to confuse flat distributions with non-flat or biased distributions. If there are 10 of yesterday’s newspapers and 10 of today’s, would we think that there is a 50/50 chance that the first customer will chose yesterday’s paper? No. It’s an obvious fallacy. But with entropy-disorder, we equivalenced the relative numbers of “equivalent” ways of arranging particles in different patterns (the flat distribution), which we associated with “order”, with probabilities of occurrence consistently correlated with entropy in a physical situation (the non-flat or biased distribution). Whew...

After performing the Entropy-Disorder Gedanken Experiment, we notice that it is not merely the number of “equivalent” arrangements that determines the probabilities, but system dynamics as well. When we add gravity and density, an intermediate arrangement like that of Figure 3b becomes the most likely, or even more like Figure 3a in the case of our Gedanken milk bottles with larger density differences. Our common sense of order and relative order corresponds here to the visual impact of the arrangements, but (one of) our mathematically based senses of order corresponds to the system state transition probabilities, which we did so poorly at determining just now. Many will long shake their heads and wonder how scientists could have held such generally independent (but situationally variantly interdependent) concepts to be the same.

Let’s also review again the “arrow of time” that “time reversal” ostensibly produced. Ostensibly, the thermodynamic arrow of time derives from the fact that we don’t see disorder becoming more ordered in the direction of time that we experience, but when we reverse time, we do see disorder becoming order. This (ostensibly) gives an asymmetry that points like an arrow in one of those directions. But...

But, the fallacies start jumping out at us. First of all, our concept of “disorder” and our concept of the relationship between entropy and “disorder” both just died an ugly death, one that does not allow us to use either increasing “disorder” or “order” as an “arrow of time”.

Can we use thermodynamic entropy divorced from disorder? NO! We saw that above that an isolated ergodic-thermodynamic system stochastically cycles eternally through all possible entropies, in either direction of time!

If there is an “arrow of time”, it will not consist of an increase in thermodynamic entropy, since this must also at times decrease, Brownianly and ergodically, nor of an increase of “disorder” or “order”. So, for our “arrow of time” to exist, there will have to be something else that makes that eternal ergodic increase-decrease-increase-decrease cycle asymmetrical. (E.g. we could have an asymmetrical stochastic “saw-tooth”-like substance.) The search for this will be very important, even if the results are negative.

To sum up:

The Entropy-Disorder Gedanken Experiment forever lays to rest any possibility that we can establish a general correlation, let alone a general equivalence, between thermodynamic entropy and disorder.
We can still gedanken special cases where we can obtain a local correlation or “equivalence” between them, but:
Our Scientific (and thus our Social-Psychological) Concept of the Equivalence, or even Correlation, of Entropy and Disorder is COMPLETELY FALSE.

After a quick, almost digressive, look at the concept of “equilibrium”, we may be ready to look at “time reversal”.

SECTIONS

Brownian Motion, “Brownian Entropy” and “Entropo-dynamics”

The “Arrow of Time”

Entropy as “Disorder”

A Gedanken Experiment

Thermodynamic Equilibrium