# Entropy is…disorder?

As I read somewhere, it said that the universe is heading toward disorder a.k.a entropy increasing.

Now as far as I know from the second law of thermodynamics it states that entropy is indeed increasing and in the end, the entropy of the universe will be maximum, so everything will evolve toward thermodynamic equilibrium (e.g same temperature everywhere in the universe).

So my question is: isn't equilibrium order? Why is entropy called a measure of disorder if more entropy means more order?

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Why do you think equilibrium means ordered? – AlphaGo Jan 18 at 12:33
yes, it's a shortcut. More entropy is more "more or equal variousness" than "more or equal disorder". The first is factual, the second subjective – igael Jan 18 at 12:43
@igael To me, the second is less than subjective ... its meaningless. At least to me it is. I've never been able to make any sense of it. I like your phrase. – garyp Jan 18 at 12:52
@OP: Note that thermal and mechanical equilibrium are two different things. The latter one usually is associated with some (spatial) ordering – Bort Jan 18 at 13:54
Essential: Entropy Demystified: The Second Law Reduced to Plain Common Sense by Arieh Ben-Naim. – Martín-Blas Pérez Pinilla Jan 19 at 7:33

I personally find the terms consistent. Think of the entropy as Boltzman proposes: $S=k \, \ln W$ Meaning high entropy states can be realized via many different configurations. Truly ordered state (assume you arrange a sculpture from atoms) can be realized via much smaller number of microscopic states. So again, equilibrium is not order - it is a mess.

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equilibrium is not order - it is a mess I might quote you for this some day. +1 – Steeven Jan 18 at 14:00
This is not true: a solid lattice is a highly ordered configuration, and is the most probable one as you cool down a gas, or liquid. Also different states of matter can coexist in certain trivial conditions and in a very stable status (i.e. water in triple point). – rmhleo Jan 18 at 15:29
@rmhleo: processes like crystallisation take place because $\Delta G= \Delta H - T\Delta S < 0$ and much lattice energy is released. – Gert Jan 18 at 16:34
@Steeven: I like Peter Atkins' "Everything always ends badly!". – Gert Jan 18 at 16:35
The trouble with using "disorder" is that it assumes a microscopic view. Equilibrium systems are complicated at the microscopic level, but simple at the macroscopic level. Indeed that is one of the ways that classical thermodynamics defines equilibrium "being adequately described by a small number of state variables" – dmckee Jan 18 at 18:08

First of all as stated by Madan Ivan: equilibrium is not order. But you can get certain systems that are in a meta-stable "local" equilibrium (here meaning that you need some energy to move it from there), for example a crystal. These can be highly ordered.

Intuitively: if you smack the crystal with a hammer it breaks to pieces. This brings your closer to the global equilibrium. In the universe as a whole there is energy exchange between such subsystems and the second law of thermodynamics states that the overall order decreases by these processes.

So I think your problem is the two uses of the word equilibrium. Meta-stable equilibria can be order while the one that is used in the second law is the global minimum.

A comment on entropy in general: there isn't just one, there is a lot of them. In thermodynamics only there are 3 distinct ones. The names I use in the following are not official, since the literature mostly does not distinguish between them.

1. The Gibbs entropy: $$S_G = -k \sum_{N} \int d \tau_N p_N \log(p_N)$$ where the sum is over all the states of the system and $p_N$ is the probability of it. It turns out that this is a constant of the equations of motion.
2. The Boltzmann entropy: $$S_B = -k \sum_{1} \int d \tau_1 p_1 \log(p_1)$$ where $p_1$ is now the one particle distribution. This entropy is just wrong, but used a lot.
3. Experimental entropy: $$\Delta S_E = \int dQ/T$$ This is the one that increases.

It can be shown that both 1. and 3. are important quantities, but the second law applies to the 3. one.

References: Unfortunately I can only link to this http://www.oxfordmartin.ox.ac.uk/event/1348 which is where I got the information from.

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No wonder you found no references, because this is wrong. In particular the Gibbs entropy is equal to the thermodynamical entropy. The Gibbs entropy never decreases, and is not a "constant of the equations of motion", except at equilibrium of course. – ederag Jan 19 at 16:30
Jayne's article, which is referenced in the wikipedia article you posted, can be found here: bayes.wustl.edu/etj/articles/gibbs.vs.boltzmann.pdf I quote from the abstract: "(5) the dynamical invariance of the Gibbs H gives a simple proof of the second law for arbitrary interparticle forces." So in fact he states (in the obscured form of H) that the Gibbs entropy is invariant. What he also shows though is that the experimental and the Gibbs entropy are equal FOR THE CANONICAL ENSEMBLE. For a general distribution he then obtains that S_E >= S_Gibbs, which is the second law. – Numrok Jan 19 at 19:44
Thanks for the link. In this paper the Gibbs entropy is claimed to be constant only in a special transformation: "Now force the system to carry out an adiabatic change of state (i.e., one involving no heat flow to or from its environment), by applying some time-dependent term in the Hamiltonian (such as moving a piston or varying a magnetic field)." This transformation is both adiabatic and reversible (and thus isentropic), hence the Gibbs entropy, like the thermodynamic one, is constant. For other transformations the Gibbs entropy may vary (like the thermodynamic one). – ederag Jan 20 at 19:43
What you are saying now is absolutely correct. The Gibbs entropy of a SUBSYSTEM may vary. But the universe as a whole always undergoes an adiabatic change. About the reversible assumption: I think it is still safe to say that the Gibbs entropy is a constant of the full quantum mechanical equations of motion (i.e. Schrödinger equation), which doesn't seem to be proven in this paper though. Thermodynamically irreversible processes wash out quantum correlations, but are still microscopically reversible. I will try to find a reference for this. – Numrok Jan 21 at 10:35
1) The entropy of an isolated system (like the universe) may increase, even if the microscopic equations of motion are reversible. Do you really disagree with that ? 2) No, in general the Gibbs entropy is not a "constant of the full quantum mechanical equations of motion". Except in some cases, for instance an isolated system undergoing a reversible transformation. 3) Do you really assume that the universe is following a reversible transformation ? – ederag Jan 21 at 17:34

What you are missing is the microscopic definition of entropy, once you know that, you will understand why people say that entropy is disorder.

Equilibrium as order

First, let's address your valid intuition that equilibrium as a form of order. Indeed, if everything is in thermal equilibrium, you just need to measure the temperature somewhere, and then you will know the temperature of everything. In our out of equilibrium, my body, my laptop, the room, outer space, all have different temperatures, and I need more information to know the state of everything, and I feel this is less "ordered" than the thermal equilibrium case.

What transpires is that less information needed corresponds to a higher degree of order. Well, let's keep that in mind for the next bit.

Entropy is microscopic disorder

In Physics, we know that the properties of macroscopic objects are determined by the motions of the particles that compose them. In particular, temperature of a gas is the disorganised jiggling of the atoms making it up.

As you increase the temperature, the atoms will move more and more erratically, and will have diverse speeds at any given time.
As you cool it, the particles will move slower and slower, until perhaps they freeze in place, forming a solid.

Which of the two - the still, regular lattice of the solid or the whizzing commotion of the particles that forms a gas - seems to you more disordered? Definitely the second. You know from thermodynamics that the gas has higher entropy than the solid. Indeed, there is a precise formula linking the macroscopic state variable $S$, entropy, and the microscopic conception of disorder I described.

Conclusion: the two ideas are reconcilable

In the projected "heat death" of the universe, everywhere there is constant temperature and density. In that sense, the universe is homogeneous and thus ordered. But microscopically - in the movements of the particles - that is the state in which there is the least order: no structure whatsoever, just a big soup of whizzing particles.

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Hmm...really interesting, I thought that in the "heat death" scenario the particles won't really move because energy can't be moved in the eventual case of thermal equilibrium. – griffinwish Jan 18 at 14:40
That's right: there is no coherent movement of energy from one region of space to another because everything is in thermal equilibrium. But at the microscopic level, there is movement. Remember that temperature is the jiggling of particles: no movement would mean absolute zero temperature. Individual particles themselves have no temperature: just position and velocity. They move around like billiard bald, colliding and exchanging energy. Thermal equilibrium means that the collisions don't have the effect of transporting energy around. – Andrea Di Biagio Jan 18 at 15:12
In fact, as the entropy increases constantly, the universe must have, started with surprisingly low entropy. The why of this is still an unresolved mystery. For the second question: Entropy is has a technical definition, and can be calculated precisely for various macroscopic configurations. Maybe you can start from here en.wikipedia.org/wiki/Entropy_(order_and_disorder) although this is definitely not the best resource. Penrose's talk linked by @rmhleo is also a very good place to get more. – Andrea Di Biagio Jan 18 at 16:07
@griffinwish Because the universe apparently started from a very ordered state. I don't think it's known why that happened. (Any state leads towards one with maximum entropy; we have less-than-maximum entropy so we must have started from a state with even less) – immibis Jan 18 at 20:18
Exactly, and I finally think I got a better understanding of this. Disorder tends to increase because there are many more disordered states than ordered ones so it is more probable for a thing to be disordered, right? Just like ice cubes in a glass and liquid water, the latter having more entropy because water molecules are flying around inside and are not standing still like in ice cubes, so there are many more ways to arrange those molecules in a disordered state thus increasing the entropy. – griffinwish Jan 18 at 20:47

Entropy is not disorder; it is a lack of information.

Consider the entropy formula $S = k_b \log \Omega$. Here, $\Omega$ is the number of microstates (sets of particle positions/momenta) corresponding to an observed macrostate (something macroscopic we can observe, like 'the gas has volume $V$ and pressure $P$). What this formula means is that the entropy is proportional to the amount of information we are missing -- the number of extra bits we would need to know, on top of knowing the macrostate, to full specify the microstate.

For example, consider heat transfer $Q$ and work $W$. Though both exchange energy, only the first increases entropy. That makes sense, because the only difference between heat transfer and work is that heat transfer is done in a disordered way. We don't know exactly how it happened, so our lack of information goes up.

Since heat transfer increases entropy, the maximum entropy is achieved at thermal equilibrium. At that point, we basically know nothing at all.

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but wouldn't thermal equilibrium mean that no heat transfer exists in the whole universe? So basically, wouldn't we know that there is nothing to know hence no entropy? – griffinwish Jan 18 at 16:02
You won't know exactly where all of the energy is. In thermal equilibrium, can you tell me where molecule #1375039 is and how fast it's moving? – knzhou Jan 18 at 16:03
yeah..probably not, thanks for the clear up – griffinwish Jan 18 at 16:09
@griffinwish However, note that this is unlikely to be a stable state - once you achieve maximum entropy, random actions will result in decreasing entropy. There's a neat little idea that that's what our universe is - a blob of low entropy that spontaneously formed in a maximum entropy "superuniverse". Energy and information are conserved, you just get local minima all the time - until the bubble of low entropy reaches maximum entropy again in a few (hundred) billion years. There's a lot of problems with that idea, but... it's pretty neat :) – Luaan Jan 18 at 17:03
wow..amazing theory. So after we reach maximum entropy we will create another blob of low entropy. Hah, that's interesting – griffinwish Jan 18 at 20:49

Entropy is a tricky concept and hard to understand. Personally I tend to avoid speaking of systems and phenomena in terms of entropy and/or temperature because they say very little of the dynamics, and I believe dynamical laws are the ones driving the universe.

When we hear that systems tend to increase entropy, we are saying there are dynamical laws driving them towards states of higher entropy. But this comes from our knowledge that for simple systems with elementary microscopic behavior (like ideal gases, or ideal liquids) when comparing two states of equilibrium, the one with higher entropy is more stable.

This might be misunderstood as an evidence that systems in general evolve by increasing entropy, which can be proven wrong. In fact the universe evolves in such a way that instead of tending to be homogeneous, is highly organised (galaxies, stars, planets, living beings).

My approach to this would be twofold: first microscopic dynamics is not elementary, which means that molecules have more degrees of freedom than we conceive when we tend to think only in terms of entropy to predict the behavior of the system. This is the same idea of Gibbs when he extended classical thermodynamics by allowing the number of molecules to change, which accounts for systems in which reactions may occur. But we can think of other types of "qualitative changes" (as I like to call them), as did Terrell Hill in his conception of Thermodynamics of Small Systems.

Secondly, I think we should not forget that the dynamics of evolution of physical system are fundamentally different from what we expect by saying that systems tend to increase their entropy, this is simply not verified, and in my opinion, misleading.

A final note in saying that Temperature, as Entropy, refers to equilibrated states and is also wrongly believed to behave the way energy does. But this is not the case: the dynamics of systems does not depend on temperature, but on the relative energies of the involved parties. Microscopically speaking, the collision dynamics depends on the relative energies or momenta, rather than on their average. Also in a non-equilibrated system, temperature (understood as mean kinetic energy) will largely fluctuate spatially before the whole system achieves the equilibrium.

PS: Sir Roger Penrose has very interesting arguments on the concept of Entropy and Universe evolution in this talk

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It is misleading to mention galaxies and living beings as examples of cases where entropy does not increase. Living being subsist only because they eat low entropy, organised, food and excrete high entropy matter. The second law states that the entropy of an isolated system always increases. A living being is emphatically not an isolated system: it is firmly embedded in the Earth. The Earth itself manages to keep a relatively stable entropy because it absorbs low entropy light from the Sun and emits high entropy infrared. Penrose himself points that out in his talk. – Andrea Di Biagio Jan 18 at 15:25
I think those examples show how the reality evolves in such a way that gives rise to highly ordered systems. This disproves the idea that growth of entropy is the direction of evolution of the systems. I agree that leaving systems is not a good example of physical one, since highly evolved organisms can actively act against physical processes. But again, this is not the point for mentioning them. Also a star and planets are more organised forms compared to matter being smeared homogeneously. – rmhleo Jan 18 at 15:37
Living beings are physical systems, and they do not act against physical processes. They are just not closed physical systems. It is true that their entropy does not increase, but that's only because they increase the entropy of the surroundings. – Andrea Di Biagio Jan 18 at 15:56
Stars and planets are more organised forms of matter, in the same sense that a stationary ball at the bottom of a well is more organised than one that is frantically bouncing around. But again, both processes involved the increase of entropy in the universe as a whole! The 2nd law is not incompatible with the creation of structure. – Andrea Di Biagio Jan 18 at 15:56
"I do not agree that all organised systems subsist by disorganising other systems." It depends on what you mean by systems. An homogeneous gas cloud that collapses under the gravitational attraction into a concentrated star and orbiting planets does so while increasing the entropy of the universe. Indeed the star is radiating heat all around, spreading countless photons in the universe. The radiation might not disrupt other "systems" ( as in other stars and planets, for example) but it is still increasing the entropy of the universe as a whole. There is no arguing with this. – Andrea Di Biagio Jan 18 at 16:13

The entropy law can be (comically) reinterpreted like "equilibrium is a state of maximum possible disorder under given physical constraints". So... things keep getting worse until it's as bad as it can get. Intuitively, large entropy means that things look more or less the same (macroscopically) for many different microscopic realizations. When the system evolves, it's statistically easy to find yourself in one of the many high-entropy states, but very rarely you can randomly stumble upon an ordered state. Imagine trying to shake a box of coins: what's the probability that you'll get all tails? The equilibrium state (you keep shaking the box - simulation of thermal motion) will be somewhere around half tails half heads, plusminus the standard deviation, typical for this system (after binomial distribution). So... disorder. In other comparison, parents all over the world know that the room only gets messier and reaches a state of chaos (this being the equilibrium state). You must put in work to make it tidy again, and it doesn't stay that way for very long.

I'm giving a common sense illustration because the physics has already been covered by other posts. People keep saying entropy is a difficult concept to grasp, but that's only if you don't explain it right.

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So correct me if I'm mistaken, the entropy of the universe is increasing because particles are always moving towards a disordered state because there are many more disordered states than ordered ones so statistically it is more probable for a system to end up disordered than ordered. Another question: why when I exercise I increase the entropy of the universe? I get that I release more energy into the surroundings but what does that have to do with increasing the disorder? – griffinwish Jan 19 at 11:21
Yes, that's correct. Whatever you do that isn't reversible, increses the entropy of the universe. To do work, you take advantage of some order (temperature difference, chemical compounds in non-equilibrium [volatile] form, potential energy...) and make it disordered in the process. By doing work, you have a side effect of waste heat (heat is the most disordered, high-entropy form of energy that can't be used for work anymore). Heat engines, for instance, operate on temperature difference, producing work, but heat up the cold reservoir, ruining the temperature difference. – orion Jan 19 at 15:48

No actually this is one perpetuating myth about entropy that even scientists themselves (and school curricula) propagate.

To answer this and dispel the myth, ask this simple question: disorder with respect to what exactly?

Why is a uniform gas disordered than a gas with two phases?

Of course a uniform gas has more (another) symmetry, in fact aquires the symmetries of the underlying environment. But so does the the two-phase gas, it has a certain symmetry (and not others) deriving from the underlying environmental process. So far so good. Where is the "disorder" exactly, and with respect to what and to whom is this a "disorder"? i think you get the point meant here.

Clearly there is a very subjective (to mention the least) concept of disorder used here which is not explained anywhere. Just stated as fact which is not.

Some take this further equating entropy with death vs life which is even more absurd. One can have a series of cages perfectly ordered, yet one will not have life in them.

Please consider this before you just accept anything thrown at you sounding scientific (while it is not)

PS

If you want the full scientific version of this answer check (especialy) the works of I. Prigogine on Entropy, Complex Dynamic Systems and Biological Systems. e.g "From Being to Becoming: Time and Complexity in the Physical Sciences"

Other schools of thermodynamics also have similar approaches and hard facts to consider. For a popular, yet somewhat thorough exposition check, for example: "The Arrow Of Time: A Voyage Through Science To Solve Time's Greatest Mystery"

To summarise:

entropy

1. is NOT disorder (mechanistic approach)
2. is NOT lack of information (bayesian/subjectivist approach),
3. is NOT contrary to evolution (inteligent design-approach)
4. is NOT simply a statistical effect (quantum-mechanical/statistical approach)
5. is NOT related solely to linear and (static) equilibrium processes, in fact entropy and (yes) the 2nd Law have been generalised (i would say simply clarified) for (dynamic) non-equilibrium / non-linear processes

abstract

In the scientific and engineering literature, the second law of thermodynamics is expressed in terms of the behavior of entropy in reversible and irreversible processes. According to the prevailing statistical mechanics interpretation the entropy is viewed as a nonphysical statistical attribute, a measure of either disorder in a system, or lack of information about the system, or erasure of information collected about the system, and a plethora of analytic expressions are proposed for the various measures. In this paper, we present two expositions of thermodynamics (both ’revolutionary’ in the sense of Thomas Kuhn with respect to conventional statistical mechanics and traditional expositions of thermodynamics) that apply to all systems (both macroscopic and microscopic, including single particle or single spin systems), and to all states (thermodynamic or stable equilibrium, nonequilibrium, and other states). .. Here entropy emerges as a microscopic nonstatistical property of matter.

Entropy is one of the most basic facts (and least understood, analysed) related directly to causality, the arrow of time, quantum-mechanics and evolution.

In fact most (if not all) time-reversible equations are wrong (ot at least crude approximations) rather than entropy and the time arrow itself.

To quote the cosmologist Arthur Eddington:

The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations - then so much the worse for Maxwell's equations. If it is found to be contradicted by observation - well, these experimentalists do bungle things sometimes. But if your theory is found to be against the Second Law of Thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.

The references given above dispel all these misconceptions.

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Terms are conventions. With a point of view from the humans we are the order. Collecting something and order it in shells is order. But I agree with you that to order something needs energy and this led to misorder and this could be a possible convention too. But it is not.

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