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One of the laws says that heat won't flow from cool to warm and at the same time this same theory claims that there is a finite (albeit tiny) chance that it will, because there is always such a microstate.

We can also have a situation where all air molecules in the room can be found in the left side of the room and none in the right side, because it is one of the microstates therefore it can happen and the entropy will drop. So how can we say that the entropy always increases when it can decrease too sometimes?

Qmechanic
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Some Student
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  • thermodynamics is empirical, that is, based on observation. Yes, there is a chance for all air molecules in a room to accumulate to one corner of the room but to observe such a behavior one needs to wait longer than the age of the universe. Can I get 10000 tails in a fair coin toss, yes, how many times I need to try? Any statistical mechanics book would clear your doubts, one of my favorites is Principles of Equilibrium Statistical Mechanics, Chowdhury and Stauffer. – physicopath Jan 08 '21 at 08:15
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    You should definitly consider studying Statistical Physics – Anthony Guillen Jan 08 '21 at 08:16
  • About breaking of second law and related one about radiaton – tryst with freedom Jan 08 '21 at 09:38
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    I don't think thermodynamics allows, for example, that heat flows from cool to warm, or that all air molecules can be in one side of a room. Thermodynamics and statistical mechanics are two different theories, with different postulates and entities to be described. Statistical mechanics is more fundamental than thermodynamics. Therefore, one can reproduce the results of thermodynamics from statistical mechanics by loosing information (taking averages, etc...). – anonymous Jan 08 '21 at 11:15
  • see https://www.amazon.com/Rational-Thermodynamics-C-Truesdell/dp/1461297370/ref=sr_1_3?dchild=1&keywords=truesdell+thermodynamics&qid=1610108300&sr=8-3 – hyportnex Jan 08 '21 at 12:19
  • @anonymus from Truesdell: "[...] By placing a sufficiently strong local source of energy at a cold spot, we certainly ought to be able to force a neighboring hot spot to grow hotter rather than colder, especially if we help the process by putting a sink of energy at the hot spot. That is, heat can flow from cold to hot, just as water can flow uphill." (page 117) As to whether air can or cannot be on one side of the room note that classical equilibrium entropy is a function of the constraints imposed on the body and not on the "state" of the body that is being constrained. – hyportnex Jan 08 '21 at 13:55

4 Answers4

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Physical laws behave differently for different scales. Even if we know the microscopic law of nature is Quantum Mechanics, we do not use them to build a bridge. The collective quantum phenomenon somehow results in a classical phenomenon when we have many atoms together.

It is the same with thermodynamics. A few molecules can have wild fluctuations and it is not very useful to define a temperature and describe how they will fill the box. Keep increasing the number of molecules and the situation will become entirely different. When you have $10^{23}$ molecules it is very useful to state a law that is supported by overwhelming probability. The probability is so overwhelmingly in favor of the law that no one has ever seen it fail for $10^{23}$ molecules. For a theory this is as good as it can get.

Liuke LYU
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I think it is a matter of probabilities. The macrostate that is seen when the system is in equilibrium has enormously large number of microstates than the macrostate where all gases are on one side of the system.

'Enormously large' still does not feel large enough, its actually mind bogglingly large. Thus, the probability of the system getting divided into two compartments spontaneously would be impossible statistically.

Niyaz Ahmed
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There are two different views on thermodynamics, two approaches.

We can build thermodynamics as a phenomenological theory based on some postulates. According to this theory, heat will not flow from a cold body to a hot one. Never ever. And in this theory there is no concept of a microstate at all. But this theory is phenomenological and approximate.

But instead of starting with the postulates of thermodynamics, we can build thermodynamics based on statistical mechanics. This theory is more precise and fundamental than phenomenological thermodynamics. And this theory predicts that sometimes heat can flow from a cold body to a hot one. That is, the second law of thermodynamics holds only statistically, on average. (However, there is a formulation of the second law of thermodynamics, which, as far as we know, is absolutely accurate: it is impossible to build a perpetual motion machine of the second kind.)

When it comes to statistical thermodynamics, I don’t know of a perfect textbook. But for a basic level I would recommend the following two books:

  • Kittel Thermal physics
  • Huang Statistical mechanics

Theirs content partially overlaps, partially complements each other. Better to start with Kittel.

warlock
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  • I like this answer ... but at a certain point I don't like the wording. "sometimes heat can flow ....". The concepts of heat and temperature are only defined for macro states / for an ensemble of microstates with different probabillities."The second law of thermodynamics holds only statistically, on average". - No, the 2nd law simply doesn't make a statement about the transition from one microstate to another. It only makes a statement about an ensemble. In that sense I feel it's a bad wording in your answer. – Quantumwhisp Jan 08 '21 at 09:37
  • "2nd law simply doesn't make a statement about the transition from one microstate to another". Of cause, it does. "Ensemble" is just a mathematical trick. In fact, every statement about an ensemble is a statement about microstates corresponding to this ensemble. – warlock Jan 08 '21 at 09:58
  • The second law of thermodynamics is not about ensembles, but about the behavior of real physical systems - for example, about a gas container. And if you take one container with gas in a certain microstate and a certain temperature and another container with gas in a certain microstate and a different temperature and make thermal contact between them, then under certain additional conditions, which in reality are always met, the direction of the energy flow will be from hot to cold container. This is what the second law is about. – warlock Jan 08 '21 at 10:15
  • Sometimes the question arises whether it is possible to determine the temperature of a gas in a certain microstate. Yes, this can be done using the Zwanzig operator technique. The Zwanzig operator maps each microstate into a macrostate, an ensemble. And if this macrostate is an equilibrium state, then it will have a certain temperature. – warlock Jan 08 '21 at 10:24
  • This is all about Gibbs approach vs Boltzmann approach. And Boltzmann approach is more physical and allows a deeper understanding. – warlock Jan 08 '21 at 10:32
  • There are some details in The physical basis of the direction of time by Zeh. – warlock Jan 08 '21 at 10:36
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You need to distinguish between the net transfer of energy in the form of heat between two bodies and the energy transfer that can occur between the individual molecules of the two bodies.

Take the simple example of heat transfer by conduction. Body A is placed in contact with body B where $T_{A}>T_{B}$ prior to contact. We know that net heat transfer only occurs spontaneously (naturally) from body A to body B until, as some point, they reach thermal equilibrium.

Now consider what is going on between the individual molecules of body A and body B. The fact that the temperature of A is higher means that the average translational kinetic energy of the molecules in A is greater than B. But the kinetic energy of the individual molecules in A and B vary about the average value of all the molecules of the bodies. This mean that the kinetic energy of some of the molecules in B may be greater than some of those in A. If those molecules happen to collide, energy is transferred from body B to A, i.e., from the lower temperature body to the higher temperature body.

But those collisions are outnumbered by the collisions between the higher kinetic energy molecules of A with the lower kinetic energy of B so that, on average, net energy is transferred from the higher temperature body A to the lower temperature body B. Or to put it another way, the probability of higher energy molecules of A colliding with lower energy molecules of B is greater then the reverse.

Hope this helps.

Bob D
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