I Are Boltzman's statistics compatible with a deterministic universe?

[Physicist248]

Are Boltzman's statistics compatible with deterministic universe? Suppose that the gas molecules in a given container are perfectly elastic objects obeying Newton's laws. Suppose further that we select the initial conditions (impulse and position of each molecule) at random. Is it true that, if the initial conditions are randomly selected, then in most cases the entropy will be increasing (but in some cases it will not)? Or do Boltzman's statistics imply inherent indeterminism/uncertainty at some level, i.e. the quantum level?

 

[PeroK]

Is it true that, if the initial conditions are randomly selected, then in most cases the entropy will be increasing (but in some cases it will not)?

Yes. In fact, the process is time symmetric, which means that in principle you could reverse any change. But, the probabilities in favour of increasing entropy are overwhelming

 

[Bill Dreiss]

Boltzmann statistics is a model of a model, the underlying model being kinetic theory, which is a deterministic model based on Newton’s laws of motion. In Boltzmann’s day, solving the equations of motion for even a few molecules was intractable, so the statistical model was devised as a substitute. However, since the advent of computers, kinetic theory has been instantiated in ideal gas simulations, a number of which can be found online. These simulations render Boltzmann’s model obsolete, since they allow us to experimentally explore the properties of an ideal gas directly and compile statistics instead of relying on abstract probabilities. They also mimic the second law behavior that we experience every day, demonstrating that the laws of motion are irreversible, contrary to common belief.

Boltzmann believed that “In nature, the tendency of transformations is always to go from less probable to more probable states”. However, according to his model this is only true for macrostates that are relatively far from the most probable macrostate. For macrostates that are nearer the most probable macrostate, it is more probable that the next macrostate will be less probable than the current one. For instance, if the system is in the most probable macrostate, the probability that the next macrostate will be less probable approaches one for large systems. This is because the probability of being in the most probable macrostate simultaneously approaches zero.

Boltzmann statistics is based on probability theory, which quantifies our subjective degree of belief or ignorance. From this perspective, probability is not a property of nature, as Boltzmann claimed, and probability theory has no bearing on whether nature is deterministic or not. On the question of whether probability is intrinsic to quantum mechanics, the jury is still out.

For details see [Link to blog with Personal Speculation redacted by the Moderators]

 

[Physicist248]

Yes. In fact, the process is time symmetric, which means that in principle you could reverse any change. But, the probabilities in favour of increasing entropy are overwhelming

OK, has somebody actually proved it? Is this a theorem of physics?

 

[Physicist248]

Boltzmann statistics is a model of a model, the underlying model being kinetic theory, which is a deterministic model based on Newton’s laws of motion. In Boltzmann’s day, solving the equations of motion for even a few molecules was intractable, so the statistical model was devised as a substitute. However, since the advent of computers, kinetic theory has been instantiated in ideal gas simulations, a number of which can be found online. These simulations render Boltzmann’s model obsolete, since they allow us to experimentally explore the properties of an ideal gas directly and compile statistics instead of relying on abstract probabilities. They also mimic the second law behavior that we experience every day, demonstrating that the laws of motion are irreversible, contrary to common belief.

Boltzmann believed that “In nature, the tendency of transformations is always to go from less probable to more probable states”. However, according to his model this is only true for macrostates that are relatively far from the most probable macrostate. For macrostates that are nearer the most probable macrostate, it is more probable that the next macrostate will be less probable than the current one. For instance, if the system is in the most probable macrostate, the probability that the next macrostate will be less probable approaches one for large systems. This is because the probability of being in the most probable macrostate simultaneously approaches zero.

Boltzmann statistics is based on probability theory, which quantifies our subjective degree of belief or ignorance. From this perspective, probability is not a property of nature, as Boltzmann claimed, and probability theory has no bearing on whether nature is deterministic or not. On the question of whether probability is intrinsic to quantum mechanics, the jury is still out.

For details see [Link to blog with Personal Speculation redacted by the Moderators]

"They also mimic the second law behavior that we experience every day, demonstrating that the laws of motion are irreversible"
Why would they be irreversible if the system is deterministic?

 

[PeroK]

OK, has somebody actually proved it? Is this a theorem of physics?

It's inherent in classical particle collisions and scattering. Every collision is equally valid in reverse. There is simply nothing inherent in the classical laws of motion that indicates time irreversibility.

It's only when you consider the probabilitistic aspect that you understand why processes are statistically irreversible (not absolutely irreversible).

 

[PeroK]

On the question of whether probability is intrinsic to quantum mechanics, the jury is still out.

It's true that there are people who will never accept QM. But, mainstream QM is inherently probabilitistic. You may as well say the jury is still out on whether the Earth is flat!

 

[Bill Dreiss]

"They also mimic the second law behavior that we experience every day, demonstrating that the laws of motion are irreversible"
Why would they be irreversible if the system is deterministic?

In section 52-2 of The Feynman Lectures on Physics, Vol. 1, Feynman says, “Next we mention a very interesting symmetry which is obviously false, i.e., reversibility in time. The physical laws apparently cannot be reversible in time, because, as we know, all obvious phenomena are irreversible on a large scale…” [my italics]. He continues with “So far as we can tell, this irreversibility is due to the very large number of particles involved, and if we could see the individual molecules, we would not be able to discern whether the machinery was working forward or backwards.” However, by attributing irreversibility to “the very large number of particles involved”, he begs the question: At what number of particles does a system switch from reversible to irreversible? But close observation of deterministic ideal gas simulations reveals that the second-law behavior holds even for a handful of molecules, in fact, for any number of molecules ≥ 2.

In section 52-4, he points out the mirror symmetry of the linear momenta in the direction of the three Cartesian coordinates and the angular momentum in two of the directions of spherical coordinates. He calls the third component of spherical motion the polar vector and illustrates mirror symmetry for this vector in Fig. 52-2, showing that a vector pointed in the northeast direction converts to a vector pointed in the northwest direction when rotated 180 degrees around the y axis. However, this is a clear misapplication of mirror symmetry, since the other mirror reversals are rotated around an axis perpendicular to the vector of motion. Therefore, mirror symmetry applied to the polar vector should rotate around an axis perpendicular the direction of the polar vector, resulting in a vector pointed toward the southwest, not northwest.

If the direction of motion is reversed from the positive direction (away from the origin) the direction will be negative (toward the origin) only until the particle reaches the origin, at which time it will become positive again. Since the final direction of motion is the same as the initial direction of motion, the process is irreversible. This is the source of the asymmetry in the laws of motion and the physical basis of the second law. Furthermore, it is apparent that the second law is not fundamental in and of itself, but an epiphenomenon of the more basic law of inertia.

For a more detailed discussion, see [Link to blog with Personal Speculation redacted by the Moderators]

 

[Bill Dreiss]

OK, has somebody actually proved it? Is this a theorem of physics?

No, it's a theorem of mathematics, an analogy with no causal connection to physics.

 

[Bill Dreiss]

It's true that there are people who will never accept QM. But, mainstream QM is inherently probabilitistic. You may as well say the jury is still out on whether the Earth is flat!

For evidence that this is a live issue, see Whitaker, Andrew. Einstein, Bohr and the Quantum Dilemma, Cambridge University Press.

 

[PeroK]

For evidence that this is a live issue, see Whitaker, Andrew. Einstein, Bohr and the Quantum Dilemma, Cambridge University Press.

Well, Andrew Whitaker may be alive, but Einstein and Bohr are definitely long dead!

 

[berkeman]

Thread closed temporarily for Moderation...

 

[berkeman]

After some cleanup, thread is reopened.

 

[Bill Dreiss]

Well, Andrew Whitaker may be alive, but Einstein and Bohr are definitely long dead!

I'm not sure how this is relevant, since their ideas are still discussed by modern physicists. While the general belief currently seems to be that Bohr was arguing in favor of an ontic interpretation of quantum probability and Einstein was arguing in favor of an epistemic interpretation, this is not true. Both recognized the epistemic nature quantum probability. It's just that Bohr thought that the uncertainty principle rendered attempts to look behind the curtain futile, while Eintein felt that if there was something there, aka hidden variables, you wouldn't find it if you didn't look.

This debate required both men to distinguish between the ontic and epistemic, a distinction that is generally lost on modern physicists. For examples of the modern conflation of the two concepts, see Lost in Math and the recent Existential Physics, both by Sabine Hossenfelder. For an article that gets to the heart of the issue of quantum probability, see page 7 of the attached file.

 

[PeroK]

This debate required both men to distinguish between the ontic and epistemic, a distinction that is generally lost on modern physicists.

Well, the alternative view is that the modern physicist has 60-70 years of experimental evidence and theoretical development over Einstein and Bohr

I don't buy into the concept that the understanding of QM has been lost in recent decades. Quite the reverse, in fact.

 

[Physicist248]

It's inherent in classical particle collisions and scattering. Every collision is equally valid in reverse. There is simply nothing inherent in the classical laws of motion that indicates time irreversibility.

It's only when you consider the probabilitistic aspect that you understand why processes are statistically irreversible (not absolutely irreversible).

I have realized that if the initial conditions were chosed at random we would be in a state of maximum entropy to begin with. I suppose that it is still possible that the universe was created with uneven distribution of thermal energy, and within this framework the microstates were chosen randomly. It seems far fetched. Maybe cosmology explains it somehow.

To me this argument that the processes are reversible in principle but not statistically does not seem convincing. God could have just as easily put everything in reverse motion. Then Boltzman's statistic and gas simulation would indicate increasing entropy while in real word we would observe decreasing entropy.

By this thought experiment I am trying to say that it is unlikely the processes are reversible in principle. Do not know if inherent randomness applies to gas molecules but the uncertainty principle does.

 

[Bill Dreiss]

Well, the alternative view is that the modern physicist has 60-70 years of experimental evidence and theoretical development over Einstein and Bohr

I don't buy into the concept that the understanding of QM has been lost in recent decades. Quite the reverse, in fact.

I didn't say that "the understanding of QM has been lost", but rather that the general ability of physicists to "distinguish between the ontic and epistemic" has. However, while the experimental evidence of QM continues to accumulate, it doesn't appear to me that theory has come that far. If you're aware of a source that goes significantly beyond Bohm, de Broglie, Feynman and Bell, please advise me. Wikipedia doesn't seem to know.

 

[PeroK]

I have realized that if the initial conditions were chosed at random we would be in a state of maximum entropy to begin with. I suppose that it is still possible that the universe was created with uneven distribution of thermal energy, and within this framework the microstates were chosen randomly. It seems far fetched. Maybe cosmology explains it somehow.

To me this argument that the processes are reversible in principle but not statistically does not seem convincing. God could have just as easily put everything in reverse motion. Then Boltzman's statistic and gas simulation would indicate increasing entropy while in real word we would observe decreasing entropy.

By this thought experiment I am trying to say that it is unlikely the processes are reversible in principle. Do not know if inherent randomness applies to gas molecules but the uncertainty principle does.

This is a science forum, so you are expected to try to understand the science and mathematics, rather than dismiss them in favour of personal preference or religious arguments.

The principle of time reversibility of the basic laws, but statistical irreversibility is simple to simulate on a computer. Your scepticism therefore, is unfounded.

 

[Physicist248]

This is a science forum, so you are expected to try to understand the science and mathematics, rather than dismiss them in favour of personal preference or religious arguments.

The principle of time reversibility of the basic laws, but statistical irreversibility is simple to simulate on a computer. Your scepticism therefore, is unfounded.

Just play your simulation backwards.

 

[PeroK]

Just play your simulation backwards.

That's the point. Each step is reversible, but the process overall shows irreversible behaviour.

The essence of science when you encounter something hard to understand is to think. Not to stop thinking.

Anyone can stop thinking. There's nothing clever in that.

 

[Physicist248]

That's the point. Each step is reversible, but the process overall shows irreversible behaviour.

The essence of science when you encounter something hard to understand is to think. Not to stop thinking.

Anyone can stop thinking. There's nothing clever in that.

Just flip the sign of all the velocities. Why are "backward" velocities less probable than "forward" velocities?

 

[Physicist248]

Boltzmann statistics is a model of a model, the underlying model being kinetic theory, which is a deterministic model based on Newton’s laws of motion. In Boltzmann’s day, solving the equations of motion for even a few molecules was intractable, so the statistical model was devised as a substitute. However, since the advent of computers, kinetic theory has been instantiated in ideal gas simulations, a number of which can be found online. These simulations render Boltzmann’s model obsolete, since they allow us to experimentally explore the properties of an ideal gas directly and compile statistics instead of relying on abstract probabilities. They also mimic the second law behavior that we experience every day, demonstrating that the laws of motion are irreversible, contrary to common belief.

Boltzmann believed that “In nature, the tendency of transformations is always to go from less probable to more probable states”. However, according to his model this is only true for macrostates that are relatively far from the most probable macrostate. For macrostates that are nearer the most probable macrostate, it is more probable that the next macrostate will be less probable than the current one. For instance, if the system is in the most probable macrostate, the probability that the next macrostate will be less probable approaches one for large systems. This is because the probability of being in the most probable macrostate simultaneously approaches zero.

Boltzmann statistics is based on probability theory, which quantifies our subjective degree of belief or ignorance. From this perspective, probability is not a property of nature, as Boltzmann claimed, and probability theory has no bearing on whether nature is deterministic or not. On the question of whether probability is intrinsic to quantum mechanics, the jury is still out.

For details see [Link to blog with Personal Speculation redacted by the Moderators]

How do these simulations work? The molecules are points or spheres? They all have the same mass, and you randomly select position and velocity for each? And they will converge to higher entropy? Just flip the signs of the velocities, and entropy will decrease. "Backward" velocities are just as probable as "forward" velocities. Something does not add up. Or am I missing something?

 

[PeroK]

Or am I missing something?

A classic example is mixing a pack of cards. If you start with a unmixed deck, then almost any sequence of mixing will jumble the cards. It is possible, of course, to mix cards and end up back where you started. But, this is statistically extremely unlikely - unless you are deliberating choosing your movements to this end.

Also, once you have a mixed deck and keep mixing, the deck stays mixed. It doesn't return to an unmixed deck. This, however, is purely statistical. With a small number of cards, you would get back to an unmixed deck with some significant probability.

Let's reduce mixing to a sequence of swapping any two cards. If you show a video of someone mixing a deck, you cannot tell by any particular swap whether the process is running forwards or backwards. Every swap is reversible. And, if you have a mixed deck (equilibrium), then again you cannot tell whether the video showing the sequence of swaps is running forwards or backwards.

But, if you have a video of a mixed deck being shuffled and returing to an unmixed deck, then you know (statistically) that the film is running backwards.

It's even more apparent if, instead of 52 cards, you have $52 \times 10^{23}$ cards. For example, the statistical likelihood of warm water separating into hot and cold water is effectively impossible. Even though every individual exchange of kinetic energy between molecules is reversible.

And, of course, when you mix hot and cold water it's statistically inevitable that you end up with warm water.

 

[vanhees71]

I didn't say that "the understanding of QM has been lost", but rather that the general ability of physicists to "distinguish between the ontic and epistemic" has. However, while the experimental evidence of QM continues to accumulate, it doesn't appear to me that theory has come that far. If you're aware of a source that goes significantly beyond Bohm, de Broglie, Feynman and Bell, please advise me. Wikipedia doesn't seem to know.

It's irrelevant for physics to distinguish between some subtle philosophical notions.

All that counts is the ability to apply a mathematical theory or model to the quantitative observations of nature, usually conducted in experiments where one investigates some aspect of nature in a controlled environment to enable the empirical test of that theory or model. I'd say these "issues" are simply resolved today with all the investigations about it with various systems (mostly photons but also neutrons, atoms, molecules, and in condensed matter (quantum dots), etc. etc.).

 

[Bill Dreiss]

How do these simulations work? The molecules are points or spheres? They all have the same mass, and you randomly select position and velocity for each? And they will converge to higher entropy? Just flip the signs of the velocities, and entropy will decrease. "Backward" velocities are just as probable as "forward" velocities. Something does not add up. Or am I missing something?

For an example, see https://phet.colorado.edu/sims/html/gas-properties/latest/gas-properties_en.html. The molecules are identical circles in 2-dimensional space. The initial conditions usually have the molecules bunched in a small region of the box. They then spontaneously disperse, increasing entropy. The simulation is strictly deterministic and probability is not involved. Since the algoritm instantiates only Newton's three deterministic laws of motion, the second-law dispersion can only be a consequence of these laws.

Which particular law is behind the dispersion? The third law, which relates to collisions, is clearly reversible. The second law, which relates to force, is not relevant since the ideal gas model assumes that no external forces or forces between the molecules exist. This leaves the first law, the law of inertia. The action of inertia is clearly visible as the source of the dispersion behind the second law.

If the simulation is reversed, the molecules will retrace their paths back to the initial conditions, temporarily decreasing entropy. However, if the simulation continues to run, the molecules will once again disperse, increasing entropy. The direction of increasing spontaneous dispersion is the direction of time.

 

[PeroK]

The action of inertia is clearly visible as the source of the dispersion behind the second law.

Which is clearly nonsense. It's the initial conditions and laws of statistical the produce dispersion.

 

[vanhees71]

Behind the second law is detailed balance or, on a fundamental level, the unitarity of the S-matrix together with some "coarse graining procedure". In the case of the standard derivation of the Boltzmann equation the latter comes in via the molecular-chaos assumption ("Stoßzahlansatz").

 

[Bill Dreiss]

Which is clearly nonsense. It's the initial conditions and laws of statistical the produce dispersion.

From Boltzmann, Ludwig. Lectures on Gas Theory (Dover Books on Physics) (p. 59):

“It is only when one reverses the velocities at time t1 that he obtains a motion for which H must increase during the time interval t1 – t0, and even then H would probably decrease again after that…” [my italics]

I’m using the same logic as Boltzmann. Do you see a flaw in our reasoning?

 

[PeroK]

I’m using the same logic as Boltzmann. Do you see a flaw in our reasoning?

Bolzmann never concluded that Newton's first law (or the law of inertia) was the cause of entropy. Your reasoning is your own.

 

[Bill Dreiss]

Bolzmann never concluded that Newton's first law (or the law of inertia) was the cause of entropy. Your reasoning is your own.

I've merely followed through with Boltzmann's and Feynman's (Post #8) observations. What is the flaw in our line of reasoning?

 

[PeroK]

I've merely followed through with Boltzmann's and Feynman's (Post #8) observations. What is the flaw in our line of reasoning?

There is no logic whatsoever in what you say. The first law is just as time-reversible as the other two. It's called statistical mechanics, not inertial mechanics.

 

[Bill Dreiss]

There is no logic whatsoever in what you say. The first law is just as time-reversible as the other two. It's called statistical mechanics, not inertial mechanics.

My observations involve kinetic theory, which preceded statistical mechanics. Feynman's line of reasoning was identical to mine and he would have reached the same conclusion as I did if he had not made a careless mistake. The crutial observation which leads to this conclusion is that if the velocities are reversed, the dispersion will resume after the initial positions are recovered. If you disagree, please explain why. Just telling me that I'm wrong gets us nowhere.

 

[PeroK]

My observations involve kinetic theory, which preceded statistical mechanics. Feynman's line of reasoning was identical to mine and he would have reached the same conclusion as I did if he had not made a careless mistake.

It takes a special type of mind, I guess, to believe something like that.

 

[PeroK]

If you disagree, please explain why. Just telling me that I'm wrong gets us nowhere.

There is nowhere to go with personal theories on PF. I've already indulged you more than I should.

 

[Physicist248]

For an example, see https://phet.colorado.edu/sims/html/gas-properties/latest/gas-properties_en.html. The molecules are identical circles in 2-dimensional space. The initial conditions usually have the molecules bunched in a small region of the box. They then spontaneously disperse, increasing entropy. The simulation is strictly deterministic and probability is not involved. Since the algoritm instantiates only Newton's three deterministic laws of motion, the second-law dispersion can only be a consequence of these laws.

Which particular law is behind the dispersion? The third law, which relates to collisions, is clearly reversible. The second law, which relates to force, is not relevant since the ideal gas model assumes that no external forces or forces between the molecules exist. This leaves the first law, the law of inertia. The action of inertia is clearly visible as the source of the dispersion behind the second law.

If the simulation is reversed, the molecules will retrace their paths back to the initial conditions, temporarily decreasing entropy. However, if the simulation continues to run, the molecules will once again disperse, increasing entropy. The direction of increasing spontaneous dispersion is the direction of time.

"If the simulation is reversed, the molecules will retrace their paths back to the initial conditions, temporarily decreasing entropy. However, if the simulation continues to run, the molecules will once again disperse, increasing entropy." Hmm ...

 

[Physicist248]

From Boltzmann, Ludwig. Lectures on Gas Theory (Dover Books on Physics) (p. 59):

“It is only when one reverses the velocities at time t1 that he obtains a motion for which H must increase during the time interval t1 – t0, and even then H would probably decrease again after that…” [my italics]

I’m using the same logic as Boltzmann. Do you see a flaw in our reasoning?

I do not like the word "probably."

 

[Bill Dreiss]

It takes a special type of mind, I guess, to believe something like that.

The essence of science when you encounter something hard to understand is to think.

 

[Bill Dreiss]

I do not like the word "probably."

I don't either, since the process he's describing is deteministic.

 

[PeroK]

I don't either, since the process he's describing is deteministic.

There is a fundamental difference between deterministic and predictable. The critical role is played by the initials conditions, which cannot be known to infinite precision. Take the three-body gravitational problem - which, despite being deterministic, exhibits chaotic behaviour.

https://en.wikipedia.org/wiki/Three-body_problem

 

[Bill Dreiss]

I do not like the word "probably."

As near as I can tell, Boltzmann's insight was first proposed by William Thompson in the paper “The Kinetic Theory of the Dissipation of Energy” (1874), which can be found in The Kinetic Theory of Gases by Stephen G. Brush. (see the highlighted text in the attachment) It's possible that this is where Boltzmann got the idea, since he was in frequent communication with the English physicists.

 

[Bill Dreiss]

I do not like the word "probably."

Here's someone else who gets it: https://www.physicsforums.com/threads/loschmidts-paradox-and-free-gas-expansion.1010383/

 

[Physicist248]

For an example, see https://phet.colorado.edu/sims/html/gas-properties/latest/gas-properties_en.html. The molecules are identical circles in 2-dimensional space. The initial conditions usually have the molecules bunched in a small region of the box. They then spontaneously disperse, increasing entropy. The simulation is strictly deterministic and probability is not involved. Since the algoritm instantiates only Newton's three deterministic laws of motion, the second-law dispersion can only be a consequence of these laws.

Which particular law is behind the dispersion? The third law, which relates to collisions, is clearly reversible. The second law, which relates to force, is not relevant since the ideal gas model assumes that no external forces or forces between the molecules exist. This leaves the first law, the law of inertia. The action of inertia is clearly visible as the source of the dispersion behind the second law.

If the simulation is reversed, the molecules will retrace their paths back to the initial conditions, temporarily decreasing entropy. However, if the simulation continues to run, the molecules will once again disperse, increasing entropy. The direction of increasing spontaneous dispersion is the direction of time.

OK, so let me put it this way. Suppose that we put gas molecules in a container. We start with all the molecules on the left having high kinetic energy, and all the molecules on the right having low kinetic energy. We run the simulation, and the kinetic energy will even out. Then we rerun the simulation with all the velocity signs reversed. The kinetic energy will still even out. Is that what you are saying?

 

[Bill Dreiss]

OK, so let me put it this way. Suppose that we put gas molecules in a container. We start with all the molecules on the left having high kinetic energy, and all the molecules on the right having low kinetic energy. We run the simulation, and the kinetic energy will even out. Then we rerun the simulation with all the velocity signs reversed. The kinetic energy will still even out. Is that what you are saying?

The simulation I linked to displays the molecular distribution in space. The energy distribution is a separate problem. Everthing I've said so far has been related to the spatial distribution, which is generally what we observe in real life. I'm not saying that you rerun the simulation from the beginning, but that after a time t1 you instantaneously reverse the directions of the molecules (don't ask me how, but this is the scenario described in the literature). The molecules will the retrace their paths until they reach the positions that they started in at time t0. This will coincide with time 2(t1 - t0). From that point the molecules will once again disperse.

 

[Physicist248]

The simulation I linked to displays the molecular distribution in space. The energy distribution is a separate problem. Everthing I've said so far has been related to the spatial distribution, which is generally what we observe in real life. I'm not saying that you rerun the simulation from the beginning, but that after a time t1 you instantaneously reverse the directions of the molecules (don't ask me how, but this is the scenario described in the literature). The molecules will the retrace their paths until they reach the positions that they started in at time t0. This will coincide with time 2(t1 - t0). From that point the molecules will once again disperse.

Energy distribution is THE problem.

 

[PeterDonis]

Is it true that, if the initial conditions are randomly selected, then in most cases the entropy will be increasing (but in some cases it will not)?

No. If you select the initial conditions entirely at random, it is overwhelmingly likely that you will select initial conditions that already have very high entropy, corresponding to thermodynamic equilibrium, and that at a macroscopic level, you will observe the system to be in such an equilibrium and not to change observably with time.

The reason we observe the second law of thermodynamics in our actual universe is that the initial conditions of our universe were not chosen randomly from all possible "states of universes". Our universe's initial conditions were such that the initial entropy was very low. That is what leaves "room" for entropy in our universe to increase and to continue increasing for a long time.

 

[PeterDonis]

Here's someone else who gets it: https://www.physicsforums.com/threads/loschmidts-paradox-and-free-gas-expansion.1010383/

Not the way you mean. See my responses in that thread and my post #45 just now in this one.

 

[Bill Dreiss]

Energy distribution is THE problem.

You're right. Boltzmann focussed on the velocity/energy distribution and made no mention of the spatial distribution. However, most of the textbooks I've reviewed have presented the example of the number of molecules in one half of a box of gas as a teaching aid. Since this example relies on the binomial distribution, it is easy to understand and illustrates many of the mathematical features of the more complicated energy distribution. Furthermore, in section 6 of his Lectures on Gas Theory, Boltzmann uses the isomorphic lottery example as the first step in his derivation of the velocity distribution.

 

[Bill Dreiss]

Correction. Boltzmann uses the isomorphic lottery example as the first step in his derivation of the entropy of the most probable macrostate. He references Maxwell on the velocity distribution.

 

[Demystifier]

Is it true that, if the initial conditions are randomly selected, then in most cases the entropy will be increasing (but in some cases it will not)?

Loosely speaking, yes. If you want to better understand it, I highly recommend https://www.amazon.com/dp/9812832254/?tag=pfamazon01-20

 

[vanhees71]

I haven't followed this thread in detail, and I hope I don't make an argument which already has been made.

The important point here is that gravitation is a long-ranged force. Arguing within Newtonian gravity, the interaction potential is going with $1/r$ only, and that makes it special for the dynamics. Dynamically the canonical Maxwell-Boltzmann equilibrium phase-space distribution $f \propto \exp(-\beta H)$ follows from Boltzmann's equation. Arguing within classical mechanics the Boltzmann equation is derived from the $N$-body Liouville equation for the multi-particle phase-space distribution by reducing it to the one-particle phase-space distribution function and treat the collisions such that they are described by cross sections under the assumption of short-ranged forces, i.e., in the dilute-gas limit the particles are mostly moving freely and only when they come close together they are scattering on each other. The mean free path or time is then much larger than the trajectory/time of the collision. Another important ingredient is the molecular-chaos assumption, i.e., you substitute the two-body phase-space distribution function occurring in the collision term by the product of the corresponding one-body phase-space-distribution functions, which neglects correlations between the particles. At this moment you also introduce a direction of time (defined as the "causal direction of time"), and this leads to the H-theorem, i.e., that entropy never decreases and the equilibrium distribution functions are completely determined by the values of the additive conserved quantities (energy, momentum, angular momentum,...) and maximum entropy under the corresponding constraints, leading to the Maxwell-Boltzmann distribution as the equilibrium distribution.

Now, in the case of long-ranged interactions you have to be more careful. E.g., if you naively put a Coulomb interaction for charged particles into the collision term to describe a plasma, the collision integrals diverge. In this electromagnetic case this can be repaired by lumping the long-ranged part of the force to the left-hand side of the Boltzmann equation by considering the particle under investigation and then split off the part of the interactions that can be described by the motion of this particle in a mean electromagnetic field due to the charge-current distribution of all other particles. The rest of the collision term than is of short-ranged nature due to screening, i.e., in a plasma around a positive ion there'll be a cloud of electrons due to the attractive Coulomb force, effectively "screening" the positive charge interacting with the other charged particles. This leads to an effective Yukawa potential for the electrostatic interaction which is short-ranged, and the collision term gets well-defined.

Now the (Newtonian) gravity is similar, i.e., you also have a long-ranged $1/r$ potential, but here the problem is that it cannot be screened since it's always attractive and not also repulsive as for like-sign electric charges. That means the correlations due to these long-ranged interactions cannot be neglected, and that's why density fluctuations in a gravitationally interacting gas tend to be not relaxing but getting larger, i.e., due to the long-ranged gravitational interaction a region of higher density attracts more matter making it even denser. That's why gravitationally bound systems like stars are formed, and that's why the distribution is not homogeneous as expected from a naive Boltzmann picture of a (dilute) gas or (Debye sreened) plasma.

This of course also holds in the relativistic case, as needed to describe neutron stars or structure formation of the universe, where the initial fluctuations shortly after the big bang are the seeds for the formation of galaxies, galaxy clusters and all that.

It's also clear that for a star to remain stable there must be some pressure acting against gravity to get a stable compact object. That's provided by the thermonuclear reactions like in our Sun, and the Pauli blocking of fermions (as in a neutron star or white dwarf). If this pressure counterbalancing the gravitational interaction, the object collapses like in a nova of a star or a supernova of a neutron star etc.