# Loschmidt's paradox and free gas expansion

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• bobdavis
In summary, Feynman says that the irreversibility of large-scale phenomena is due to the large number of particles involved, and that, in principle, a system could be made reversible by restricting the number of particles.
bobdavis
Following up on a previous discussion: https://www.physicsforums.com/threa...entum-in-a-closed-system.1009693/post-6570341

An uncontained system of particles interacting only by elastic collision is the same as a gas undergoing free expansion. If, for simplicity, the particles are assumed to be identical with equal mass, then the evolution of the system can be treated as if the particles pass through each other without colliding and can be drawn as just the straight world-lines extending from the initial velocity vectors of each particle (both backward and forward in time). There are finitely many particles, so finitely many world-lines and finitely many points of intersection, which all occur within a finite time-interval, before which some particles may be drifting toward each other and after which the particles are all simply drifting away from each other. According to thermodynamics, the expansion of the gas is an irreversible process corresponding to an increase in entropy.

Loschmidt's paradox suggests that it's paradoxical for an irreversible process to arise from time-symmetric dynamics. But in this example the dynamics are time-symmetric and the irreversible expansion appears to happen in either direction. In the normal-time picture there is a time before which there are no intersections of world-lines, and in the reversed-time picture this becomes the time after which there are no intersections and the particles are all drifting away from each other. In either case, leading up to the region of collisions it might appear as though the system is spontaneously compressing over the long-term but eventually after traversing the region of collisions the system will be spontaneously expanding again, and irreversibly in the sense that the system can never spontaneously compress again.

This simple example seems to suggest that there's nothing inherently paradoxical about an irreversible process arising from time-symmetric dynamics, excepting that time-symmetric dynamics might allow a finite duration after any particular starting time where the system can appear to be temporarily running an irreversible process in reverse, but will eventually get past it and permanently fall into an actual normal-time irreversible process.

Is this a correct understanding of applying and resolving Loschmidt's paradox for the example of free gas expansion?

bobdavis said:
Following up on a previous discussion: https://www.physicsforums.com/threa...entum-in-a-closed-system.1009693/post-6570341

An uncontained system of particles interacting only by elastic collision is the same as a gas undergoing free expansion. If, for simplicity, the particles are assumed to be identical with equal mass, then the evolution of the system can be treated as if the particles pass through each other without colliding and can be drawn as just the straight world-lines extending from the initial velocity vectors of each particle (both backward and forward in time). There are finitely many particles, so finitely many world-lines and finitely many points of intersection, which all occur within a finite time-interval, before which some particles may be drifting toward each other and after which the particles are all simply drifting away from each other. According to thermodynamics, the expansion of the gas is an irreversible process corresponding to an increase in entropy.

Loschmidt's paradox suggests that it's paradoxical for an irreversible process to arise from time-symmetric dynamics. But in this example the dynamics are time-symmetric and the irreversible expansion appears to happen in either direction. In the normal-time picture there is a time before which there are no intersections of world-lines, and in the reversed-time picture this becomes the time after which there are no intersections and the particles are all drifting away from each other. In either case, leading up to the region of collisions it might appear as though the system is spontaneously compressing over the long-term but eventually after traversing the region of collisions the system will be spontaneously expanding again, and irreversibly in the sense that the system can never spontaneously compress again.

This simple example seems to suggest that there's nothing inherently paradoxical about an irreversible process arising from time-symmetric dynamics, excepting that time-symmetric dynamics might allow a finite duration after any particular starting time where the system can appear to be temporarily running an irreversible process in reverse, but will eventually get past it and permanently fall into an actual normal-time irreversible process.

Is this a correct understanding of applying and resolving Loschmidt's paradox for the example of free gas expansion?
I believe the following is consistent with your post.

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.

bobdavis said:
Following up on a previous discussion: https://www.physicsforums.com/threa...entum-in-a-closed-system.1009693/post-6570341

An uncontained system of particles interacting only by elastic collision is the same as a gas undergoing free expansion. If, for simplicity, the particles are assumed to be identical with equal mass, then the evolution of the system can be treated as if the particles pass through each other without colliding and can be drawn as just the straight world-lines extending from the initial velocity vectors of each particle (both backward and forward in time). There are finitely many particles, so finitely many world-lines and finitely many points of intersection, which all occur within a finite time-interval, before which some particles may be drifting toward each other and after which the particles are all simply drifting away from each other. According to thermodynamics, the expansion of the gas is an irreversible process corresponding to an increase in entropy.

Loschmidt's paradox suggests that it's paradoxical for an irreversible process to arise from time-symmetric dynamics. But in this example the dynamics are time-symmetric and the irreversible expansion appears to happen in either direction. In the normal-time picture there is a time before which there are no intersections of world-lines, and in the reversed-time picture this becomes the time after which there are no intersections and the particles are all drifting away from each other. In either case, leading up to the region of collisions it might appear as though the system is spontaneously compressing over the long-term but eventually after traversing the region of collisions the system will be spontaneously expanding again, and irreversibly in the sense that the system can never spontaneously compress again.

This simple example seems to suggest that there's nothing inherently paradoxical about an irreversible process arising from time-symmetric dynamics, excepting that time-symmetric dynamics might allow a finite duration after any particular starting time where the system can appear to be temporarily running an irreversible process in reverse, but will eventually get past it and permanently fall into an actual normal-time irreversible process.

Is this a correct understanding of applying and resolving Loschmidt's paradox for the example of free gas expansion?
As near as I can tell, this idea 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 and Scotch physicists.

#### Attachments

• Thompson p.355 in Brush.pdf
741.8 KB · Views: 95
bobdavis said:
If, for simplicity, the particles are assumed to be identical with equal mass, then the evolution of the system can be treated as if the particles pass through each other without colliding
No, it can't, since elastic collisions can change the directions of both particles. You need to rethink your entire analysis in the light of this correction.

bobdavis said:
According to thermodynamics, the expansion of the gas is an irreversible process corresponding to an increase in entropy.
Yes.

bobdavis said:
Loschmidt's paradox suggests that it's paradoxical for an irreversible process to arise from time-symmetric dynamics.
Yes, but we now understand that this is wrong. The time asymmetry does not arise from the dynamics but from the special nature of the initial conditions. You chose initial conditions with the particles all bunched up in a very small region of space, surrounded by emptiness. That is a very, very unlikely condition, and its time evolution will, as you note (though you have the details of the reasoning wrong), cause the system of particles to expand, occupying more and more space as time passes. That is the reason why the time evolution increases entropy: because there are many, many more ways for the particles to occupy a larger region of space than a smaller one, and you chose a special initial condition where they occupied a small space.

Demystifier and PeroK
Bill Dreiss said:
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?
No, he doesn't, because the point he is making is not what you claim. The point he is making, as is clear if you read the rest of the section in question, is that with a small enough number of particles, we can control the system accurately enough to set it up with the conditions exactly reversed, and watch the dynamics proceed exactly as the time reverse of the original system. So we can confirm that the dynamics themselves are time reversible. The reference to "the very large number of particles" really just means that for such systems we can't track each particle individually and we can't control what each particle does individually, so all we have are statistics; and then you have to look at the choice of initial conditions to see whether you expect time asymmetry (per my most in response to the OP just now).

Bill Dreiss said:
this is a clear misapplication of mirror symmetry
Bill Dreiss said:
it is apparent that the second law is not fundamental in and of itself, but an epiphenomenon of the more basic law of inertia.
These are personal speculations on your part and are off limits here.

PeterDonis said:
No, he doesn't, because the point he is making is not what you claim. The point he is making, as is clear if you read the rest of the section in question, is that with a small enough number of particles, we can control the system accurately enough to set it up with the conditions exactly reversed, and watch the dynamics proceed exactly as the time reverse of the original system. So we can confirm that the dynamics themselves are time reversible. The reference to "the very large number of particles" really just means that for such systems we can't track each particle individually and we can't control what each particle does individually, so all we have are statistics; and then you have to look at the choice of initial conditions to see whether you expect time asymmetry (per my most in response to the OP just now).
These are personal speculations on your part and are off limits here.
It seems impossible to exactly reverse the motion of the particles for a real gas due to the butterfly effect and the fact that the average molecule moves at about half a kilometer per second at room temperature. In place of speculating on the hypothetical results of such an experiment, we can observe the behavior of deterministic ideal gas simulations like https://phet.colorado.edu/sims/html/gas-properties/latest/gas-properties_en.html. Notice that the virtual gas disperses in the same way for any number of particles from 1000 down to 2. Controlling a “very large number of particles” in such a simulation would require no modification of the code and would be limited only by computer capacity. The second-law behavior of the resulting virtual gas would not differ in kind from that composed of a smaller number of virtual particles. Attached is an algorithm that exactly reverses the velocities, sending them back to their initial positions. Upon reaching that point, the virtual gas will once again expand, as was anticipated in 1874 by Wm. Thompson in a paper which can be found in Brush’s The Kinetic Theory of Gases (see attachment). What he deduced parallels my logic regarding Feynman’s polar vector. From observation of these ideal gas simulations, it is obvious that the dispersal of the virtual molecules is a consequence of the law of inertia coded into the algorithms.

At the top of page 52-7. He says “it is just because there are three dimensions in space that we can associate the quantity with a direction perpendicular to that plane.” [my italics] Comparing this rule to his example in Fig. 52-2 reveals an inconsistency. This is deduction, not speculation.

As to my statement relating inertia to the second law, I clearly overgeneralized. I should qualify my conclusion with the preface “For the ideal gas simulations I examined, it is apparent…”

Thank you for your comments. They have inspired me to increase my understanding.

## 1. What is Loschmidt's paradox?

Loschmidt's paradox, also known as the reversibility paradox, is a thought experiment that challenges the second law of thermodynamics. It questions how the irreversible process of gas expansion, which leads to an increase in entropy, can be reconciled with the time-reversal symmetry of the underlying physical laws.

## 2. What is free gas expansion?

Free gas expansion refers to the process of a gas expanding into a larger volume without any external work being done on the gas. This results in an increase in the volume of the gas and a decrease in its density, leading to a decrease in temperature and an increase in entropy.

## 3. How does Loschmidt's paradox relate to free gas expansion?

Loschmidt's paradox arises when considering the time-reversal symmetry of the underlying physical laws in free gas expansion. While the expansion process is irreversible and leads to an increase in entropy, the underlying physical laws are time-reversible. This paradox highlights the limitations of our understanding of the relationship between microscopic and macroscopic phenomena.

## 4. What are some proposed solutions to Loschmidt's paradox?

There are several proposed solutions to Loschmidt's paradox, including the idea that the second law of thermodynamics is statistical in nature and only applies to large systems, and that the initial conditions of the gas play a crucial role in determining the direction of time. Other solutions involve considering the role of quantum mechanics and the concept of quantum entanglement.

## 5. How does Loschmidt's paradox impact our understanding of thermodynamics?

Loschmidt's paradox challenges our current understanding of thermodynamics and the relationship between microscopic and macroscopic phenomena. It highlights the need for further research and exploration into the fundamental laws of physics and their implications for the behavior of gases and other systems. It also raises questions about the limitations of our current theories and the potential for new discoveries and breakthroughs in this field.

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