Resolving Paradoxes: Loschmidt, Zermelo, and the H-theorem

1. The Paradoxes Formally Stated

The H-theorem asserts that $dH/dt\le 0$, implying a monotonic approach to equilibrium. However, the classical mechanics that underpins the kinetic theory possesses properties that seem to directly contradict this conclusion.

The paper "An introduction to statistical mechanics via the Kac ring" by Gottwald and Oliver provides a concise statement of these objections.

Let's summarize the two objections:

1. Loschmidt's Reversibility Paradox: The fundamental laws of mechanics (e.g., Newton's or Hamilton's equations) are time-reversal invariant. This means for any evolution of the system where $H$ decreases, one can construct a new initial state—by reversing the velocities of all particles—from which the system will evolve by retracing its steps, causing $H$ to increase. This directly contradicts the claim that $H$ must always decrease.

2. Zermelo's Recurrence Paradox: This objection invokes the Poincaré recurrence theorem. It states that an isolated mechanical system confined to a finite volume of phase space will, after a sufficiently long time, return arbitrarily close to its initial state. This implies that if $H$ decreases from its initial value, it must eventually increase to return to (or near) that initial value. Therefore, $H(t)$ cannot be a monotonically decreasing function for all time.

Both paradoxes highlight a genuine tension: how can irreversible macroscopic behavior emerge from reversible and recurrent microscopic dynamics?

2. A Toy Model: The Kac Ring

To make these abstract paradoxes concrete, Mark Kac developed a brilliant toy model. The Kac ring is a simple, deterministic system that is explicitly reversible and recurrent, yet for which a macroscopic description appears to be irreversible. This model allows us to see exactly where the statistical assumption comes into play and why it leads to an apparent paradox.

The key insight from the Kac ring is that the irreversible macroscopic equation $\Delta (t)=(1-2\mu {)}^t\Delta (0)$ is not a true representation of the microscopic evolution for any specific ring configuration. It is an approximation derived by assuming that the state of the system is always "chaotic" or random enough that the fraction of black balls in front of markers is always equal to the overall fraction of black balls. This assumption—the Stoßzahlansatz—discards information about the precise correlations building up in the system, and in doing so, it introduces irreversibility. The true microscopic evolution is, of course, perfectly reversible and will recur after at most $2N$ steps, at which point the Stoßzahlansatz must fail spectacularly.

3. Resolution: The Statistical Nature of the Second Law

The resolution to both paradoxes lies in a profound conceptual shift, initiated by Boltzmann himself in response to these very criticisms. The H-theorem, and by extension the second law of thermodynamics, is not a deterministic mechanical theorem but a statistical one. It describes not what must happen, but what is overwhelmingly probable to happen.

This intellectual pivot is what marks the transition from kinetic theory to what we now call statistical mechanics. The paper by Brown and Myrvold details this crucial development in Boltzmann's thinking.

With this statistical viewpoint, we can now resolve the paradoxes one by one.

3.1 Resolving Loschmidt's Paradox

Loschmidt's objection is valid: if you could prepare the velocity-reversed state, the system would indeed evolve towards lower entropy (higher H). Boltzmann's famous response, which you saw in the Villani video, was "Go and reverse them!". This is more than just a dismissal; it points to the heart of the matter.

• The Special Nature of the Reversed State: The process of a gas evolving towards equilibrium involves collisions that create correlations between the outgoing particles. The Stosszahlansatz assumes that particles are uncorrelated before colliding. A typical, randomly chosen microstate will satisfy this pre-collision chaos. However, if you let the system evolve and then perfectly reverse all velocities, you create a highly atypical and specially prepared state. In this state, the particles are now correlated before their next collision, precisely in a way that will cause them to evolve back to the initial, lower-entropy configuration.

• Failure of the Stosszahlansatz: For this special, time-reversed state, the Stosszahlansatz is false. In fact, an "anti-Stosszahlansatz" holds. The derivation of the H-theorem, which relies on the standard Stosszahlansatz, is therefore not applicable to this trajectory. A derivation using the anti-chaos assumption would instead prove $dH/dt\ge 0$.

The resolution is thus probabilistic: an astronomically vast majority of microscopic states corresponding to a non-equilibrium macrostate will evolve towards equilibrium. The set of "anti-thermodynamic" microstates, like the one prepared by velocity reversal, is non-zero but has a vanishingly small (formally, measure zero) probability of being realized by chance. The second law describes the behavior of typical initial conditions, not exquisitely fine-tuned ones.

3.2 Resolving Zermelo's Paradox

The recurrence paradox is resolved by considering the timescales involved. While the Poincaré recurrence theorem is mathematically correct, the recurrence times for macroscopic systems are physically meaningless.

• Astronomical Timescales: For a system with $N\sim 10^{23}$ particles, the Poincaré recurrence time ${\tau }_P$ is hyper-exponentially large, typically estimated to be on the order of ${\tau }_P\sim ({\tau }_c{)}^{10^{23}}$, where ${\tau }_c$ is a typical collision time. This number is so vast that it dwarfs the age of the universe ($\sim 10^{10}$ years) by an incomprehensible margin.

• Physical vs. Mathematical Truth: While it is mathematically true that the system will eventually recur, this will not happen on any timescale relevant to a physical experiment or observation. As Cédric Villani noted in the video you watched, "This will simply not occur on a time scale that is reasonable".

The H-theorem correctly describes the system's behavior over any physically relevant duration. The eventual recurrences predicted by Zermelo are just one example of the fantastically rare statistical fluctuations that are possible but will, with overwhelming probability, never be observed.

4. Synthesis: Coarse-Graining and the Arrow of Time

The ultimate resolution lies in understanding that macroscopic observables represent a form of coarse-graining over the microscopic details. We don't track every particle; we measure pressure, temperature, or the distribution function $f(\mathbf{r},\mathbf{p},t)$.

The Gottwald and Oliver paper provides a beautiful summary of this modern perspective, distinguishing between two ways of looking at the evolution of the ensemble.

As the paper explains, both the Loschmidt and Zermelo paradoxes lose their bite when we adopt this statistical viewpoint.

• Loschmidt's paradox vanishes: The "arrow of time" is defined by the process of coarse-graining and the loss of microscopic information. Preparing a time-reversed state is equivalent to preparing a non-typical initial state where microscopic correlations have been macroscopically engineered to conspire against the most probable evolution. The H-theorem describes the evolution of typical states, which lack such conspiracies.
• Zermelo's paradox vanishes: The Boltzmann approach, based on the Stoßzahlansatz, is an approximation that is valid only as long as statistical correlations remain weak. Over the immense Poincaré recurrence time, these correlations would build up, and the approximation would fail. But on any realistic timescale, the approximation holds, and the system behaves irreversibly.

Conclusion

Today we have confronted two of the most significant conceptual challenges to the kinetic theory of gases. By working through their resolutions, we have uncovered the true nature of the second law as derived from mechanics.

Key Takeaways:

• Loschmidt's and Zermelo's paradoxes highlight the apparent contradiction between macroscopic irreversibility and the time-reversible, recurrent nature of microscopic mechanics.
• The resolution to Loschmidt's paradox is that the H-theorem describes the evolution of typical microstates. Time-reversed trajectories are possible but require starting from a set of highly atypical, correlated initial conditions for which the Stosszahlansatz is invalid.
• The resolution to Zermelo's paradox is based on timescales. The Poincaré recurrence time for any macroscopic system is so astronomically large as to be physically irrelevant.
• The fundamental insight is that the second law of thermodynamics is a statistical law, not a deterministic one. It describes the overwhelmingly probable tendency of a system to evolve from a less probable macrostate (one with a smaller phase-space volume) to a more probable one (one with a larger phase-space volume). Irreversibility is an emergent property of this statistical description.

We have now established a solid microscopic foundation for the approach to equilibrium. In the next lesson, we will move from these deep foundational questions to a more practical tool for analyzing transport phenomena. We will introduce the relaxation-time approximation to the Boltzmann equation, which will allow us to calculate concrete physical quantities like conductivity and viscosity.