Math 2.0

Statistical mechanics, structure of matter Kinetic theory Problem

Loschmidt's paradox

Builds on Hamiltonian systemBoltzmann equation

Problem

Consider a gas of NN particles whose motion is a Hamiltonian system with Hamiltonian H(q,p)=∑i∣pi∣22m+U(q)H(q, p) = \sum_i \frac{\abs{p_i}^2}{2m} + U(q). This dynamics is reversible: if t↦(q(t),p(t))t \mapsto (q(t), p(t)) is a solution, then so is t↦(q(−t),−p(−t))t \mapsto (q(-t), -p(-t)). Let SS be a function of the state that is even in the momenta, such as an entropy computed from positions and speeds. Take a trajectory along which SS increases on a time interval. The reversed trajectory is also a motion of the gas, and along it SS decreases. The H-theorem says that the entropy of a dilute gas governed by the Boltzmann equation never decreases. The paradox: how can such a theorem be derived from a dynamics that is reversible in time?

Remarks

The objection was raised by Josef Loschmidt in 1876 against Boltzmann's derivation of the H-theorem. The standard answer is that the irreversibility does not come from the dynamics alone. It enters through the molecular chaos assumption in the derivation of the Boltzmann equation. That assumption takes the velocities of two particles about to collide to be independent, but not those of two particles that have just collided. This assumption is asymmetric in time and holds for typical, not all, initial states; the reversed state of a typical state is very special. Lanford's theorem makes this precise for hard spheres in the Boltzmann–Grad limit, for a short time. A related objection, due to Ernst Zermelo, uses the Poincaré recurrence theorem. The Kac ring is a simple model in which both the paradox and its answer can be checked by hand.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Loschmidt%27s_paradox
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AboutH-theorem
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Changed3 November 2026

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