We define a classical gas as one described by classical dynamics including electrodynamics. Details of collisions are considered as are interactions with external electromagnetic and gravitational fields. Classical molecules have continuous internal degrees of freedom including vibration and rotation.

A quantum gas adds electron spin, Fermi-Dirac and Bose-Einstein statistics for anti-symmetric and symmetric particles, but most importantly it treats the kinetics of collisions - particle scattering - quantum mechanically. Quantum molecules have complex discrete internal structures. We ignore relativistic effects.

Gases can be studied from three perspectives - thermodynamics, statistical mechanics, and kinetic theory (which follows the details of molecular interactions). They can also be treated classically or quantum mechanically.

Boltzmann's original (1872) formulation of the H-theorem was based on the kinetics of molecular collisions, but it contained an unjustifiable assumption (Stosszahlansatz) about the absence of correlations of molecular velocities before and after the collisions (molecular chaos) that introduced probability, perhaps inadvertently, into his calculations.

We will show that Boltzmann's original assumption of molecular chaos may be justified for quantum gases of particles with accessible internal energy states.

With a combination of kinetic theory and probabilistic assumptions, Boltzmann derived the famous expression that he identified with Rudolf Clausius' thermodynamic entropy.

Five years later, Boltzmann responded to a brief remark by his Vienna colleague Josef Loschmidt that a gas with the same molecular positions but opposite velocities should show an entropy decrease, since classical mechanics is time reversible.

Boltzmann immediately agreed with Loschmidt and speculated that perhaps one could not prove the entropy increase with a purely mechanical analysis. In its place, he gave a new defense of the H-theorem (eq.2), using only the assumption of equiprobable microstates.

The entropy of a non-equilibrium macrostate is proportional to the logarithm of the number of microstates consistent with the macrostate description. Microstates compatible with thermal equilibrium outnumber non-equilibrium microstates by many orders of magnitude. Assuming that transitions between microstates are all equally likely (essentially the ergodic hypothesis as Ehrenfest named it), non-equilibrium macrostates quickly evolve to thermal equilibrium.

Macrostates that depart from equilibrium contain information (negative entropy) that in principle can be observed and measured. This includes cases of states prepared by experimenters, including removing a barrier between unlike gases, or perfume in a bottle at time zero. In these cases, quantitatively calculable information in the initial state is completely lost at equilibrium.

Increasing entropy can be equated to decreasing information.

This is the core idea of statistical mechanics, whether the gas is ideal, classical, or quantum.

However, both the reversibility objection and the recurrence objection are still valid. Given special initial conditions corresponding to the highly improbable "time-reversed" velocities, the entropy will decrease.

Pierre-Simon Laplace had postulated a super intelligence who can predict the future, given perfect knowledge of the positions and velocities of material particles together with their force laws. But given the essential impossibility of preparing an initial state for a test of this or Loschmidt's hypothesis, Boltzmann thought it not a practical objection.

Extreme sensitivity to initial conditions was appreciated by James Clerk Maxwell as early as 1865 [ref?], when he noted the occurrence of singular points in hydrodynamical flows and argued that something like them in the mind might allow living creatures to escape from strict determinism. Modern computer simulations of ideal gases confirm Maxwell's and Boltzmann's intuitions, showing that miniscule errors in original positions lead very quickly to randomness in the distributions.

"Chaos theory" is the deterministic mathematical formalism that describes the dynamics of physical systems near singular points in their motions where infinitesimal differences in position or velocity lead to exponentially large differences at later times. It does not involve quantum uncertainty.

Zermelo's recurrence objection was based on Poincare's studies of the n-body problem, which indicated that recurrence would be "quasi-periodic." Poincare also discovered exceptional non-recurrent paths, which have been shown to be infinite in number yet with zero probability, a "set of measure zero" in modern terminology ¹.

• Boltzmann's H-theorem is valid
• Loschmidt's reversibility objection is valid
• Zermelo's recurrence objection is valid

Boltzmann was as aware of the ideal gas approximations as any modern scientist. He was the first to include external forces (a gravitational field). He attempted kinetic calculations for polyatomic molecules. And he knew that collisions with real container walls are likely to be inelastic and non-specular, adding randomness to the time evolution of the gas.

Boltzmann also knew that the exactness of deterministic laws of classical mechanics themselves went "beyond experience." No observational evidence justified their perfection. Newton himself doubted analytical perfection, in part for theological reasons that such perfection limited God's powers.

Loschmidt's reversibility objection forced Boltzmann to see that some configurations of position and velocity might lead to to a reduction in entropy, but Boltzmann appreciated the practical impossibility of preparing such a state. He thought fluctuations of local reductions in the entropy would occur. But he argued they would be short-lived in a non-ideal gas, given the effects of random perturbations from external forces.

In modern times, calculations have shown that even tiny amounts of matter at stellar distances can alter the trajectories of classical particles. A gram of matter at the distance of Sirius can cause a particle to miss a predicted collision after as few as 50 collisions. [Berry?]

Faced with Zermelo's recurrence objection, Boltzmann calculated the recurrence time for even a small number of particles and showed that it exceeded by many orders of magnitude the likely age of the universe. For all practical purposes, recurrence to a prepared state (with all particles in half the container volume for example) was impossible.

• Boltzmann's H-theorem is valid
• Loschmidt's reversibility objection still has some merit. But not only the gas particle velocities need to be reversed, all the events in the universe must reverse, including the cosmic expansion. This ties the arrow of time to both thermodynamics and the universe expansion.
• Zermelo's recurrence objection remains valid

Just as Hamilton's classical equations of motion are time reversible, Erwin Schrodinger's reformulation of them as the equation of motion for the probability amplitude ψ(r, t) of a quantum state is time reversible. If H is the Hamiltonian,

Quantum statistical mechanics embodies the equiprobability assumptions about phase-space volumes (the ergodic hypothesis in both Boltzmann and Gibbs formulations) into the Fermi Golden Rule and Master Equation, which says that transition probabilities from microstate i to microstate j are equal to the reverse transition from j to i. The matrix element P_ij is the complex conjugate of P_ji.

Quantum kinetic theory treats the collisions of gas molecules as problems in quantum scattering. In this case, inelastic collisions could include Raman scattering that changes the internal quantum states of the colliding molecules. Moreover, colliding atoms could combine to form molecules, emitting the binding energy as radiation.

In these cases, the resulting dissociation at a later time, however short, would mean that information has been created and destroyed. As a consequence, the kind of quantum coherence needed for time reversibility of the Schrodinger equation is lost. Boltzmann's original guess that velocities of particles after collisions are random would be justified.

• Boltzmann's H-theorem is valid, a consequence of Fermi's Golden Rule (equality of forward and reverse transition probabilities between microstates) and the Master Equation.
• Loschmidt's reversibility objection is not valid. Time reversal of the Schrodinger equation for many particles requires perfect coherence of the probability amplitudes. "Time reversibility" is irrelevant in systems that create and destroy information and thus destroy coherence.
• Zermelo's recurrence objection is valid but does not have even quasi-periodicity. In quantum mechanics, the probability of recurrence has nothing to do with kinetic theory. It is simply the statistical mechanical probability of the initial macrostate.

But the addition of molecular collisions analyzed as quantum scattering processes that create and destroy information, with temporary and local entropy decreases (fluctuations), strengthens the H-theorem by vitiating Loschmidt's reversibility objection. In addition, it replaces Zermelo's recurrence objection with the unavoidable but essentially vanishingly small probability that any prepared system should return to its initial conditions after a finite and physically significant time.