H-theorem Totally Explained

Everything about H-theorem totally explained

In thermodynamics, the H-theorem, introduced by Boltzmann in 1872, describes the increase in the entropy of an ideal gas in an irreversible process, by considering the Boltzmann equation. It appears to predict an irreversible increase in entropy, despite microscopically reversible dynamics. This has led to much discussion.

Boltzmann's H-theorem

The quantity H is defined as the integral over velocity space : ť ).

But the two brackets will have the same sign, so each contribution to dS/dt can't be negative.
Therefore ť

Delta S geq 0

for an isolated system.
The same mathematics is sometimes also presented for classical systems, considering probability flows between coarse-grained cells in the phase space (for example, Tolman (1938)).

Critique

Several criticisms can be made of the above "proof", for example by Gull (1989):

It relies on the use of approximate quantum mechanics (Fermi's golden rule), not necessarily valid for large perturbations.
Are the probabilities to be considered as representing N independent systems of 1 particle, or as applying to 1 system of N particles? If it's the former, then it's ignoring the inter-particle correlations between the systems after collisions, explaining the information loss. The 1-particle entropy also ignores many-body effects in the potential energy, so bears little relation to the entropy of any real gas.
On the other hand, treated properly, an N-particle system has N-particle states. An isolated system will presumably sit in one of its N-particle microstates and make no transitions at all.

Analysis

At the heart of the H-theorem is the replacement of 1-state to 1-state deterministic dynamics by many-state to many-state Markovian mixing, with information lost at each Markovian transition.
   Gull is correct that, with the powers of Laplace's demon, one could in principle map forward exactly the ensemble of the original possible states of the N-particle system exactly, and lose no information. But this wouldn't be very interesting. Part of the program of statistical mechanics, not least the MaxEnt school of which Gull is an enthusiastic proponent, is to see just how much of the detail information in the system one can ignore, and yet still correctly predict experimentally reproducible results.
   The H-theorem's program of regularly throwing information away, either by systematically ignoring detailed correlations between particles, or between particular sub-systems, or through systematic regular coarse-graining, leads to predictions such as those from the Boltzmann equation for dilute ideal gases or from the recent entropy-production fluctuation theorem, which are useful and reproducibly observable. They also mean that we've learnt something qualitative about the system, and which parts of its information are useful for which purposes, which is additional beyond even the full specification of the microscopic dynamical particle trajectories.
   (It may be interesting that having rounded on the H-theorem for not considering the microscopic detail of the microscopic dynamics, Gull then chooses to demonstrate the power of the extended-time MaxEnt/Gibbsian method by applying it to a Brownian motion example - a not so dissimilar replacement of detailed deterministic dynamical information by a simplified stochastic/probabilistic summary!)
   However, it's an assumption that the H-theorem's coarse-graining isn't getting rid of any 'interesting' information. With such an assumption, one moves firmly into the domain of predictive physics: if the assumption goes wrong, it may produce predictions which are systematically and reproducibly wrong.

Further Information

Get more info on 'H-theorem'.

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