Equilibrium states are used as limit states to define thermodynamically reversible processes. When these processes are understood in terms of statistical physics, these limit states can change with time due to thermal fluctuations. For macroscopic systems, the changes are insignificant on ordinary time scales and what little change there is can be suppressed by macroscopically negligible, entropy-creating dissipation. For systems of molecular sizes, the changes are large on short time scales. They can only sometimes be suppressed with significant entropy-creating dissipation, and this entropy creation is unavoidable if any process is to proceed to completion. As a result, at molecular scales, thermodynamically reversible processes are impossible in principle. Unlike the macroscopic case, they cannot be realized even approximately when we account for all sources of dissipation, and argumentation invoking them on molecular scales can lead to spurious conclusions.

## References

To see this, note that for small changes we have $dFsys=d(Usys\u2212TSsys)=dUsys\u2212TdSsys=dUsys+dUenv=dUtot=0$, where the heat passed to the environment in the reversible process is *dQ*_{sys} *= TdS*_{sys}, which equals the energy change in the environment *dU*_{env}.

For a small, reversible change we have *dF*_{sys} *= dU*_{sys} − *ΤdU*_{sys} *= dU*_{sys} − *dQ* = −*dW*, so that −*dF*_{sys}*/dx = dW/dx = X*.

To connect with the usual statement of the canonical distribution, if *V*_{ph,tot} is the volume of the full phase space accessible to the system, then the canonical distribution is *p* = exp(−*E/kT*)/ *Z*(*V*_{ph,tot}) and the probability that the system is in subvolume *V*_{ph} is equal to *Z*(*V*_{ph})/ *Z*(*V*_{ph,tot}).

The intermediate states can never be completely inaccessible or the process could not proceed. Rather the process design must be such as to make them accessible only with arbitrarily small probability.

*P*(λ

_{fin})/

*P*(λ

_{init})] give slightly higher dissipation than the corresponding formulas (22) and (23) of an earlier paper [

_{fin})/P(λ

_{init})]. The latter formulae presumed that the process ends in a way that prevents return to the initial state. In the absence of a non-dissipative way of preventing this return, the newer formulae provide a better limit.

These two probabilities are to be read as follows: over the longer term in which the gas-piston system fully explores the phase space accessible to it, it comes to an equilibrium with probability *P*(*h*_{0}) of the initial compressed *h*-state and probability *P*(*h*_{1}) of the final, expanded h-state.

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