Work performed under the auspices of the United States Department of Energy, contract number DE-AC07-76-IDO1570.
Journal of Non-Equilibrium Thermodynamics
Time reversal, Numerical analysis, Gibbs' equation, Solid-liquid interfaces -- Mathematical models
In a recent paper of the same title [J. Non-Equilib. Thermodyn., 15 (1990), 151], Liboff observed that the fine-grained Gibbs entropy of a canonical Hamiltonian system remains constant in time even for Hamiltonians that are not even in momenta and consequently violate time-reversal invariance (TRI). Here we extend this observation to non-canonical Hamiltonian systems, including systems with singular Poisson tensors and pseudo-Hamiltonian systems that violate the Jacobi identity. Necessary and sufficient conditions are given for the Gibbs entropy to be constant in such systems. The concept of TRI is not in general meaningful for such systems, but it is shown that systems with constant entropy are always microscopically reversible in the Poincare recurrence sense, which implies that H- (Lyapunov) functions do not exist. This result applies as a special case to canonical systems, regardless of whether or not they obey TRI. A distinction should therefore be drawn between microscopic reversibility and TRI.
J.D. Ramshaw, "Microscopic Irreversibility and Gibbs Entropy," J. Non-Equilib. Thermodyn. 16, 33 (1991)