Positivity of entropy production in nonequilibrium statistical mechanics (Q1285243)

Let \(f:M\to M\) be a \(C^1\)-diffeomorphism on a compact Riemannian manifold. Let \(\rho\) be a density (with respect to the Riemannian volume \(dx)\) and \(\rho_1=\rho\circ f^{-1}\) where \(J\) denotes the Jacobian of \(f\). The entropy gain is defined by \[ \int\rho_1 \log\rho_1-\rho\log \rho dx =\int \rho\log J dx. \] Taking vague limits of \(\rho_n dx\) where \(\rho_n\) is the iterated density obtained from the above formula, one defines the entropy production of a vague limit \(\mu\) by \(e_f(\mu)= -\int\log J d\mu\). The paper discusses this notion, extended also to noninvertible maps and diffusions. In particular, formulas for \(e_f(\mu)\) are derived and it is shown that \(e_f(\mu)\geq 0\) when \(\mu\) is an SBR measure or a certain absolutely continuous measure.