The zeroth law of thermodynamics and volume-preserving conservative system in equilibrium with stochastic damping

We propose a mathematical formulation of the zeroth law of thermodynamics and develop a stochastic dynamical theory, with a consistent irreversible thermodynamics, for systems possessing sustained conservative stationary current in phase space while in equilibrium with a heat bath. The theory generalizes underdamped mechanical equilibrium: dx = g dt + {-D del phi dt + root 2D dB (t)}, with del . g = 0 and {...} respectively representing phase-volume preserving dynamics and stochastic damping. The zeroth law implies stationary distribution u(ss)(x) = e(-phi(x)). We find an orthogonality del phi . g = 0 as a hallmark of the system. Stochastic thermodynamics based on time reversal (t, phi, g) -> (-t, phi, -g) is formulated: entropy production e(p)(#)(t) = -dF(t)/dt; generalized heat h(d)(#)(t) = -dU(t)/dt, U(t) = integral(Rn) phi(x)u(x, t) dx being internal energy, and free energy F (t) = U (t) + integral(Rn) u(x, t) In u(x, t) dx never increases. Entropy follows dS/dt = e(p)(#) - h(d)(#). Our formulation is shown to be consistent with an earlier theory of P. Ao. Its contradistinctions to other theories, potential-flux decomposition, stochastic Hamiltonian system with even and odd variables, Klein-Kramers equation, Freidlin-Wentzell's theory, and GENERIC, are discussed. (C) 2013 Elsevier B.V: All rights reserved.