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

•A mathematical theory for stochastic underdamped dynamical equilibrium with conservative current.•Irreversible thermodynamics with time symmetry (t,ϕ,g)→(−t,ϕ,−g)(t,ϕ,g)→(−t,ϕ,−g): stationary potential ϕ, current g.•Mathematical formulation of the zeroth law of thermodynamics in the context of stochastic dynamics.•Cyclic dynamics is either underdamped equilibrium or overdamped NESS, depends on t reversal symmetry.

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=gdt+{−D∇ϕdt+2DdB(t)}, with ∇⋅g=0∇⋅g=0 and {⋯}{⋯} respectively representing phase-volume preserving dynamics and stochastic damping. The zeroth law implies stationary distribution uss(x)=e−ϕ(x)uss(x)=e−ϕ(x). We find an orthogonality ∇ϕ⋅g=0∇ϕ⋅g=0 as a hallmark of the system. Stochastic thermodynamics based on time reversal (t,ϕ,g)→(−t,ϕ,−g)(t,ϕ,g)→(−t,ϕ,−g) is formulated: entropy production ep#(t)=−dF(t)/dt; generalized “heat” hd#(t)=−dU(t)/dt, U(t)=∫Rnϕ(x)u(x,t)dx being “internal energy”, and “free energy” F(t)=U(t)+∫Rnu(x,t)lnu(x,t)dx never increases. Entropy follows dSdt=ep#−hd#. 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–Wentzellʼs theory, and GENERIC, are discussed.