The Zeroth Law of Thermodynamics and Volume-Preserving Conservative Dynamics with Equilibrium 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: $dx=gdt+{-D\nabla\phi dt+\sqrt{2D}dB(t)}$,with$\nabla\cdot g=0$and${\cdots}$ respectively representing phase-volume preserving dynamics and stochastic damping. The zeroth law implies stationary distribution $u^{ss}(x)=e^{-\varphi (x)}$. We find an orthogonality $\mathrm{\nabla }\varphi \cdot g=0$ as a hallmark of the system. Stochastic thermodynamics based on time reversal $(t,\varphi ,g)\to (-t,\varphi ,-g)$ is formulated: entropy production $e_p^{\mathrm{\#}}(t)=-dF(t)\mathrm{/}dt$; generalized “heat” $h_d^{\mathrm{\#}}(t)=-dU(t)\mathrm{/}dt$, $U(t)={\int }_{{\mathbb{R}}^n}\varphi (x)u(x,t)dx$ being “internal energy”, and “free energy” $F(t)=U(t)+{\int }_{{\mathbb{R}}^n}u(x,t)\ln u(x,t)dx$ never increases. Entropy follows $\frac{dS}{dt}=e_p^{\mathrm{\#}}-h_d^{\mathrm{\#}}$. 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.