  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 $u^{ss}(x)=e^{-\phi(x)}$. We find an
orthogonality $\nabla\phi\cdot g=0$ as a hallmark of the system. Stochastic
thermodynamics based on time reversal
$\big(t,\phi,g\big)\rightarrow\big(-t,\phi,-g\big)$ is formulated: entropy
production $e_p^{\#}(t)=-dF(t)/dt$; generalized "heat" $h_d^{\#}(t)=-dU(t)/dt$,
$U(t)=\int_{\mathbb{R}^n} \phi(x)u(x,t)dx$ 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 $\frac{dS}{dt}=e_p^{\#}-h_d^{\#}$. 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.
