Volume-Preserving Conservative Dynamics with Equilibrium Stochastic Damping    [PDF]

Hong Qian
We show that Ao's stochastic process [ J. Phys. A: Math. Gen. 37 L25 (2004)] is consistent with a phase-volume preserving conservative dynamical system in equilibrium with a stochastic damping satisfying detailed balance: $dq=(\gamma-D\nabla\phi)dt + d\xi(t)$ where $\nabla\cdot\gamma=0$, < \xi(t)\xi(t')> =2D\delta(t-t')$, and stationary density$u^{ss}(q)=e^{-\phi(q)}$. We show that a hallmark of such systems is an orthogonality between the gradient of the stationary density and the conservative rotation in phase space:$\nabla\phi\cdot\gamma=0$. A stochastic thermodynamics based on time reversal$(t,\gamma)\rightarrow (-t,-\gamma)$yields a novel formula for heat dissipation$h_d^*(t)=-\int_{R^n}\nabla\phi\cdot D\nabla ln(u(q,t)e^{\phi})dq$and entropy production$e_p^*(t)=-dF(t)/dt$: The "free energy"$F(t)$never increases. Th entropy balance equation$\frac{dS}{dt}=e_p^*-h_d^*$is also recovered. The relations among Ao's stochastic process, a Hodge-like potential-flux decomposition, stochastic Hamiltonian system with Klein-Kramers equation, and the theory of GENERIC, are discussed. When$\phi\$ is given, three different types of time reversal leading to different open-system thermodynamics are suggested.
View original: http://arxiv.org/abs/1206.7079