A Hamiltonian-Entropy Production Connection in the Skew-symmetric Part of a Stochastic Dynamics

The infinitesimal transition probability operator for a continuous-time discrete-state Markov process, $\mathcal{Q}$, can be decomposed into a symmetric and a skew-symmetric parts. As recently shown for the case of diffusion processes, while the symmetric part corresponding to a gradient system stands for a reversible Markov process, the skew-symmetric part, $\frac{d}{dt}u(t)=\text{mcA}u$, is mathematically equivalent to a linear Hamiltonian dynamics with Hamiltonian $H=1\mathrm{/}2u^T({\text{mcA}}^T\text{mcA}{)}^{1\mathrm{/}2}u$. It can also be transformed into a Schr"{o}dinger-like equation $\frac{d}{dt}u=i\mathcal{H}u$ where the “Hamiltonian” operator $\mathcal{H}=-i\text{mcA}$ is Hermitian. In fact, these two representations of a skew-symmetric dynamics emerge natually through singular-value and eigen-value decompositions, respectively. The stationary probability of the Markov process can be expressed as $\mathrm{\parallel }{u}_i^s{\mathrm{\parallel }}^2$. The motion can be viewed as “harmonic” since $\frac{d}{dt}\mathrm{\parallel }u(t)-\vec{c}{\mathrm{\parallel }}^2=0$ where $\vec{c}=(c,c,\mathrm{.}\mathrm{.}\mathrm{.},c)$ with $c$ being a constant. More interestingly, we discover that $$\text{Tr}$(\mcA^T\mcA)=\sum_{j,\ell=1}^n \frac{(q_{j\ell}\pi_\ell-q_{\ell j}\pi_j)^2}{\pi_j\pi_{\ell}}$, whose right-hand-side is intimately related to the entropy production rate of the Markov process in a nonequilibrium steady state with stationary distribution $\{{\pi }_j\}$. The physical implication of this intriguing connection between conservative Hamiltonian dynamics and dissipative entropy production remains to be further explored.