2012/05/30 by Hong Qian, Qian, Hong
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Statistical Mechanics (cond-mat.stat-mech) #cond-mat.stat-mech #math-ph #math.DS #math.MP
paper · pdf · doi:10.48550/arxiv.1205.6552
18 pages
arxiv created 2013/04/06 · arxiv updated 2013/04/09
The infinitesimal transition probability operator for a continuous-time discrete-state Markov process, 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, (d)/(dt)u(t)=\mcA u, is mathematically equivalent to a linear Hamiltonian dynamics with Hamiltonian H=1/2uT(\mcAT\mcA)1/2u. It can also be transformed into a Schrödinger-like equation (d)/(dt)u=iHu where the "Hamiltonian" operator H=-i\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 ‖usi‖2. The motion can be viewed as "harmonic" since (d)/(dt)‖u(t)-c‖2=0 where c=(c,c,...,c) with c being a constant. More interestingly, we discover that \textrmTr(\mcAT\mcA)=∑j,ℓ=1n \frac(qjℓπ_ℓ-qℓ jπj)2πjπℓ, whose right-hand-side is intimately related to the entropy production rate of the Markov process in a nonequilibrium steady state with stationary distribution \πj\. The physical implication of this intriguing connection between conservative Hamiltonian dynamics and dissipative entropy production remains to be further explored.