The entropy production and circulation of diffusion processes on manifolds

Non-equilibrium of a general stochastic system of coupled oscillators: entropy production rate and rotation numbers

“…Specific results did appear in some special stochastic processes [2,10,12] and Axiom-A systems [6], where in the latter case the EPR is decomposed into the linear combination of the Lyapunov characteristic exponents. Specific results did appear in some special stochastic processes [2,10,12] and Axiom-A systems [6], where in the latter case the EPR is decomposed into the linear combination of the Lyapunov characteristic exponents.…”

Mathematical formalism for isothermal linear irreversibility

“…The reversibility under this potential condition has been announced for the general nonlinear equation (1) but without proof [19,11], and discussed in [3,18] for the equation on compact manifold. In this paper, we provide a simple proof for the linear case.…”

“…Statement (i) is also equivalent to A −1 Bx having a potential function U (x) = 1 2 x T A −1 Bx and R n e −U (x) dx < ∞. The reversibility under this potential condition has been discussed for the general nonlinear equation (1.1) but without proof (Qian 1998;Qian & Qian 1985), and studied in Guo et al . (1998) and Qian & Wang (1999) for the equation on a compact manifold.…”

“…Remark 2: The statement (i) is also equivalent to A −1 Bx having a potential function U (x) = 1 2 x T A −1 Bx and R n e −U(x) dx < ∞. The reversibility under this potential condition has been announced for the general nonlinear equation (1) but without proof [19,11], and discussed in [3,18] for the equation on compact manifold. In this paper, we provide a simple proof for the linear case.…”

Energy and entropy equations in coupled nonequilibrium thermal mechanical diffusive chemical heterogeneous system

“…There have been extensive studies to discuss the coupling problem in a complex heterogeneous system [1][2][3][4][5][6][7][8][9][10]. In the current textbooks and literatures, the energy equation is shown as q _ u ¼ r : _ e À r Á q; e ij ¼ ðu i;j þ u j;i Þ=2; or dU ¼…”