Potential and flux field landscape theory. II. Non-equilibrium thermodynamics of spatially inhomogeneous stochastic dynamical systems

Abstract

We have established a general non-equilibrium thermodynamic formalism consistently applicable to both spatially homogeneous and, more importantly, spatially inhomogeneous systems, governed by the Langevin and Fokker-Planck stochastic dynamics with multiple state transition mechanisms, using the potential-flux landscape framework as a bridge connecting stochastic dynamics with non-equilibrium thermodynamics. A set of non-equilibrium thermodynamic equations, quantifying the relations of the non-equilibrium entropy, entropy flow, entropy production, and other thermodynamic quantities, together with their specific expressions, is constructed from a set of dynamical decomposition equations associated with the potential-flux landscape framework. The flux velocity plays a pivotal role on both the dynamic and thermodynamic levels. On the dynamic level, it represents a dynamic force breaking detailed balance, entailing the dynamical decomposition equations. On the thermodynamic level, it represents a thermodynamic force generating entropy production, manifested in the non-equilibrium thermodynamic equations. The Ornstein-Uhlenbeck process and more specific examples, the spatial stochastic neuronal model, in particular, are studied to test and illustrate the general theory. This theoretical framework is particularly suitable to study the non-equilibrium (thermo)dynamics of spatially inhomogeneous systems abundant in nature. This paper is the second of a series.

Potential landscape and flux field theory for turbulence and nonequilibrium fluid systems

“…(5) describes a Langevin-type stochastic field dynamics tracing a stochastic trajectory in the state space (velocity field configuration space). The corresponding ensemble dynamics governing the evolution of the probability distribution functional of the velocity field is described by the functional Fokker-Planck equation (FFPE) [33,36,37]:…”

“…As a consequence, for nonequilibrium fluid systems without detailed balance, we obtain from Eq. (11), by dividing both sides with P s [u], the force decomposition equation in the potential landscape and flux field theory [36,37]:…”

“…The global dynamics of nonequilibrium spatially extended systems has been investigated on the basis of the potential-flux form of the nonequilibrium driving force [36]. A set of nonequilibrium thermodynamic equations applicable to both spatially homogeneous and inhomogeneous systems has been established within this framework [37], which has synthesized and generalized much of the work based on a stochastic approach to nonequilibrium thermodynamics.…”

“…The potential landscape and flux field theory has extended the range of application of the potential landscape and flux framework to spatially extended systems (fields) and deepened the understanding of certain aspects in the theoretical framework [35,36,37]. The global dynamics of nonequilibrium spatially extended systems has been investigated on the basis of the potential-flux form of the nonequilibrium driving force [36].…”

“…It is particularly suited for the study of the global dynamics and nonequilibrium thermodynamics of stochastic field systems governed by the Langevin and Fokker-Planck field equations [35,36,37]. The potential landscape and flux framework, which has its historical origin in the energy landscape theory in protein folding dynamics, was initially developed for nonequilibrium biological systems and has been applied extensively in that area and beyond [38,39,40,41].…”