Mathematical Theory Of Nonequilibrium Thermodynamics Of Switching Diffusion Processes

In the past 20 years,the tremendous progress of experimental techniques has enabled the scientists to observe the randomness in molecular scale.Due to the development of biology and modern thermodynamics,pure jumping process or pure diffusion process would not be able to describe some complex systems,such as molecular motors.The suitable mathematical framework to model a molecular motor should be the one that combines the chemical master equation and Langevin equation into one system,which leads to a switching diffusion process(SDP for short).A switching diffusion process is a widely used stochastic model in industry,economy and biology etc.,to describe complex systems.In recent years,more and more topics about the SDP have been studied,including ergodicity,recurrence,reversibility,etc.Especially in 2004,Fuxi Zhang calculated the entropy production rates of the stationary inhomogeneous SDP and gave several equivalent conditions of reversibility of the SDP,by constructing the minimal SDP.But the complete theory of nonequilibrium thermodynamics of the SDP have not been extensively studied yet,especially the interaction and combination of the two components of an SDP.Firstly,we propose the decomposition of the entropy production rate in one-dimensional and high-dimensional SDPs,based on the flux decomposition.And we also give the corresponding decompositions of entropy change and heat dissipation rate,which are consistent with each other.The non-negativity of each decomposed component of entropy production rate is proved,which leads to two detailed Clausius inequalities.However,similar decompositions of the housekeeping heat dissipation rate and free energy dissipation rate cannot guarantee the non-negativity of each decomposed component.Hence,we modify this decomposition with the flow of exponential relative information under steady-state fluxes,resulting in another decomposition with all non-negative components.Furthermore,we get the marginal distribution of each component and describe the system into three subsystems of mesostate subsystem,microstate subsystem and remaining subsystem by the method of coarse-graining.Then we provide the nonequilibrium thermodynamics of one-dimensional and high-dimensional SDPs under the perspectives of coarse-graining and exchange of information between the chemical kinetics and mechanical motion,resulting in several other decompositions of entropy production rate.The present paper is divided into two parts,one for one-dimensional SDP and another for high-dimensional SDP according to the state space of the diffusion component.The structures and results of the two parts are parallel and the derivation,calculation and proof of high-dimensional case are more complex than one-dimensional case because the derivation of matrix is more complex than that of scalar.At last,we give the summary and prospect,which point out the direction for our future research.