2015.7.15 Nonequilibrium thermodynamics in phase space
题目1：Nonequilibrium thermodynamics in phase space
题目1： Mathematical methods for stochastic dynamics
报告人： 钱纮 教授
(1) Nonequilibrium thermodynamics in phase space
Nonequilibrium thermodynamics (NET) concerns transport processes in systems out of global equilibrium. Macroscopic NET has found wide applications in chemical and mechanical engineering, biological physics and biophysical chemistry, heterogeneous materials with interfaces, and thermal radiation.On a mesoscopic level and in terms of statistical descriptions of dynamics,various transport phenomena, such as chemical reactions, ionic transport, thermo-chemo-mechanics, diffusion, etc., can all be quantitatively described in terms of a single variable: a flux that expresses transport of probabilities in phase space. Mesoscopic stochastic NET in its abstract form, therefore, attains a universal formulation as a branch of applied probability. In the present review, we introduce this mesolevel formulation of NET through simple examples in chemistry. Several fundamental insights can be traced back to T. L. Hill's pioneering work on nonequilibrium steady state cycle kinetics on graphs. The statistical mechanics of Onsager's reciprocal relations is elucidated. A local equilibrium assumption is needed to apply the abstract theory in phase space to laboratory measurements. Chemo-mechanical, thermo-mechanical, and enzyme catalyzed thermo-chemical and electro-chemical energy transductions are discussed. With rigorous mathematical basis, mesoscopic stochastic NET provides fundamental concepts needed for understanding complex processes in chemistry, physics and biology. It gives the foundation for designing nano-devices that perform useful functions through energy transduction.
(2) Modern Mathematical methods for stochastic dynamics
We outline an attempt to lay the groundwork for understanding stochastic dynamical descriptions of biological, and other complex processes in terms of a discrete-state space, discrete-time random dynamical system (RDS), or random transformation approach. Such mathematics is not new for continuous systems, but the discrete state space formulation significantly reduces the technical requirements for its introduction to a much broader audiences. In particular, we establish some elementary contradistinctions between Markov chain (MC) and RDS descriptions of a stochastic dynamics. It is shown that a given MC is compatible with many possible RDS, and we study in particular the corresponding RDS with maximum metric entropy. Specifically, we show an emergent behavior of an MC with a unique absorbing and aperiodic communicating class, after all the trajectories of the RDS synchronizes. In biological modeling, it is now widely acknowledged that stochastic dynamics is a more complete description of biological reality than deterministic equations; here we further suggest that the RDS description could be a more refined description of stochastic dynamics than a Markov process. Possible applications of discrete-state RDS are systems with fluctuating law of motion, or environment, rather than inherent stochastic movements of individuals.