Non-equilibrium Statistical Mechanics Based on the Free Energy Landscape and Its Application to Glassy Systems

Extending the concept of the Ginzburg-Landau theory of phase transition to non-equilibrium systems, I present a free energy landscape (FEL) formalism of non-equilibrium statistical mechanics and show that the FEL formalism provides a framework for unified description of thermodynamic and dynamic properties of non-equilibrium systems. I first show that a conditional free energy φ(T,V,N, {Ri }) can be defined as a function of configuration {Ri } of a given average position of atoms so that the probability of finding the configuration {Ri} is in proportion to exp[-φ(T,V,N, {Ri })/kbT]. Thermodynamic quantities in quasi-equilibrium states are given by their average over the configuration, and the temperature dependence of the FEL manifests itself in the temperature derivatives of thermodynamic quantities. As an example, I discuss the entropy and the specific heat, focusing on the contributions due to configuration and the temperature dependence of the FEL, and show that an additional contribution due to the temperature dependence of the FEL exists in the specific heat. I generalize the FEL formalism so that time dependent phenomena can be analyzed in a frame work similar to the time-dependent Ginzburg-Landau theory. I introduce a time-dependent probability function of configuration and describe its time dependence by a Fokker-Planck equation which guarantees that the probability function satisfies the initial condition and the proper long-time limit. The time dependence of a physical quantity is given by its average over the time-dependent distribution function. In order to show the robustness of the FEL formalism in explaining thermodynamic and dynamic effects in a unified frame work, I discuss several phenomena found in super-cooled liquids on the basis of the FEL formalism which includes glass transition singularities, slow relaxations, cooling rate dependence of the specific heat, the ac specific heat, temperature dependence of the crystallization time and the temperature modulation spectroscopy.