Levy diffusion in a force field, huber relaxation kinetics, and nonequilibrium thermodynamics: H theorem for enhanced diffusion with Levy white noise
A characteristic functional approach is suggested for Levy diffusion in disordered systems with external force fields. We study the overdamped motion of an ensemble of independent particles and assume that the force acting upon one particle is made up of two additive components: a linear term generated by a harmonic potential and a second term generated by the interaction with the disordered system. The stochastic properties of the second term are evaluated by using Huber's approach to complex relaxation [Phys. Rev. B 31, 6070 (1985)]. We assume that the interaction between a moving particle and the environment can be expressed by the contribution of a large number of relaxation channels, each channel having a very small probability of being open and obeying Poisson statistics. Two types of processes are investigated: (a) Levy diffusion with static disorder for which the fluctuations of the random force are frozen and last forever and (b) diffusion with strong dynamic disorder and independent Levy fluctuations (Levy white noise). In both cases we show that the probability distribution of the position of a diffusing particle tends towards a stationary nonequilibrium form. The characteristic functional of concentration fluctuations is evaluated in both cases by using the theory of random point processes. For large times the fluctuations of the concentration field are stationary and the corresponding probability density functional can be evaluated analytically. In this limit the fluctuations depend on the distribution of the total number of particles but are independent of the initial positions of the particles. We show that the logarithm of the stationary probability functional plays the role of a nonequilibrium thermodynamic potential, which has a structure similar to the Helmholtz free energy in equilibrium thermodynamics: it is made up of the sum of an energetic component, depending on the external mechanical potential, and of an entropic component, depending on the concentration field. We show that the conditions for the existence and stability of the nonequilibrium steady state, which emerges for large times, can be expressed in terms of the stochastic potential. For Levy white noise the average concentration field can be expressed as the solution of a fractional Fokker-Planck equation. We show that the stochastic potential is a Lyapunov function of the fractional Fokker-Planck equation, which ensures that all transient solutions for the average concentration field tend towards a unique stationary form.