Mo. Vlad et al., Levy diffusion in a force field, Huber relaxation kinetics, and nonequilibrium thermodynamics: H theorem for enhanced diffusion with Levy white noise, PHYS REV E, 62(2), 2000, pp. 1743-1763
A characteristic functional approach is suggested for Levy diffusion in dis
ordered systems with external force fields. We study the overdamped motion
of an ensemble of independent particles and assume that the force acting up
on one particle is made up of two additive components: a linear term genera
ted by a harmonic potential and a second term generated by the interaction
with the disordered system. The stochastic properties of the second term ar
e evaluated by using Huber's approach to complex relaxation [Phys. Rev. B 3
1, 6070 (1985)]. We assume that the interaction between a moving particle a
nd the environment can be expressed by the contribution of a large number o
f relaxation channels, each channel having a very small probability of bein
g open and obeying Poisson statistics. Two types of processes are investiga
ted: (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 Livy fluctuations (Levy white noise). In b
oth cases we show that the probability distribution of the position of a di
ffusing particle tends towards a stationary nonequilibrium form. The charac
teristic functional of concentration fluctuations is evaluated in both case
s by using the theory of random point processes. For large times the fluctu
ations of the concentration field are stationary and the corresponding prob
ability density functional can be evaluated analytically. In this limit the
fluctuations depend on the distribution of the total number of particles b
ut 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 th
e Helmholtz free energy in equilibrium thermodynamics: it is made up of the
sum of an energetic component, depending on the external mechanical potent
ial, and of an entropic component, depending on the concentration field. We
show that the conditions for the existence and stability of the nonequilib
rium steady state, which emerges for large times, can be expressed in terms
of the stochastic potential. For Livy white noise the average concentratio
n field can be expressed as the solution of a fractional Fokker-Planck equa
tion. We show that the stochastic potential is a Lyapunov function of the f
ractional Fokker-Planck equation, which ensures that all transient solution
s for the average concentration field tend towards a unique stationary form
.