Mo. Vlad et al., NONEQUILIBRIUM FLUCTUATION-DISSIPATION RELATIONS FOR INDEPENDENT RANDOM RATE-PROCESSES WITH DYNAMICAL DISORDER, Journal of mathematical physics, 37(2), 1996, pp. 803-835
A class of rate processes with dynamical disorder is investigated base
d on the two following assumptions: (a) the system is composed of a ra
ndom number of particles (or quasiparticles) which decay according to
a first-order kinetic law; (b) the rate coefficient of the process is
a random function of time with known stochastic properties. The formal
ism of characteristic functionals is used for the direct computation o
f the dynamical averages. The suggested approach is more general than
the other approaches used in the literature: it is not limited to a pa
rticular type of stochastic process and can be applied to any type of
random evolution of the rate coefficient. We derive an infinity of exa
ct fluctuation-dissipation relations which establish connections among
the moments of the survival function and the moments of the number of
surviving particles. The analysis of these fluctuation-dissipation re
lations leads to the unexpected result that in the thermodynamic limit
the fluctuations of the number of particles have an intermittent beha
vior. The moments are explicitly evaluated in two particular cases: (a
) the random behavior of the rate coefficient is given by a non-Markov
ian process which can be embedded in a Markovian process by increasing
the number of state variables and (b) the stochastic behavior of the
rate coefficient is described by a stationary Gaussian random process
which is generally non-Markovian. The method of curtailed characterist
ic functionals is used to recover the conventional description of dyna
mical disorder in terms of the Kubo-Zwanzig stochastic Liouville equat
ions as a particular case of our general approach. The fluctuation-dis
sipation relations can be used for the study of fluctuations without m
aking use of the whole mathematical formalism. To illustrate the effic
iency of our method for the analysis of fluctuations we discuss three
different physicochemical and biochemical problems. A first applicatio
n is the kinetic study of the decay of positrons or positronium atoms
thermalized in dense fluids: in this case the time dependence of the r
ate coefficient is described by a stationary Gaussian random function
with an exponentially decaying correlation coefficient. A second appli
cation is an extension of Zwanzig's model of ligand-protein interactio
ns described in terms of the passage through a fluctuating bottle neck
; we complete the Zwanzig's analysis by studying the concentration flu
ctuations. The last example deals with jump rate processes described i
n terms of two independent random frequencies; this model is of intere
st in the study of dielectric or conformational relaxation in condense
d matter and on the other hand gives an alternative approach to the pr
oblem of protein-ligand interactions. We evaluate the average survival
function in several particular cases for which the jump dynamics is d
escribed by two activated processes with random energy barriers. Depen
ding on the distributions of the energy barriers the average survival
function is a simple exponential, a stretched exponential, or a statis
tical fractal of the inverse power law type. The possible applications
of the method in the field of biological population dynamics are also
investigated. (C) 1996 American Institute of Physics.