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Notes: A model for multichannel parallel relaxation is suggested based on the following assumptions: (a) an individual channel is characterized by a set of continuous state variables; the corresponding relaxation rate is a function of the state variables as well as of the time interval for which the channel is open; (b) the number of channels is a random variable described by a correlated point process defined in the space of state parameters of an individual channel. Analytical expressions for the generating functional of the overall relaxation rate and for the average survival function are derived in terms of the generating functional of the point process. The general formalism is applied to the problem of direct energy transfer from excited donors to acceptors in fractal systems with dynamic disorder. It is assumed that the number of acceptors obeys a Poissonian distribution law with a constant average density in a df-dimensional fractal structure embedded in a ds-dimensional Euclidean space (ds=1,2,3) and that an individual relaxation rate is an inverse power function of the distance between the acceptor and the donor molecules.The dynamic disorder is described in terms of three different functions: the rate ω(t) of opening of a channel at time t, the attenuation function cursive-phi(t) of the reactivity of an individual channel at time t, and the probability density ψ(t) of the time interval within which a channel is open. Several particular cases corresponding to different functions ω(t), cursive-phi(t), and ψ(t) are investigated. The static disorder corresponds to a survival function of the stretched exponential type exp[−(Ωt)β] with 1(approximately-greater-than)β(approximately-greater-than)0. For very strong dynamic disorder there is no attenuation of reactivity, the opening time is infinite and the survival function is given by a compressed exponential exp[−const.t1+β], 1(approximately-greater-than)β(approximately-greater-than)0. The other cases analyzed correspond to a slowly decreasing attenuation function and to an exponential distribution of the opening time, respectively; for them the efficiency of relaxation is between the ones corresponding to the two extreme cases of static and very strong dynamic disorder. The general conclusion is that the passage from static to the dynamic disorder results in an increase of the efficiency of the relaxation process. © 1995 American Institute of Physics.

Notes: The asymptotic behavior of multichannel parallel relaxation processes for systems with dynamical disorder is investigated in the limit of a very large number of channels. An individual channel is characterized by a state vector x which, due to dynamical disorder, is a random function of time. A limit of the thermodynamic type in the x-space is introduced for which both the volume available and the average number of channels tend to infinity, but the average volume density of channels remains constant. Scaling arguments combined with a stochastic renormalization group approach lead to the identification of two different types of universal behavior of the relaxation function corresponding to nonintermittent and intermittent fluctuations, respectively. For nonintermittent fluctuations a dynamical generalization of the static Huber's relaxation equation is derived which depends only on the average functional density of channels, ρ[W(t′)]D[W(t′)], the channels being classified according to their different relaxation rates W=W(t′), which are random functions of time. For intermittent fluctuations a more complicated relaxation equation is derived which, in addition to the average density of channels, ρ[W(t′)]D[W(t′)], depends also on a positive fractal exponent H which characterizes the fluctuations of the density of channels. The general theory is applied for constructing dynamical analogs of the stretched exponential relaxation function. For nonintermittent fluctuations the type of relaxation is determined by the regression dynamics of the fluctuations of the relaxation rate. If the regression process is fast and described by an exponential attenuation function, then after an initial stretched exponential behavior the relaxation process slows down and it is not fully completed even in the limit of very large times. For self-similar regression obeying a negative power law, the relaxation process is less sensitive to the influence of dynamical disorder. Both for small and large times the relaxation process is described by stretched exponentials with the same fractal exponent as for systems with static disorder. For large times the efficiency of the relaxation process is also slowed down by fluctuations. Similar patterns are found for intermittent fluctuations with the difference that for very large times and a slow regression process a crossover from a stretched exponential to a self-similar algebraic relaxation function occurs. Some implications of the results for the study of relaxation processes in

Notes: The dependence of solution behavior to perturbations of the initial function (IF) in a class of nonlinear differential delay equations (DDEs) is investigated. The structure of basins of attraction of multistable limit cycles is investigated. These basins can possess complex structure at all scales measurable numerically although this is not necessarily the case. Sensitive dependence of the asymptotic solution to perturbations in the initial function is also observed experimentally using a task specific electronic analog computer designed to investigate the dynamics of an integrable first-order DDE.

Notes: Here cell population dynamics in which there is simultaneous proliferation and maturation is considered. The resulting mathematical model is a nonlinear first-order partial differential equation for the cell density u(t,x) in which there is retardation in both temporal (t) and maturation variables (x), and contains three parameters. The solution behavior depends on the initial function cursive-phi(x) and a three component parameter vector P=(δ,λ,r). For strictly positive initial functions, cursive-phi(0)(approximately-greater-than)0, there are three homogeneous solutions of biological (i.e., non-negative) importance: a trivial solution ut≡0, a positive stationary solution ust, and a time periodic solution up(t). For cursive-phi(0)=0 there are a number of different solution types depending on P: the trivial solution ut, a spatially inhomogeneous stationary solution unh(x), a spatially homogeneous singular solution us, a traveling wave solution utw(t,x), slow traveling waves ustw(t,x), and slow traveling chaotic waves uscw(t,x). The regions of parameter space in which these solutions exist and are locally stable are delineated and studied.

Notes: Numerical solutions to a model equation that describes cell population dynamics are presented and analyzed. A distinctive feature of the model equation (a hyperbolic partial differential equation) is the presence of delayed arguments in the time (t) and maturation (x) variables due to the nonzero length of the cell cycle. This transport like equation balances a linear convection with a nonlinear, nonlocal, and delayed reaction term. The linear convection term acts to impress the value of u(t,x=0) on the entire population while the death term acts to drive the population to extinction. The rich phenomenology of solution behaviour presented here arises from the nonlinear, nonlocal birth term. The existence of this kinetic nonlinearity accounts for the existence and propagation of soliton-like or front solutions, while the increasing effect of nonlocality and temporal delays acts to produce a fine periodic structure on the trailing part of the front. This nonlinear, nonlocal, and delayed kinetic term is also shown to be responsible for the existence of a Hopf bifurcation and subsequent period doublings to apparent "chaos'' along the characteristics of this hyperbolic partial differential equation. In the time maturation plane, the combined effects of nonlinearity, nonlocality, and delays leads to solution behaviour exhibiting spatial chaos for certain parameter values. Although analytic results are not available for the system we have studied, consistency and validation of the numerical results was achieved by using different numerical methods. A general conclusion of this work, of interest for the understanding of any biological system modeled by a hyperbolic delayed partial differential equation, is that increasing the spatio-temporal delays will often lead to spatial complexity and irregular wave propagation. © 1996 American Institute of Physics.

Notes: A class of rate processes with dynamical disorder is investigated based on the two following assumptions: (a) the system is composed of a random 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 formalism of characteristic functionals is used for the direct computation of the dynamical averages. The suggested approach is more general than the other approaches used in the literature: it is not limited to a particular type of stochastic process and can be applied to any type of random evolution of the rate coefficient. We derive an infinity of exact 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 relations leads to the unexpected result that in the thermodynamic limit the fluctuations of the number of particles have an intermittent behavior. The moments are explicitly evaluated in two particular cases: (a) the random behavior of the rate coefficient is given by a non-Markovian 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 characteristic functionals is used to recover the conventional description of dynamical disorder in terms of the Kubo–Zwanzig stochastic Liouville equations as a particular case of our general approach. The fluctuation–dissipation relations can be used for the study of fluctuations without making use of the whole mathematical formalism.To illustrate the efficiency of our method for the analysis of fluctuations we discuss three different physicochemical and biochemical problems. A first application is the kinetic study of the decay of positrons or positronium atoms thermalized in dense fluids: in this case the time dependence of the rate coefficient is described by a stationary Gaussian random function with an exponentially decaying correlation coefficient. A second application is an extension of Zwanzig's model of ligand–protein interactions described in terms of the passage through a fluctuating bottle neck; we complete the Zwanzig's analysis by studying the concentration fluctuations. The last example deals with jump rate processes described in terms of two independent random frequencies; this model is of interest in the study of dielectric or conformational relaxation in condensed matter and on the other hand gives an alternative approach to the problem of protein–ligand interactions. We evaluate the average survival function in several particular cases for which the jump dynamics is described by two activated processes with random energy barriers. Depending on the distributions of the energy barriers the average survival function is a simple exponential, a stretched exponential, or a statistical fractal of the inverse power law type. The possible applications of the method in the field of biological population dynamics are also investigated. © 1996 American Institute of Physics.

Notes: Abstract A model for erythroid production based on a continuous maturationproliferation scheme is developed. The model includes a simple control mechanism operating at the proliferating cell level, and analytic solutions for the time dependent response of the model are derived. Using this model, the response of the erythron to a massive depletion of the proliferating cell compartment (due for example to cytostatic drugs or radiation) is calculated. It is demonstrated that a therapeutic measure designed to decrease the erythroid precursor maturation velocity may considerably ameliorate the deleterious effects of proliferating cell destruction. One way to decrease the erythroid cell maturation rate would be by having the patient breathe in an oxygen enriched atmosphere.