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Notes: We investigate stationary and nonstationary probability densities for a weakly forced nonlinear physical or chemical system that displays self-oscillations in the absence of forcing. The period and amplitude of forcing are taken as adjustable constraints. We consider a homogeneous reaction system described by a master equation. Our method of solution is based on the Wentzel–Kramers–Brillouin (WKB) expansion of the probability density with the system size as the expansion parameter. The first term in this expansion is the stochastic potential (eikonal). In the absence of forcing, the probability density is logarithmically flat on the limit cycle. With periodic forcing, the phenomenon of phase locking can occur whereby a stable cycle, which is close to the unforced cycle, adopts a constant relative phase to the forcing. A saddle cycle also exists and has a different constant relative phase. For such phase-locked solutions, the distribution over the relative phases is peaked on the stable cycle and exhibits a logarithmically flat region (a plateau) that originates on the saddle cycle. This plateau is due to a nonzero relative phase slippage: large fluctuations from the stable cycle over the saddle cycle are overwhelmingly more probable in a certain relative phase direction, which depends upon the location of the parameters within an entrainment region. This distribution of relative phases is logarithmically equivalent to that of a Brownian particle in a periodic potential with a constant external force in the strong damping and weak noise limits. For parameter values outside of an entrainment region (for which a quasiperiodic solution exists), the distribution in relative phase is logarithmically flat. For this regime, we investigate the evolution of an initially localized density and show that the width grows proportionally with the square root of time. The proportionality factor depends upon both the position (phase) on the cross section of the peak of the density and the distance in parameter space from the boundary of the entrainment region. For parameter values that approach the boundary of an entrainment region, this proportionality factor tends to infinity. We also determine an expression for the first order correction to the stochastic potential for both entrained and quasiperiodic solutions. A thermodynamic interpretation of these results is made possible by the equality of the stochastic potential with an excess work function. © 1998 American Institute of Physics.

Notes: The eikonal (WKB) approximation is applied to a stationary one-dimensional master equation describing an arbitrary reaction mechanism. The uniqueness of a nontrivial (fluctuational) eikonal solution is proven. Consistent eikonal and exact analytical solutions are obtained for systems with an arbitrary, but unique step size of stochastic transitions. An analytical eikonal solution for the stationary probability density for systems with mixed step sizes of 1 and 2 is also obtained and found to differ significantly from the systems with a uniform step size, particularly in the case of multiple stationary states. © 1995 American Institute of Physics.

Notes: We examine the effect of periodic and discrete perturbations on the phase of an oscillatory chemical reaction system near a Hopf bifurcation. Discrete perturbations reset the phase of the oscillation and periodic perturbations entrain the frequency of the oscillation for perturbation frequencies in a small range about each rational multiple of the natural frequency. These phase responses may be determined from time series of a single essential species. The new phase resulting from discrete perturbations and the relative phase between the oscillation and the forcing of an entrained oscillation are described by the same response function, which is a simple sinusoid. We show that for single species perturbations, the amplitude and phase offset of this response function equal the magnitude and the argument, respectively, of the corresponding component of the adjoint eigenvector of the Jacobi matrix (that corresponds to a pure imaginary eigenvalue). These phase response methods are simpler than quenching studies for determining the adjoint eigenvectors, and in addition yield the local isochrons of the periodic orbit. © 1995 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: For a nonequilibrium system described at the mesoscopic level by the master equation, we prove that the probability of fluctuations about a steady state is governed by a thermodynamic function, the "excess work.'' The theory applies to systems with one or more nonequilibrium steady states, for reactions in a compartment that contains intermediates Xj of variable concentration, along with a reactant A and product B whose concentrations are held constant by connection of the reaction chamber to external reservoirs. We use a known relation between the stationary solution Ps(X) of the master equation and an underlying stochastic Hamiltonian H: to logarithmic accuracy, the potential that gives Ps(X) is the stochastic action S evaluated along fluctuational trajectories, obtained by solving Hamilton's equations of motion starting at a steady state. We prove that the differential action dS equals a differential excess work dφ0, and show that dφ0 can be measured experimentally in terms of total free energy changes for the reaction compartment and the reservoirs. Thus we connect the probability of concentration fluctuations in an open reaction compartment to thermodynamic functions for the entire closed system containing the compartment. The excess work dφ0 is the difference between the total free energy change for a specified change in the quantities of A, X, Y, and B in the state of interest, and the free energy change for the same changes in species numbers, imposed on the same system in a reference state (A,X0,Y0,B).The reference-state concentration for species Xj is derived from the momentum pj canonically conjugate to Xj along the fluctuational trajectory. For systems with linear rate laws, the reference state (A,X0,Y0,B) is the steady state, and φ0 is equivalent to the deterministic excess work φdet* introduced in our previous work. For nonlinear systems, (A,X0,Y0,B) differs from the deterministic reference state (A,X*,Y*,B) in general, and φ0≠φdet*. If the species numbers change by ±1 or 0 in each elementary step and if the overall reaction is a conversion A→X→Y→B, the reference state (A,X0,Y0,B) is the steady state of a corresponding linear system, identified in this work. In each case, dφ0 is an exact differential. Along the fluctuational trajectory away from the steady state, dφ0(approximately-greater-than)0. Along the deterministic kinetic trajectory, dφ0≤0, and φ0 is a Liapunov function. For two-variable systems linearized about a steady state, we establish a separate analytic relation between Ps(X), φdet*, and a scaled temperature T*. © 1995 American Institute of Physics.