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Notes: The thermodynamic and stochastic theory of chemical systems far from equilibrium is extended to reactions in inhomogeneous system for both single and multiple intermediates, with multiple stationary states coupled with linear diffusion. The theory is applied to the two variable Selkov model coupled with diffusion, in particular to the issue of relative stability of two stable homogeneous stationary states as tested in a possible inhomogeneous experimental configuration. The thermodynamic theory predicts equistability of such states when the excess work from one stationary state to the stable inhomogeneous concentration profile equals the excess work from the other stable stationary state. The predictions of the theory on the conditions for relative stability are consistent with solutions of the deterministic reaction-diffusion equations. In the following article we apply the theory again to the issue of relative stability for single-variable systems, and make comparison with numerical solutions of the reaction-diffusion equations for the Schlögl model, and with experiments on an optically bistable system where the kinetic variable is temperature and the transport mechanism is thermal conduction.

Notes: The eikonal approximation (instanton technique) is applied to the problem of large fluctuations of the number of species in spatially homogeneous chemical reactions with the probability density distribution described by a master equation. For both autocatalytic and nonautocatalytic reactions, the analysis of the distribution about a stable stationary state and of the transitions between coexisting stable states comes, to logarithmic accuracy, to the analysis of Hamiltonian dynamics of an auxiliary dynamical system. The latter can be done explicitly in a few cases, including one-species systems, systems with detailed balance, and systems close to the bifurcation points where the number of the stable states changes. In the last case, the fluctuations display universal features, and, for saddle-node bifurcation points, the logarithm of the probability of escape from the metastable state (per unit time) is proportional to the distance to the bifurcation point (in the parameter space) raised to the power 3/2. We compare the eikonal approximation for the stationary distribution of a master equation to Monte Carlo numerical solutions for two chemical two-variable systems with multiple stationary states, where none of the cited restrictions exists. For one of the systems in the pattern of optimal paths we observe caustics emanating from the saddle point.

Notes: A random path integral representation of the Ross–Hunt–Hunt thermodynamic and stochastic theory is given for chemical reactions far from equilibrium in the case of constant-step and one-variable processes. An explicit analytical expression for the chemical Lagrangian is presented. A connection is made between the thermodynamic fluctuation–dissipation regimes characteristic to the process and the chemical Lagrangian. The path integral formalism is used to prove the validity of fluctuation regression hypothesis and to derive two variational principles for the most probable and average paths, respectively. The most probable path corresponds to the absolute maximum of the Lagrangian and the average path corresponds to the minimum value of the information gain obtained by observing a certain average path. For nonlinear regimes these two variational principles generally give distinct results; they are identical only in the vicinity of a stable steady state. An eikonal approximation is suggested for evaluating time-dependent probability distributions which reduces the integration of Master Equations to two quadratures. The suggested eikonal approximation leads to a proportionality between the species-specific free energy of the system and the extremal value of the time integral of the chemical Lagrangian. This relationship is similar to the expression of the mechanical action in terms of the Lagrangian in classical mechanics. Most results derived in this paper for one variable can be extended to multivariable systems. Finally a comparison is made with other stochastic approaches to nonequilibrium thermodynamics.

Notes: A new fluctuation–dissipation relation is suggested for constant step, one intermediate chemical processes far from equilibrium. It establishes a relationship between the net reaction rate t˜(x), the probability diffusion coefficient D(x) in the composition space, and the species-specific affinity A(x): t˜(x)=2D(x)tanh[−A(x)/2kT], where x is the concentration of the active intermediate, k is Boltzmann's constant, and T is the absolute temperature. The theory is valid for nonlinear fluctuations of arbitrary size. For macroscopic systems the fluctuation–dissipation relation may be viewed as a force-flux relationship. We distinguish four fluctuation–dissipation regimes which correspond to the decrease of the absolute value of the species-specific affinity. The passage from high ||A(x)|| to small ||A(x)|| corresponds to a crossover from a linear dependence of the species-specific dissipation rate φ(overdot)(x) on ||A(x||)||, φ(overdot)(x)∼−||A(x)||, to a square one: φ(overdot)(x)∼−A2(x). A main feature of the fluctuation–dissipation relation is its symmetry with respect to the contributions of the forward and backward chemical processes to fluctuation and relaxation. Two new physical interpretations of the probability diffusion coefficient are given: one corresponds to a measure of the strength of fluctuations at a steady state, and the other to a measure of the instability of a given fluctuation state. The dispersion of the number q of reaction events in a given time interval is given by a generalized Einstein relation: 〈Δq2〉=2VD(x)t, where V is the volume of the system. The diffusion coefficient D(x) is proportional to the reciprocal value of the mean age 〈τ(x)〉 of a fluctuation state characterized by the concentration x: D(x)=1/[2V〈τ(x)〉]. These interpretations are not related to the use of a Fokker–Planck approximation of the chemical Master Equation.

Notes: A generalized thermodynamic description of one-variable complex chemical systems is suggested on the basis of the Ross, Hunt, and Hunt (RHH) theory of nonequilibrium processes. Starting from the stationary solution of a chemical Master Equation, two complimentary, related sets of generalized state functions are introduced. The first set of functions is derived from a generalized free energy FX, and is used to compute the moments of stationary and non-Gaussian concentration fluctuations. Exact expressions for the cumulants of concentration are derived; a connection is made between the cumulants and the fluctuation–dissipation relations of the RHH theory. The second set of functions is derived from an excess free energy φ(x); it is used to express the conditions of existence and stability of nonequilibrium steady states. Although mathematically distinct, the formalisms based on the FX and φ(x) functions are physically equivalent: both lead to the same type of differential expressions and to similar global equations. A comparison is made between the RHH and Keizer's theory of nonequilibrium processes. An appropriate choice of the integration constants occurring in Keizer's theory is made for one-variable systems. The main differences between the two theories are: the constraints for the two theories are different; the stochastic and thermodynamic descriptions are global in RHH, whereas Keizer's theory is local. However, both theories share some common features. Keizer's fluctuation–dissipation relation can be recovered by using the RHH approach; it is valid even if the fluctuations are nonlinear.If the thermodynamic constraints are the same, then Keizer's theory is a first-order approximation of RHH; this approximation corresponds to a Gaussian description of the probability of concentration fluctuations. Keizer's theory is a good approximation of RHH in the vicinity of a stable steady state: near a steady state the thermodynamic functions of the two theories are almost identical; the chemical potential in the stationary state is of the equilibrium form in both theories. Keizer's theory gives a very good estimate of the absolute values of the peaks of the stationary probability density of RHH theory. Away from steady states the predictions of the two theories are different; the differences do not vanish in the thermodynamic limit. The shapes of the tails of the stationary probability distributions are different; and hence the predictions concerning the relative stability are different for the two theories.