Random paths and fluctuation–dissipation dynamics for one-variable chemical systems far from equilibrium

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Abstract
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.