Path integral solutions of stochastic equations for nonlinear irreversible processes: The uniqueness of the thermodynamic Lagrangian

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Abstract
The conditional propagator associated with a generalized Langevin equation or Fokker–Planck equation is represented as a functional integral over paths, each exponentially weighted according to the time integral of the thermodynamicLagrangian along the path. A principal result of this paper is the proof that the thermodynamicLagrangian is unique for single‐variable processes or for multivariable processes in flat spaces, contrary to earlier statements that the Lagrangian depends on an arbitrary parameter, often denoted α. Fixing the parameter α is equivalent to choosing a stochastic calculus (α = 0 corresponds to the Itô calculus and α = 1/2 to the Stratonovich calculus), but consistent operations with any stochastic calculus selected lead to the same Lagrangian. The Euler–Lagrange equations derived by a variational minimization of the thermodynamic action (the time integral of the Lagrangian) are also unique; these equations give the most heavily weighted path between any two fixed points in the space of stochastic variables. For linear Gaussian processes, the global path of least action from a point is identical to the phenomenological path. When Laplace’s method can be used to approximate the path integral, the conditional propagator can be written as a product of a thermodynamic factor and a kinetic factor. The thermodynamic factor is large if a process is thermodynamically favorable, but the kinetic factor may be small if large changes in the stochastic variables are required on a short time scale.

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