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

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 thermodynamic Lagrangian along the path. A principal result of this paper is the proof that the thermodynamic Lagrangian 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.