Thermodynamics of the space of trajectories governed by a combination of two additive boundary functionals

Abstract

This paper extends the formalism of stochastic path-space thermodynamics by systematically expanding the space of thermodynamic variables with a spectrum of boundary and relative functionals of random processes. Generalizing the approaches introduced in preprints arXiv:2607.24078 and arXiv:2609.08477, we investigate the fluctuation statistics of a one-dimensional Ornstein-Uhlenbeck process with an asymmetric linear drift and a step penalty potential at the origin, which models the waiting phase of a molecular Brownian motor. Utilizing the Feynman-Kac formalism and the parabolic cylinder theory, a numerical scheme based on matching logarithmic derivatives via the secant method is constructed. A high-precision eigenvalues of the effective Hamiltonian is obtained, which define the cumulant generating function. We formulate a modified path-space fluctuation theorem of the Gallavotti-Cohen type for the conjugate drift current at a fixed integral occupation time of the system in the dissipative half-plane. The physical and thermodynamic significance of the new variables is elucidated. A combination of"residence time"and"integral current"functionals is applied to describe an ion channel sensitive to mechanical or electrical stimuli.