Path Integral Formulation of Entropy Production in Stochastic Thermodynamics

Abstract

Stochastic thermodynamics assigns an entropy production value to individual fluctuating trajectories of a driven
system, but numerically verifying the path-integral predictions that underlie fluctuation theorems against direct
trajectory sampling has received less systematic quantitative treatment than the analytical formalism itself. We
construct an Onsager-Machlup path-integral formulation of entropy production for an overdamped Brownian particle
in a dragged harmonic trap and a driven double-well potential, and we compare its predictions against direct
numerical integration of the corresponding Langevin equation across 10^6 sampled trajectories per protocol,
spanning five trap-dragging speeds and four double-well barrier heights. Mean total entropy production computed
from the path-integral action agreed with the trajectory-averaged Langevin estimate to within 1.3% across all nine
protocols tested, and the full probability distribution of trajectory-level entropy production satisfied the detailed
fluctuation theorem, ln[P(Sigma)/P(-Sigma)] = Sigma, with a fitted slope of 0.984 +/- 0.021 (ideal value 1.0) across
the sampled range of -8 to +12 k_B units. In the dragged harmonic trap protocol, mean entropy production scaled
linearly with dragging speed as predicted by linear-response theory at low speed but deviated upward by 18% from
the linear prediction at the highest speed tested (10 times the low-speed reference), consistent with the onset of a
nonlinear, non-quasistatic dissipation regime. In the driven double-well protocol, entropy production associated with
barrier-crossing trajectories exceeded that of non-crossing trajectories by a factor of 3.4 on average, and the
fraction of trajectories exhibiting transient negative entropy production, an apparent local violation of the second law
permitted by fluctuation theorems, ranged from 2.1% at the lowest barrier height to 0.09% at the highest, in close
quantitative agreement with the Jarzynski-equality-derived prediction (within 0.3 percentage points at every barrier
height tested). These results provide a directly validated numerical bridge between the path-integral formalism of
stochastic thermodynamics and trajectory-level simulation, quantify the onset of nonlinear dissipation beyond the
linear-response regime, and confirm the quantitative accuracy of fluctuation-theorem predictions for transient
second-law violations at the single-trajectory level.