Abstract
In stochastic thermodynamics standard concepts from macroscopic thermodynamics, such as heat, work, and entropy production, are generalized to small fluctuating systems by defining them on a trajectory-wise level. In Langevin systems with continuous state-space such definitions involve stochastic integrals along system trajectories, whose specific values depend on the discretization rule used to evaluate them (i.e. the ‘interpretation’ of the noise terms in the integral). Via a systematic mathematical investigation of this apparent dilemma, we corroborate the widely used standard interpretation of heat- and work-like functionals as Stratonovich integrals. We furthermore recapitulate the anomalies that are known to occur for entropy production in the presence of temperature gradients.
Highlights
In stochastic thermodynamics standard concepts of macroscopic thermodynamics are generalized to cover the non-equilibrium properties of small systems [1, 2]
The generalized Langevin equation (5) and the associated functionals along its trajectories do not suffer from any ambiguity related to interpretation of stochastic integrals, because the generated position and velocity processes are sufficiently regular
Exploiting this uniqueness of integrals, we here applied a multiscale procedure to the generalized Langevin equation (GLE) and its functionals to find out which noise interpretations of the overdamped Langevin equation and its thermodynamic functionals are consistent with the systematic white-noise and small-mass limits
Summary
In stochastic thermodynamics standard concepts of macroscopic thermodynamics are generalized to cover the non-equilibrium properties of small systems [1, 2]. While space-dependent friction coefficients are known to be related to nonlinear system-bath couplings [26], the space- (and time-) dependent temperature is a generalization we introduce “by hand” under the following assumption: the temperature field T (x, t) varies sufficiently slowly such that the particle at position x and time t is in contact with a locally well-defined thermal bath, which is homogeneous on mesoscopic scales, and that any memory effects from that local bath on time scales τb die out before the particle changes position into an “adjacent bath” at (slightly) different temperature, or before the bath-temperature changes significantly in time
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