2 research outputs found
The Measure-theoretic Identity Underlying Transient Fluctuation Theorems
We prove a measure-theoretic identity that underlies all transient
fluctuation theorems (TFTs) for entropy production and dissipated work in
inhomogeneous deterministic and stochastic processes, including those of Evans
and Searles, Crooks, and Seifert. The identity is used to deduce a tautological
physical interpretation of TFTs in terms of the arrow of time, and its
generality reveals that the self-inverse nature of the various trajectory and
process transformations historically relied upon to prove TFTs, while necessary
for these theorems from a physical standpoint, is not necessary from a
mathematical one. The moment generating functions of thermodynamic variables
appearing in the identity are shown to converge in general only in a vertical
strip in the complex plane, with the consequence that a TFT that holds over
arbitrary timescales may fail to give rise to an asymptotic fluctuation theorem
for any possible speed of the corresponding large deviation principle. The case
of strongly biased birth-death chains is presented to illustrate this
phenomenon. We also discuss insights obtained from our measure-theoretic
formalism into the results of Saha et. al. on the breakdown of TFTs for driven
Brownian particles
Fluctuation Theorems for Entropy Production and Heat Dissipation in Periodically Driven Markov Chains
Asymptotic fluctuation theorems are statements of a Gallavotti-Cohen symmetry
in the rate function of either the time-averaged entropy production or heat
dissipation of a process. Such theorems have been proved for various general
classes of continuous-time deterministic and stochastic processes, but always
under the assumption that the forces driving the system are time independent,
and often relying on the existence of a limiting ergodic distribution. In this
paper we extend the asymptotic fluctuation theorem for the first time to
inhomogeneous continuous-time processes without a stationary distribution,
considering specifically a finite state Markov chain driven by periodic
transition rates. We find that for both entropy production and heat
dissipation, the usual Gallavotti-Cohen symmetry of the rate function is
generalized to an analogous relation between the rate functions of the original
process and its corresponding backward process, in which the trajectory and the
driving protocol have been time-reversed. The effect is that spontaneous
positive fluctuations in the long time average of each quantity in the forward
process are exponentially more likely than spontaneous negative fluctuations in
the backward process, and vice-versa, revealing that the distributions of
fluctuations in universes in which time moves forward and backward are related.
As an additional result, the asymptotic time-averaged entropy production is
obtained as the integral of a periodic entropy production rate that generalizes
the constant rate pertaining to homogeneous dynamics