Abstract
We propose a field-theoretic thermodynamic uncertainty relation as an extension of the one derived so far for a Markovian dynamics on a discrete set of states and for overdamped Langevin equations. We first formulate a framework which describes quantities like current, entropy production and diffusivity in the case of a generic field theory. We will then apply this general setting to the one-dimensional Kardar–Parisi–Zhang equation, a paradigmatic example of a non-linear field-theoretic Langevin equation. In particular, we will treat the dimensionless Kardar–Parisi–Zhang equation with an effective coupling parameter measuring the strength of the non-linearity. It will be shown that a field-theoretic thermodynamic uncertainty relation holds up to second order in a perturbation expansion with respect to a small effective coupling constant. The calculations show that the field-theoretic variant of the thermodynamic uncertainty relation is not saturated for the case of the Kardar-Parisi-Zhang equation due to an excess term stemming from its non-linearity.
Highlights
The thermodynamic uncertainty relation (TUR) in a non-equilibrium steady state (NESS) provides a bound on the entropy production in terms of mean and variance of an arbitrary current [1]
In our manuscript we use these current-like observables and we focus on the original TUR
We will show that the thermodynamic uncertainty relation from (24) holds for the KPZ equation driven by Gaussian white noise in the weak-coupling regime
Summary
The thermodynamic uncertainty relation (TUR) in a non-equilibrium steady state (NESS) provides a bound on the entropy production in terms of mean and variance of an arbitrary current [1]. A temporal coarse-graining procedure is described, which allows the formulation of a discrete Markov jump process in terms of an overdamped Langevin equation for the mesoscopic states of the model These authors observe that for purely dissipative dynamics the TUR is saturated. We present a field-theoretic equivalent to the TUR Such a thermodynamic uncertainty relation for general field-theoretic Langevin equations may prove helpful in further understanding complex dynamics like turbulence for fluid flow or non-linear growth processes, described by the stochastic Navier-Stokes equation
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