Field-Theoretic Thermodynamic Uncertainty Relation

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.

Numerical Study of the Thermodynamic Uncertainty Relation for the KPZ-Equation

“…For other field-theoretic formulations of stochastic thermodynamic concepts, in particular the stochastic entropy production, see, e.g., [27,28,29]. The derivation of the KPZ-TUR in [25] relies on a perturbational approximation in a small effective coupling parameter of the KPZ non-linearity. This approach is quite generally applicable to stochastic field-theoretic overdamped Langevin equations.…”

Quantitative analysis of non-equilibrium systems from short-time experimental data

“…As we discuss in the supplemental material however, there is another component of the entropy production, related to the work that the flow does against the confining potential [37,62]. This component, which does indeed depend on the value of u d , is not estimated by our inference scheme, due to the fact that u d is a field (corresponding to the velocities of the molecules of the thermal bath) which is odd under time reversal, for which the TUR does not hold [46,52,[63][64][65]]. Hence we expect that the values of σ we find close to the bubble are underestimates of the true value.…”

Variational approach to KPZ: Fluctuation theorems and large deviation function for entropy production

“…(19) (by considering arbitrary, however reasonable, forms of P 1 and P 2 ) as well as the LDF scaling, together with the analysis of possible relations allowing us to determine-or at least to give a range to bound-the critical exponents (for instance, as numerically evaluated in Refs. [37][38][39] or to determine general thermodynamic uncertainty relations 25 will be the subject of forthcoming work.…”

“…As indicated in Ref. 25, we do not explicitly state an expression for the Jacobian ensuing from the variable transformation from noises to heights, as it turns out that such a Jacobian does not contribute to the entropy relations. For the inverse or backward trajectories, we have…”

“…21,22 Until very recently, these two aspects have not been analyzed together, that is, studies on the stochastic thermodynamics for the KPZ model are scarce. [23][24][25] At variance with the one dimensional study just done in Ref. 24, we want here to exploit the variational approach indicated above and introduced in Refs.…”