Abstract
We investigate the thermodynamic uncertainty relation for the (1+1) dimensional Kardar–Parisi–Zhang (KPZ) equation on a finite spatial interval. In particular, we extend the results for small coupling strengths obtained previously to large values of the coupling parameter. It will be shown that, due to the scaling behavior of the KPZ equation, the thermodynamic uncertainty relation (TUR) product displays two distinct regimes which are separated by a critical value of an effective coupling parameter. The asymptotic behavior below and above the critical threshold is explored analytically. For small coupling, we determine this product perturbatively including the fourth order; for strong coupling we employ a dynamical renormalization group approach. Whereas the TUR product approaches a value of 5 in the weak coupling limit, it asymptotically displays a linear increase with the coupling parameter for strong couplings. The analytical results are then compared to direct numerical simulations of the KPZ equation showing convincing agreement.
Highlights
Over the last years there has been remarkable progress in field theory with regard to the Kardar–Parisi–Zhang (KPZ) dynamics [1] on the one hand and in stochastic thermodynamics with respect to the thermodynamic uncertainty relation (TUR) on a discrete set of states [2,3,4] on the other hand
We have given an analytical description of the thermodynamic uncertainty relation depending on the coupling strength of the KPZ non-linearity, see (82)
In particular we showed that equaltime correlation functions, in the present case the steady state current J and the entropy production rate σ, can be obtained exactly via functional integration using the known steady state probability density functional ps[h] of the (1 + 1) dimensional KPZ equation, see (16) and (17), respectively
Summary
Over the last years there has been remarkable progress in field theory with regard to the Kardar–Parisi–Zhang (KPZ) dynamics [1] on the one hand and in stochastic thermodynamics with respect to the thermodynamic uncertainty relation (TUR) on a discrete set of states [2,3,4] on the other hand.
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