The Two Scaling Regimes of the Thermodynamic Uncertainty Relation for the KPZ-Equation

We investigate the thermodynamic uncertainty relation for the $$(1+1)$$ ( 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.

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

A general framework for the field-theoretic thermodynamic uncertainty relation was recently proposed and illustrated with the $$(1+1)$$ ( 1 + 1 ) dimensional Kardar–Parisi–Zhang equation. In the present paper, the analytical results obtained there in the weak coupling limit are tested via a direct numerical simulation of the KPZ equation with good agreement. The accuracy of the numerical results varies with the respective choice of discretization of the KPZ non-linearity. Whereas the numerical simulations strongly support the analytical predictions, an inherent limitation to the accuracy of the approximation to the total entropy production is found. In an analytical treatment of a generalized discretization of the KPZ non-linearity, the origin of this limitation is explained and shown to be an intrinsic property of the employed discretization scheme.

Large fluctuations of the KPZ equation in a half-space

We investigate the short-time regime of the KPZ equation in 1+11+1 dimensions and develop a unifying method to obtain the height distribution in this regime, valid whenever an exact solution exists in the form of a Fredholm Pfaffian or determinant. These include the droplet and stationary initial conditions in full space, previously obtained by a different method. The novel results concern the droplet initial condition in a half space for several Neumann boundary conditions: hard wall, symmetric, and critical. In all cases, the height probability distribution takes the large deviation form P(H,t) \sim \exp( - \Phi(H)/\sqrt{t})P(H,t)∼exp(−Φ(H)/t) for small time. We obtain the rate function \Phi(H)Φ(H) analytically for the above cases. It has a Gaussian form in the center with asymmetric tails, |H|^{5/2}|H|5/2 on the negative side, and H^{3/2}H3/2 on the positive side. The amplitude of the left tail for the half-space is found to be half the one of the full space. As in the full space case, we find that these left tails remain valid at all times. In addition, we present here (i) a new Fredholm Pfaffian formula for the solution of the hard wall boundary condition and (ii) two Fredholm determinant representations for the solutions of the hard wall and the symmetric boundary respectively.