Dynamic Correlators of Fermi-Pasta-Ulam Chains and Nonlinear Fluctuating Hydrodynamics

Abstract

We study the equilibrium time correlations for the conserved fields of classical anharmonic chains and argue that their dynamic correlator can be predicted on the basis of nonlinear fluctuating hydrodynamics. In fact, our scheme is more general and would also cover other one-dimensional Hamiltonian systems, for example, classical and quantum fluids. Fluctuating hydrodynamics is a nonlinear system of conservation laws with noise. For a single mode, it is equivalent to the noisy Burgers equation, for which explicit solutions are available. Our focus is the case of several modes. No exact solution has been found so far, and we rely on a one-loop approximation. The resulting mode-coupling equations have a quadratic memory kernel and describe the time evolving 3×3 correlator matrix of the locally conserved fields. Long time asymptotics is computed analytically, and finite time properties are obtained through a numerical simulation of the mode-coupling equations.

Fluctuating Hydrodynamics Approach to Equilibrium Time Correlations for Anharmonic Chains

“…When starting this project together with Christian Mendl, in the summer of 2012 we spent many days in numerically simulating the mode coupling equations with initial conditions S αα (j, 0) = δ αα δ 0j . Only a few plots are in print [38], simply because there is such a large parameter space and it is not clear where to start and where to end. Still, for our own understanding this period was extremely helpful.…”

Crossover between different universality classes: Scaling for thermal transport in one dimension

“…Transport in one dimension has, for a long time, been realized to be anomalous in most cases [1,2], with signatures of a universal power-law scaling of transport coefficients, among which the heat transport has been extensively investigated in the recent decades, both by various theoretical techniques, such as the renormalization group [3], mode coupling [4,5] or cascade [6][7][8], nonlinear fluctuating hydrodynamics [9][10][11], and Lévy walks [12][13][14][15]; and also by computer simulations [16][17][18][19][20][21][22][23][24][25][26]. For all studied cases two main scaling exponents have been given the most focus, i.e., α describing the divergence of heat conductivity with space size L as L α and γ characterizing the space(x)-time(t) scaling of heat spreading density ρ(x, t) as t −1/γ ρ(t −1/γ x, t).…”

Shocks, Rarefaction Waves, and Current Fluctuations for Anharmonic Chains

“…For the Riemann problem their explicit form is not needed. Setting D = (∂ ρ , ∂ v ), one obtains for the change of c σ along the vector fields ψ σ , 10) and strictly positive for ρ > 0.…”

“…We study here Fermi-Pasta-Ulam type anharmonic chains with domain-wall initial conditions and are not aware of any previous systematic study. The structure of equilibrium timecorrelations for such chains has been elucidated only recently [9,10]. In particular one now understands the link to anomalous transport which is most directly observed when coupling the chain to thermal reservoirs at distinct temperatures, see [11].…”