A stochastic model of anomalous heat transport: analytical solution of the steady state

Abstract

Abstract. We consider a one-dimensional harmonic crystal with conservative noise, in contact with two stochastic Langevin heat baths at different temperatures. The noise term consists of collisions between neighbouring oscillators that exchange their momenta, with a rate γ. The stationary equations for the covariance matrix are exactly solved in the thermodynamic limit (N → ∞). In particular, we derive an analytical expression for the temperature profile, which turns out to be independent of γ. Moreover, we obtain an exact expression for the leading term of the energy current, which scales as 1/ √ γN . Our theoretical results are finally found to be consistent with the numerical solutions of the covariance matrix for finite N .

Anomalies and absence of local equilibrium, and universality, in one-dimensional particles systems

“…[27,28]. Taking of k 2 = 2 does not yield profiles which approximate the theoretical one of [27] more closely than k 2 = 1. The only result which is robust against all the modifications of the dynamics, which we have investigated, is the validity of the linear relation (5), which must then play an important role in the identification of universality classes and in the equivalence of thermostats in 1-d systems.…”

“…The dependence on p is quite strong although, in both cases, the profiles settle around the unperturbed ones, when p decreases. The cases with lower temperature difference more rapidly collapse on a unique shape, different from the theoretical T of [27]. Figure 14, shows that increasing the rigidity of the chains, reduces the effect of the parameter p. Comparing Figs.…”

“…the discussion in Ref. [27,46]. In any event, the strong and irregular dependence of the steady states of both deterministic and stochastic 1-d systems, on microscopic details of the dynamics, as well as the violations of Fourier Law, reveal the absence of genuine local thermodynamic equilibrium and of diffusion, see e.g.…”

“…To investigate this question, we have analyzed the case with k 2 = 1, T ℓ = 320, T r = 20, θ ℓ = θ r = 1, for N = 100, 200, 250, 300, 400. Differently from [27], we found no evidence of the N −1/3 rate of convergence towards the asymptotic profile T (x) and, in most of the chain, the difference between our profiles and T (x) was observed to grow with N. Therefore, as shown in Appendix A, either a scaling regime towards T (x) sets in at N larger than 400, or our asymptotic profiles will not correspond to those of [27]. On the other hand, Appendix A shows that our profiles converge rapidly to a given asymptotic profile, which is not odd.…”

Nonequilibrium generalised Langevin equation for the calculation of heat transport properties in model 1D atomic chains coupled to two 3D thermal baths

“…In order to obtain a diffusive transport regime, one has to introduce any form of anharmonic effects in the system 10,[12][13][14]16,17,22,24,26,27,[29][30][31][32][33][34][35][36][37][38] . Diffusive transport is also obtained when extra local stochastic processes 9,10,15,17,18,26,32,41 or extra collision processes 42,43 are introduced. A vibrational mode coupling in classical systems 45 is also responsible for diffusive transport.…”