Anomalous heat equation in a system connected to thermal reservoirs

Abstract

We study anomalous transport in a one-dimensional system with two conserved quantities in presence of thermal baths. In this system we derive exact expressions of the temperature profile and the two point correlations in steady state as well as in the non-stationary state where the later describe the relaxation to the steady state. In contrast to the Fourier heat equation in the diffusive case, here we show that the evolution of the temperature profile is governed by a non-local anomalous heat equation. We provide numerical verifications of our results. AbstractWe here provide the details of calculations for the results presented in the main text. DYNAMICAL OPERATORS AND DISCRETE EQUATIONSThe explicit expression of the deterministic part of the Fokker Planck (FP) equation given by operators L l , L b , defined as

Anomalous Heat Transport in One Dimensional Systems: A Description Using Non-local Fractional-Type Diffusion Equation

“…The exact results of Eqs. (34) have been verified in [69] from direct numerical solution of Eqs. (30,31) and it was noted that density profiles were similar to the temperature profiles seen in AHT.…”

“…For the finite open system, solving the above equations exactly seems to be difficult. However, it was observed numerically [34] that for large N the temperature field T i (t) scales as T i (t) = T i N , t N 3/2 and the correlation field C i,j (t) scales as C i,j (t) = 1 √ N C |i−j| √ N , i+j 2N , t N 3/2 , i = j. Inserting these into (54), and expanding in powers of 1/ √ N , we find at leading order the following equations…”

“…In [30], it has been numerically shown that indeed j ∼ 1/ √ N . Recently, an understanding of the open system was achieved using the fractional equation description, which we now discuss [34]. An aspect that we will point out here is that the fractional-equation-type description in the open-set up is strongly dependent on boundary conditions (fixed or free or mixed).…”

“…It is important to note that non-linear temperature profiles can also be obtained in case of diffusive transport if the thermal conductivity κ is temperature-dependent and ∆T is large. On the other hand, for many systems with AHT, one finds a strongly non-linear temperature profile even when ∆T is made arbitrary small [5,10,11,26,34,36,65]. Quite often the profiles are characterized by divergent slopes at the boundaries.…”

“…(55,56,57), while the steady state part satisfies these equations but with ∂ τ T ss (v) = 0. The boundary conditions for the steady state part are given by [34] C ss (u, z → 0) = 0, C ss (u → ∞, z) = 0, C ss (u = 0, z) = J/2.…”