Energy transport in an Ising disordered model

Abstract

We introduce a new microcanonical dynamics for a large class of Ising systems isolated or maintained out of equilibrium by contact with thermostats at different temperatures. Such a dynamics is very general and can be used in a wide range of situations, including disordered and topologically inhomogenous systems. Focusing on the two-dimensional ferromagnetic case, we show that the equilibrium temperature is naturally defined, and it can be consistently extended as a local temperature when far from equilibrium. This holds for homogeneous as well as for disordered systems. In particular, we will consider a system characterized by ferromagnetic random couplings J ij ∈ [1 − ǫ, 1 + ǫ]. We show that the dynamics relaxes to steady states, and that heat transport can be described on the average by means of a Fourier equation. The presence of disorder reduces the conductivity, the effect being especially appreciable for low temperatures. We finally discuss a possible singular behaviour arising for small disorder, i.e. in the limit ǫ → 0.

Dynamical and stationary critical behavior of the Ising ferromagnet in a thermal gradient

“…So, each column of the sample evolves, at its own temperature, towards its stationary configuration (t → ∞), and in particular we focus our attention on the critical region. According to equation (8) Figure 2 shows the time dependence of the second-order cumulant (see equations (5) and (14) On the other hand, by starting the simulations with fully disordered initial configurations such that m 0 = 0, which corresponds to samples in contact with a thermal bath at T = ∞ that are then suddenly annealed to a desired thermal gradient, we measured the time evolution of the fluctuations of the order parameter that under equi- librium conditions is identified with the magnetic susceptibility, as shown in figure 4.…”

“…As already established in the field of critical phenomena, the so-called cumulants (see equations (5) and (6)) are suitable functions of the moments of the order parameter distribution function whose pre-scaling factor, disregarding high-order finite-size scaling corrections, is independent of the system size. So, plots of the cumulants versus the control parameter (i.e., the temperature of the gradient thermal bath) should exhibit a common intersection point, as it is indicated with full straight vertical On the other hand, plots of the fluctuations of the order parameter, which are identified with the susceptibility in standard measurements, show peaks close to criticality as expected (see figure 9).…”

“…This physical situation is often addressed by means of analitical and numerical methods where the control parameter asuumes a fixed value. However, here we considered an alternative approach by assuming that along a given spatial direction, the system undergoes a well established gradient of the control parameter, such that the critical point lies between the extreme values, e.g., T 1 < T c < T 2 , where one has a thermal gradient between temperatures T 1 and T 2 , and the critical temperature T c is within that range [1,2,3,4,5].Of course, a standard reversible system, usually studied under equilibrium conditions, will be out of equilibrium under the gradient constraint applied to the control parameter. However, we will show below that this situation is no longer a shortcoming for the application of methods and theories already developed for the study of equilibrium systems.…”

“…It is worth mentioning that the above simulation studies [1,2,3,5] of the Ising system are precisely focused on the calculation of the thermal conductivity, among other relevant observables, such as the energy profiles of the systems transversal sections. Also, in all cases the discussed results were obtained under stationary conditions.…”

“…In fact, in the presence to thermal fluctuations it is expected that transport properties may exhibit non-trivial spatial patterns due to the non-equilibrium nature of the system under study [5], and under these conditions the thermal conductivity depends on the temperature (apart from other relevant parameters of the model). In particular, for the Ising system in a thermal gradient, Agliari et al [5] reports that K(T ) exhibits a peak slightly above criticality, whose position is independent of both the lattice size and the temperature difference between the extrems of the sample. Therefore, the system that we study in the present paper is not simply an Ising model where different temperatures are imposed at the bundaries, but it describes a system where the temperature gradient is present in the heat bath itself and the transport properies of the system are determined also by the bath and not only by the Ising dynamics.…”