Nonequilibrium two-dimensional Ising model with stationary uphill diffusion

Abstract

Usually, in a nonequilibrium setting, a current brings mass from the highest density regions to the lowest density ones. Although rare, the opposite phenomenon (known as "uphill diffusion") has also been observed in multicomponent systems, where it appears as an artificial effect of the interaction among components. We show here that uphill diffusion can be a substantial effect, i.e., it may occur even in single component systems as a consequence of some external work. To this aim we consider the two-dimensional ferromagnetic Ising model in contact with two reservoirs that fix, at the left and the right boundaries, magnetizations of the same magnitude but of opposite signs.We provide numerical evidence that a class of nonequilibrium steady states exists in which, by tuning the reservoir magnetizations, the current in the system changes from "downhill" to "uphill". Moreover, we also show that, in such nonequilibrium setup, the current vanishes when the reservoir magnetization attains a value approaching, in the large volume limit, the magnetization of the equilibrium dynamics, thus establishing a relation between equilibrium and nonequilibrium properties.

Uphill migration in coupled driven particle systems

“…In this work we propose a simple model of independent random walkers on two rings, in which migration between driven and nondriven rings is observed and fully explained in terms of the typical times spent by particles in the two regions. The observed mass transport across different regions of the domain shares some interesting features with the uphill diffusion phenomenon discussed in [3][4][5] in the context of lattice gas or spin models, and with particle migration on a half-driven ladder [6].…”

Emergence of stationary uphill currents in 2D Ising models: the role of reservoirs and boundary conditions

“…This phenomenon was indeed observed and reproduced in [9] by performing extensive Monte Carlo simulations of the model (note that no violation of the thermodynamic principles occurs therein, as the total energy of the system is not a conserved quantity). A step forward was then taken in [5], in which a similar phenomenology was also observed in a 2D Ising model on a square lattice coupled to two external magnetization reservoirs attached at the right and left boundaries (whose magnetizations are equal, respectively, to m + > 0 and m − = −m + ). The presence of a stationary uphill current is induced by the reservoirs, whose updating mechanism at the boundaries breaks the condition of local detailed balance, cf.…”

“…1.6]. In particular, the authors show that, in presence of a ferromagnetic phase transition, a certain critical value m crit marks the transition between two different regimes (referred to, in [5], as the stable and metastable one), and characterized, respectively by downhill and uphill currents, when m + is, respectively, larger or smaller than m crit .…”

“…In [5] it is claimed that the critical value m crit , evaluated at β = 1, can also be estimated by measuring the magnetization value m eq (that approaches, in the large volume limit, the critical value m β in Eq. (2) [19,26]) evaluated at the rightmost column of an Ising model in equilibrium conditions (i.e.…”

A lattice model for active-passive pedestrian dynamics: a quest for drafting effects

“…With regards to the building up of the aerodynamic drag, it would also be interesting to verify the onset of the drafting phenomenon in lattice gas models in the presence of non-standard transport regimes leading to uphill diffusion of particles; see [18][19][20][21][22] for the study of such transport mechanisms.…”