Interacting nonequilibrium systems with two temperatures

We investigate a simplified model of two fully connected magnetic systems maintained at different temperatures by virtue of being connected to two independent thermal baths while simultaneously being interconnected with each other. Using generating functional analysis, commonly used in statistical mechanics, we find exactly soluble expressions for their individual magnetization that define a two-dimensional nonlinear map, the equations of which have the same form as those obtained for densely connected equilibrium systems. Steady states correspond to the fixed points of this map, separating the parameter space into a rich set of nonequilibrium phases that we analyze in asymptotically high and low (nonequilibrium) temperature limits. The theoretical formalism is shown to revert to the classical nonequilibrium steady state problem for two interacting systems with a nonzero heat transfer between them that catalyzes a phase transition between ambient nonequilibrium states.

Figure 1

Visualization of the model as two planes $\sigma$ and $\tau$. The long-range intraplane interaction is $J$ and the short-range interplane interaction is $J^{\prime }$.

Figure 2

Fixed points of the magnetization $m^{\sigma }$ for $J=2$ and ${\tilde{\beta }}_{\sigma }=3$.

Figure 3

Dynamics of the 2D map of the magnetization for ${\beta }_{\tau }={10}^{-2}$, ${\beta }_{\sigma }={10}^2$, $J=-5$, and $J^{\prime }={10}^{-2}$. The period-2 cycle in $m_{\sigma }$ sets in already at the first iteration, while $m_{\tau }$ remains at zero mainly due to its very high temperature.

Figure 4

Partial phase diagram for the limiting cases of high and low temperatures. The figure shows only the first quadrant because the other three are simply its symmetrical images reflected about the axes. On the dashed axis, the corresponding magnetization is zero because the temperature is infinite.

Figure 5

Stability of the GP phase for high temperatures (small $\beta$).

Figure 6

Basin of attraction for values $J=2$, $J^{\prime }=1$, ${\beta }_{\sigma }={10}^3$, and ${\beta }_{\tau }=0$.