Nonequilibrium statistical mechanics of a two-temperature Ising ring with conserved dynamics

The statistical mechanics of a one-dimensional Ising model in thermal equilibrium is well-established, textbook material. Yet, when driven far from equilibrium by coupling two sectors to two baths at different temperatures, it exhibits remarkable phenomena, including an unexpected “freezing by heating.” These phenomena are explored through systematic numerical simulations. Our study reveals complicated relaxation processes as well as a crossover between two very different steady-state regimes.

Figure 1

Sketch of the two-temperature Ising ring model studied in this paper. Whereas exchanges across solid (blue online) borders are accepted with the ordinary Metropolis rate for a system at temperature $T$, exchanges across dashed (red online) borders are accepted with the corresponding rate for a system at temperature $T_w$.

Figure 2

Log-binning relaxation of the energy density $\langle \mathcal{H}\rangle /L$ for the standard Ising system at temperature $T=1$ and two different lattice sizes $L$. In this plot (as well as in the other figures below showing log-binning relaxation) the average of the quantity of interest is sampled at intervals between $2^x$ and $2^{x+1}$ MCS. The equilibrium value of $\langle \mathcal{H}\rangle /L$ is already very close to the value $-tanh\left(1\right)\approx -0.76$ of the infinite lattice. The data result from averaging over an ensemble of 40 to 100 independent realizations. Error bars are comparable to the sizes of the symbols.

Figure 3

Log-binning relaxation of the energy density for the standard Ising model with $L=512$. The data result from averaging over an ensemble of 40 to 100 independent realizations. Error bars are comparable to the sizes of the symbols.

Figure 4

Time trace of the standard Ising system after it reached equilibrium at $T=1$. The plot shows the evolution of the system over a total of 2000 MCS after an initial relaxation period of $2^{20}$ MCS, where two consecutive horizontal lines are separated by 10 MCS. The system size is $L=128$.

Figure 5

$G\left(r\right)$ for the standard equilibrium Ising ring with Kawasaki dynamics for a system of length $L=127$. $L$ is chosen to be odd to ensure a symmetric result about a given reference point. The lines refer to the approximation (12).

Figure 6

Equilibrium distributions of the normalized window magnetization $m$ for the standard Ising model with $L=128$ and $w=32$.

Figure 7

Time trace of the $2J$ model after it reached equilibrium at $T=1$. The plot shows the evolution of the system over a total of 2000 MCS after an initial relaxation period of $2^{20}$ MCS, where two consecutive horizontal lines are separated by 10 MCS. The system size is $L=128$. The boundaries between the regions are highlighted in cyan (gray).

Figure 8

Equilibrium distributions of the normalized window magnetization $m$ for the $2J$ Ising model with $L=128$ and $w=32$. Distributions for different temperatures are shown.

Figure 9

Time traces of the two-temperature model at different values of $T$. The traces are from a system of length $L=128$, with a window of size $w=32$ wherein the system is in contact with a reservoir at temperature $T_w=\infty$. For the left system we have $T=1$, whereas for the right system $T=1.75$. The boundaries between the regions are highlighted in cyan (gray). Both plots show the evolution of the system over a total of 2000 MCS after an initial relaxation period of $2^{20}$ MCS, where two consecutive vertical lines are separated by 10 MCS.

Figure 10

(a) Log-binning relaxation of the average energy density for $T=1$ with fixed system size $L=512$ and varying window size $w$ (yielding different values for the ratio $R=w/L$; see legend). The dashed vertical line is placed at the point at which the green curve with $w=128$ exits the metastable state, which is equivalent to ${(w/2)}^3={64}^3=2^{18}$. (b) Log-binning relaxation of the average energy density for fixed ratio $R=w/L=1/4$ and $T=1$. The dashed vertical lines are placed to correspond with ${(w/2)}^3$ for their respective curves (leftmost curve located at ${16}^3$, rightmost curve located at ${128}^3$).

Figure 11

Upper panel: Relaxation of the normalized average window magnetization for $R=1/4$ and $T=1$; see main text. Here the dashed vertical lines correspond with ${\left(2w\right)}^3$. Lower panel: The same for the standard Ising model.

Figure 12

Log-binning relaxation of the average energy density for $L=128$, $T=1$, and various values of the ratio $R=w/L$.

Figure 13

$G((w+1)/2,r)$ for the two-temperature Ising ring with $L=127$, $w=31$, and various values of $T$. Here the lattice site $i=(w+1)/2$ is the site in the middle of the window of width $w$. The end of the window is indicated by the dashed vertical line.

Figure 14

Average bond energy $u_i$ at site $i$ for $L=128$, $T=1$, and various ratios $R=w/L$. The solid magenta (gray) horizontal line indicates the equilibrium energy of the standard model at $T=1$. The dashed vertical lines are placed to demonstrate the end of the window region for the $R=1/4$ curve. For all curves the borders of the infinite temperature window are marked by minima in $u_i$ followed by discontinuous transitions to maxima outside the window.

Figure 15

Distributions for the normalized window magnetization $m$ with system parameters $L=128$ and $R=1/4$ ($w=32$). Results for different values of $T$ are shown.

Figure 16

Distributions for the normalized window magnetization $m$ with $L=128$ and $T=1$ and various window sizes.

Figure 17

Two cycles of configurations illustrating irreversibility and detailed balance violation in the $2T$ model (a, b). By comparison, these cycles are reversible in the $2J$ model (c, d). See text for explanation of symbols.