Usefulness of an equal-probability assumption for out-of-equilibrium states: A master equation approach

We examine the effectiveness of assuming an equal probability for states far from equilibrium. For this aim, we propose a method to construct a master equation for extensive variables describing nonstationary nonequilibrium dynamics. The key point of the method is the assumption that transient states are equivalent to the equilibrium state that has the same extensive variables, i.e., an equal probability holds for microscopic states in nonequilibrium. We demonstrate an application of this method to the critical relaxation of the two-dimensional Potts model by Monte Carlo simulations. While the one-variable description, which is adequate for equilibrium, yields relaxation dynamics that are very fast, the redundant two-variable description well reproduces the true dynamics quantitatively. These results suggest that some class of the nonequilibrium state can be described with a small extension of degrees of freedom, which may lead to an alternative way to understand nonequilibrium phenomena.

Figure 1

Time evolutions of order parameter $m=2n_0-1$ (multiplied by $t^{1/2}$) in the all-to-all coupling Ising model. The decimal logarithm is shown on the horizontal axis. The result of the exact solution is denoted by “master eq.” and that obtained from the free-energy gradient is denoted by “free energy.” While the long-time behaviors of the two results are the same as $m\propto t^{-1/2}$, the short-time dynamics is significantly different.

Figure 2

(a) Energy dependence of velocity for the one-variable description. Colors denote temperatures. (b) Velocity observed in the KMC. Colors denote temperatures. Nonmonotonic behaviors are observed and finite-size dependence appears near the zero-crossing points of $V_E$.

Figure 3

(a) Time evolution of energy. Colors denote temperatures. The decimal logarithm is shown on the horizontal axis. The results of the one-variable description are denoted by lines, and those of KMC simulations are denoted by symbols. While the equilibrium values are the same, the short-time dynamics is significantly different between the two methods. (b) Relaxation of energy at the critical temperature. The decimal logarithm is shown on both axes. Colors denote temperatures. Three groups are shown: the KMC (symbol), the one-variable description (dashed line), and the two-variable description (solid line).

Figure 4

Confirmation of the power law at $\beta ={\beta }_c$. (a) The microcanonical velocity and (b) the microcanonical temperature. It is shown that $|V_E|$ and $|\beta -{\beta }_c|$ converge to finite values by multiplying $x^{1.5}$, where $x=|E-E_c|/N$, and taking the limit $x\to 0$. This means the two quantities are proportional to $|E-E_c{|}^{1.5}$.

Figure 5

(a) Time evolution of the internal energy. The results using the two-variable description (lines) and those obtained by the KMC method (symbols) are in good agreement. (b) Time evolution of the order parameter. The line with the slope $\beta /z\nu =0.0602$ [17] is also shown. The two-variable description reproduces the critical behavior well.

Figure 6

The trajectory in $(E\times N_0)$ space of ${\beta }_{\mathrm{ext}}=1.010>{\beta }_c$ with various initial conditions. The fixed point is indicated by the open circle at $(E,N_0)/N=(0.370,0.805)$. The dashed line denotes $h(E,N_0)=0$.

Figure 7

(a) Trajectories for the system $L=128$. The results obtained by the integration of the master equation are denoted by lines, and those obtained by the KMC method are denoted by symbols. The results of the two-variable description reproduce the true dynamics significantly well. The contour line $h(E,N_0)=0$ (dashed line) is also shown. (b) The contour line of $\beta \left(\vec{A}\right)$.