Violation of detailed balance accelerates relaxation

Recent studies have experienced the acceleration of convergence in Markov chain Monte Carlo methods implemented by the systems without detailed balance condition (DBC). However, such advantage of the violation of DBC has not been confirmed in general. We investigate the effect of the absence of DBC on the convergence toward equilibrium. Surprisingly, it is shown that the DBC violation always makes the relaxation faster. Our result implies the existence of a kind of thermodynamic inequality that connects the nonequilibrium process relaxing toward steady state with the relaxation process which has the same probability distribution as its equilibrium state.

Figure 1

Since ${\lambda }_1^W={\lambda }_1^S=0$ and $d\det (\lambda E_N-W)/d\lambda {{|}_{\lambda =0}\ge d\det (\lambda E_N-S)/d\lambda |}_{\lambda =0}$, $\det (\lambda E_N-W)$ (the dashed line) is smaller than $\det (\lambda E_N-S)$ (the solid line) for negative $\lambda$ sufficiently close to $\lambda =0$. Since $|\det (\lambda E_N-W)|=\det (\mathrm{Re}\lambda E_N-S)$ does not hold in the region ${\lambda }_2^S<\mathrm{Re}\lambda <0$ according to (4), the two lines cannot cross each other in this region. The filled and dotted circles denote ${\lambda }_2^S$ and ${\lambda }_2^W$, respectively.

Figure 2

Relaxations of $\langle m\left({\tau }_{\mathrm{MC}}\right)\rangle$ in the 1D Ising model for several values of asymmetry parameter $\gamma$. The upper and lower figures depict the cases of $\beta J=0.1$ and $0.02$, respectively.