Langevin dynamics neglecting detailed balance condition

An improved method for driving a system into a desired distribution, for example, the Gibbs-Boltzmann distribution, is proposed, which makes use of an artificial relaxation process. The standard techniques for achieving the Gibbs-Boltzmann distribution involve numerical simulations under the detailed balance condition. In contrast, in the present study we formulate the Langevin dynamics, for which the corresponding Fokker-Planck operator includes an asymmetric component violating the detailed balance condition. This leads to shifts in the eigenvalues and results in the acceleration of the relaxation toward the steady state. The numerical implementation demonstrates faster convergence and shorter correlation time, and the technique of biased event sampling, Nemoto-Sasa theory, further highlights the efficacy of our method.

Figure 1

(Left) Orbits of the duplicated system until $t=5$. The horizontal and vertical axes denote the mean of the location. The red curve denotes the case of $\gamma =0$ and the blue one represents that of $\gamma =10.0$. (Right) Time evolution of the mean of the location. The horizontal axis represents time, and the vertical axis denotes the mean of the location. From top to bottom, we plot the cases of $\gamma =0$ (red crosses) and $\gamma =10.0$ (blue and purple circles).

Figure 2

Relaxation in magnetization (left panel) and internal energy (right panel) in KT phase. The horizontal axis represents time, and the vertical axis represents magnetization and internal energy. The cases of $\gamma =0$ (red crosses) and $\gamma =10.0$ (blue tilted crosses and purple squares) are plotted.