Martingale drift of Langevin dynamics and classical canonical spin statistics

Abstract

The martingale characterizes a kind of fairness or unbiased nature of the stochastic process which is associated with another stochastic process. If $x_t$ evolves according to the Langevin equation whose mean drift is $a_t$ as function of $x_t,$ and that $a_t$ as induced stochastic process is martingale in turn associated with the former process, then we show that the amplitude of $a_t$ is the Langevin function, which is originally the canonical response of a single classical Heisenberg spin under static field. Furthermore, the asymptotic limit of $x_t/t$ obeys the ensemble statistics of such Heisenberg spin.

Figures (3)

• Figure 1: 2D trajectories from ${\vec{x}}_0=(0_+,0_+).$ (Left) Close-up view near the origin. (Right) Whole trajectories up to $t=16.$
• Figure 2: 2D trajectories from ${\vec{x}}_0=(x_0,0_+)$ with (a) $x_0=0$, (b) $x_0=1$, (c) $x_0=2$ and (d) $x_0=4.$ In all cases the duration is up to $t=40$ and we used the identical set of noise histories for $\vec{\xi}_t$ in (\ref{['eq:eqL']})
• Figure 3: The 2D orientational distributions of $\lim_{t\to\infty}({{\vec{x}}_t}/{t})={\vec{a} } ^{\,*}_\infty$ for the different starting points, $x_0=0,1,2,$ and $4,$ from top to bottom. We represent the orientation of ${\vec{a} } ^{\,*}_\infty$ using $\cos\theta,$ where $\theta$ is the angle between ${\vec{a} } ^{\,*}_\infty$ and ${\vec{x}}_0=(x_0,0),$ and we show the distributions by the empirical cumulative probability. Numerically the ensemble of ${\vec{a} } ^{\,*}_\infty$ is approximated by 3000 realisations of ${\vec{x}}_t/t|_{t=40}.$ The red dashed curves represent the canonical equilibrium distribution of a single unitary spin under the non-dimensionalized field ${\vec{x}}_0,$ see (\ref{['eq:cancumul']}) in the main text.