Fractional translational diffusion of a Brownian particle in a double well potential

The fractional translational diffusion of a particle in a double-well potential (excluding inertial effects) is considered. The position correlation function and its spectrum are evaluated using a fractional probability density diffusion equation (based on the diffusion limit of a fractal time random walk). Exact and approximate solutions for the dynamic susceptibility describing the position response to a small external field are obtained. The exact solution is given by matrix continued fractions while the approximate solution relies on the exponential separation of the time scales of the fast “intrawell” and low overbarrier relaxation processes associated with the bistable potential. It is shown that knowledge of the characteristic relaxation times for normal diffusion allows one to predict accurately the anomalous relaxation behavior of the system for all relevant time scales.

Figure 1

Potential $V\left(y\right)=A{y}^2+B{y}^4$ for various values of $A∕\left(2B^{1∕2}\right)=-\sqrt{2}$ (curve 1), $-1$ (2), $-1∕\sqrt{2}$ (3), 0 (4), and $1$ (5).

Figure 2

The real ${\chi }^{\prime }$ and imaginary ${\chi }^{″}$ parts of the dynamic susceptibility vs. $\omega \tau$ for the barrier height $Q=6$ and various values of the fractional exponent $\sigma =1$ (normal diffusion), 0.8, and 0.6. Solid lines are the matrix continued fraction solution. Filled circles: the bimodal approximation, Eq. (17). Dashed and dotted lines: the low and high frequency asymptotes, Eqs. (14, 15).

Figure 3

${\chi }^{\prime }$ and ${\chi }^{″}$ vs. $\omega \tau$ for $\sigma =0.8$ and various values of $Q=3$, 6, and 8. Key as in Fig. 2.