Semiclassical master equation in Wigners phase space applied to Brownian motion in a periodic potential

The quantum Brownian motion of a particle in a cosine periodic potential $V\left(x\right)=-V_0\cos (x∕x_0)$ is treated using the master equation for the time evolution of the Wigner distribution function $W(x,p,t)$ in phase space $(x,p)$. The dynamic structure factor, escape rate, and jump-length probabilities are evaluated via matrix continued fractions in the manner customarily used for the classical Fokker-Planck equation. The escape rate so yielded is compared with that given analytically by the quantum-mechanical reaction rate solution of the Kramers turnover problem. The matrix continued fraction solution substantially agrees with the analytic solution.

Figure 1

The real and imaginary parts of the normalized dynamic structure factor $\tilde{S}(k,\omega )∕\tilde{S}(k,0)$ vs $\omega \eta$ for various values of barrier parameter $g=5$, 7, 9, and 11; the damping coefficient ${\gamma }^{\prime }=10$ and $k=0.2$. Solid and dashed lines: the matrix continued fraction solution with $\Lambda =0.02$ and $\Lambda =0$ (classical case), respectively. Stars and open diamonds: Eq. (25) with $\Lambda =0.02$ and $\Lambda =0$, respectively.

Figure 2

The normalized longest relaxation time $\tau ∕\eta$ vs ${\gamma }^{\prime }$ for the barrier parameter $g=5$ and various values of the quantum parameter $\Lambda =0$ (classical case), 0.01, 0.02, and 0.03. Solid lines: the universal equation (17). Dashed lines: the IHD equation (21). Open circles: the matrix continued fraction solution of the master equation (4). Symbols: the matrix continued fraction solution of Eq. (4) with the constant diffusion coefficient $D_{pp}=\gamma m∕\beta$.

Figure 3

$\tau ∕\eta$ vs ${\gamma }^{\prime }$ for various values of the barrier parameter $g=5$, 7, 9, and 11. Solid and dotted lines: the universal equation (17) for $\Lambda =0.02$ and $\Lambda =0$ (classical case), respectively. Dashed lines: the IHD equation (21) for $\Lambda =0.02$. Open circles: the matrix continued fraction solution of Eq. (4). Symbols: the matrix continued fraction solution of Eq. (4) with the constant diffusion coefficient $D_{pp}=\gamma m∕\beta$.

Figure 4

The jump-length probabilities $P_n$ and $P_n^M$ for the barrier parameter $g=5$, damping parameter ${\gamma }^{\prime }=0.1$, and two values of the quantum parameter $\Lambda =0$ (up and down triangles; classical case) and $\Lambda =0.02$ (stars and crosses). Up triangles and stars: Eq. (14); down triangles and crosses: Eq. (24).