Figure 1

(Color online) Schematic picture of the Brownian particle in the tilted periodic potential $U\left(x\right)=V\left(x\right)-Fx$, defined in Eqs. (1, 17) below. $L$ denotes the period of the potential $V\left(x\right)$, and $L_r$ stands for the length of free slide between the barriers.Reuse & Permissions

Figure 2

(Color online) Average velocity $⟨v⟩$ (a), effective diffusion $D_{eff}$ (b), and Peclet number Pe (c) are drawn versus tilt $F$. Parameters of the potential are $\epsilon =100$, $\Delta =10$, which result in a barrier height of $\mathrm{\Delta }V=0.1$ in the absence of a tilt $(F=0)$ and a critical tilt $F_c\simeq 0.6$ above which all barriers disappear. The rescaled inverse temperature is set to $\beta =100$, corresponding to a small quantum parameter ${\lambda }_0\simeq 0.00115129$. The mean velocity and Peclet number are always larger in the quantum case, even if the classical effective diffusion is smaller (as is the case for $F\lesssim 0.25$). With panel (d) we depict the corresponding classical, $U\left(x\right)$ (dashed line), quantum effective, $U_{eff}\left(x\right)$ (solid line), and thermodynamic, $\psi \left(x\right)$ (dotted line) tilted potentials, respectively. The position-dependent diffusion coefficient is plotted in panel (e).Reuse & Permissions

Figure 3

(Color online) Average velocity (a), effective diffusion (b), and Peclet number (c) versus the inverse temperature ${\beta }_0$. The chosen parameters are bias $F=0.2$ and potential parameters $\epsilon =100$ and $\Delta =10$. The two arrows in panel (c) mark the characteristic temperatures where the system starts to markedly deviate from the classical behavior (see text for details). The region of intermediate temperatures is located between the downward- and upward-pointing arrows. It bridges between the regions of freely sliding Brownian particle dynamics, characterized by the Peclet number $\mathrm{Pe}=FL{\beta }_0$ [see the thin, dotted line in (c)] and Poisson-like behavior with $\mathrm{Pe}=2$. Within this region the Peclet number assumes its maximal value.Reuse & Permissions

Figure 4

The influence of quantum corrections is illustrated as a function of bias $F$ vs inverse temperature ${\beta }_0$ by the relative difference between the quantum $\left(Q\right)$ and the corresponding classical $\left(C\right)$ values of the current $({⟨v⟩}^Q-{⟨v⟩}^C)∕{⟨v⟩}^C$ (a), the effective diffusion $(D_{eff}^Q-D_{eff}^C)∕D_{eff}^C$ (b), and the Peclet number $({\mathrm{Pe}}^Q-{\mathrm{Pe}}^C)∕{\mathrm{Pe}}^C$ (c), respectively. The positive values of the depicted quantities indicate the relevant role of quantum effects.Reuse & Permissions

Figure 5

(Color online) The influence of the quantum corrections on the evolution of the coarse-grained probability density function $P(x,t)$ (see discussion in [22]) is illustrated for the same set of parameters as in Fig. 2 and for different forces $F=0.2$ and 0.3. Note that in the quantum case the velocity is always larger; therefore, the maximum of the density is located at a more distant position as compared to the classical case. The width of the density differs depending on the driving parameters and thereby reflects the effective diffusion strength.Reuse & Permissions