Anomalous mobility of Brownian particles in a tilted symmetric sawtooth potential

Overdamped motion of Brownian particles in a 1D periodic system with a simple symmetric sawtooth potential subjected to both unbiased thermal noise and spatially nonhomogeneous three-level colored noise is considered analytically. Upon application of a tilting force the particles exhibit anomalous transport properties, namely, absolute negative mobility, negative differential mobility, and the phenomenon of hypersensitive differential response. It is established that the mobility (differential mobility included) depends nonmonotonically on the parameters (switching rate, amplitude, and temperature) of nonequilibrium and thermal noises. The necessary conditions for various anomalous transport properties are found.

Figure 1

Schematic representation of different states and their transitions in the model (6) with the sawtooth-like potential (10) at low temperatures. The lines depict the net potentials $V_n\left(x\right)=V\left(x\right)-Fx-z_{n}fx$ with $z_1=-1$, $z_2=0$, and $z_3=1$. Unbiased transitions with a switching rate $\nu$ can take place between the discrete states, but only at specific positions, namely, in the interval $x∊(0,\frac{1}{2})$, modulo 1 between $V_2$ and $V_3$, and in the interval $x∊(\frac{1}{2},1)$ modulo 1 between $V_2$ and $V_1$. All quantities are dimensionless with scaling by Eqs. (4, 5). (a) The case of $F=1$, $f=4$. (b) The case of $F=1.6$, $f=3$.

Figure 2

The current $J$ vs the applied force $F$ in the region of anomalous mobility. (a) The general case [Eq. (A1)]. Dashed line (1): $D={10}^{-2}$, $\nu ={10}^{-2}$, and $f=1.9$. Dotted line (2): $D={10}^{-2}$, $\nu ={10}^{-2}$, and $f=3$. Solid line (3): $D={10}^{-7}$, $\nu ={10}^3$, and $f=3$. Note that in the cases of curves (2) and (3) the phenomenon of absolute negative mobility occurs. (b) The case of an adiabatic limit [Eqs. (A3, A4, A5, A6, A7)]. Here the temperature $D$ is zero and the curves correspond to (1) $f=1.5$, (2) $f=3$, (3) $f=5$, and (4) $f=7$. All quantities are dimensionless with scaling by Eqs. (4, 5).

Figure 3

The mobility $m_0$ vs temperature $D$ at various noise amplitudes $f$ in the case of an adiabatic limit [Eq. (24)]. Note that in the case of $f<2$ the mobility is positive for all values of $D>0$. At large values of the temperature, $D>1$,the mobility $m_0$ saturates to the value 1. The dots were computed by means of the exact equation (A1) with $f=6$, $\nu ={10}^{-6}$. All quantities are dimensionless with scaling by Eqs. (4, 5).

Figure 4

The mobility $m_0$ vs the switching rate $\nu$ at various temperatures $D$ and $f=3$. The curves with $D=0.1$ and with $D=0.01$ are computed from the exact formula (A1) for the current $J$. The line with $D=0$ corresponds to the asymptotic equation (28). All quantities are dimensionless with scaling by Eqs. (4, 5).

Figure 5

The current $J$ vs applied force $F$ in the region of negative differential resistance. The curves are computed from the exact equation (A1) at $f=3$. Solid line (1): $D=3\times {10}^{-9}$, $\nu ={10}^8$. Solid line with filled dots (2): $D={10}^{-6}$, $\nu ={10}^{12}$. Dotted line (3): $D=5\times {10}^{-3}$, $\nu ={10}^9$. Dashed line (4): $D={10}^{-5}$, $\nu ={10}^3$. Dashed-dotted line (5): $D=3\times {10}^{-9}$, $\nu ={10}^4$. The filled dots were computed by means of the asymptotic formula (A8). Note the hypersensitive response (jumps) of the current in the cases of curves (1), (2), and (5). All quantities are dimensionless with scaling by Eqs. (4, 5).