Absolute Negative Mobility Induced by Thermal Equilibrium Fluctuations

A novel transport phenomenon is identified that is induced by inertial Brownian particles which move in simple one-dimensional, symmetric periodic potentials under the influence of both a time periodic and a constant, biasing driving force. Within tailored parameter regimes, thermal equilibrium fluctuations induce the phenomenon of absolute negative mobility (ANM), which means that the particle noisily moves backwards against a small constant bias. When no thermal fluctuations act, the transport vanishes identically in these tailored regimes. ANM can also occur in the absence of fluctuations on grounds which are rooted solely in the complex, inertial deterministic dynamics. The experimental verification of this new transport scheme is elucidated for the archetype symmetric physical system: a convenient setup consisting of a resistively and capacitively shunted Josephson junction device.

Figure 1

In panel (a) we present a small part of the bifurcation diagram of the noiseless system as a function of the amplitude $a\in (3.8,4.2)$. The ordinate indicates the stroboscopic velocity in the asymptotic long time limit. The system is biased by the positive force $f=0.1$. The other parameter values are $\gamma =0.9$ and $\omega =4.9$. The straight line at $v\approx 0.9$ corresponds to a stable locked state. It persists for all shown values of the driving amplitude. For amplitudes up to $a\approx 4.04$ a running state with negative average velocity coexists. In panel (b) the corresponding orbit is depicted for $a=3.99$. It is a period 2 state which stays within one potential well in the first period and moves to the left neighboring state in the second period. In panel (c) a realization of the stochastic system is displayed for $a=4.2$ and $D_0=0.001$. Its average velocity is negative. In the inset we show a typical part of the stochastic trajectory (thick line) which contributes to the transport. For several periods of the driving force it closely follows a deterministic unstable periodic orbit (thin line). This orbit lies on the unstable branch emanating from a pitchfork bifurcation at $a\approx 3.99$ of the running state shown in panel (b). A part of the bifurcation diagram including the first bifurcation of the period two orbit to a stable period four orbit (solid line) and the mentioned unstable period two orbit (dashed line) is shown in the magnification in panel (a).

Figure 2

The average velocity $⟨⟨v⟩⟩$ of an inertial Brownian particle described by Eq. (3) is depicted as a function of the external force $f$ for the deterministic (dashed line) and noisy (solid line) dynamics. The system parameters are $a=4.2$, $\omega =4.9$, $\gamma =0.9$, $D_0=0$ (dashed line), and $D_0=0.001$ (solid line). The absolute mobility defined as $⟨⟨v⟩⟩/f$ assumes a negative value for the noisy system in the range $|f|<0.17$. The most pronounced ANM occurs for small absolute values of the bias $f$. For the bias force $f\in (-f_{\mathrm{stall}},f_{\mathrm{stall}})$, with $f_{\mathrm{stall}}\approx \pm 0.17$, the Brownian particle moves opposite to the applied bias $f$.

Figure 3

The mobility coefficient $\mu =(\partial ⟨⟨v⟩⟩/\partial f)(f=0)$ depicted versus the dimensionless temperature strength, $D_0\propto T$, for three values of the cosine-driving strength $a$; the strength $a=4.2$ (solid line) corresponds to thermal noise-induced ANM. The driving strength $a=5.1$ (dashed line) corresponds to a regime exhibiting deterministic ANM, and $a=4.4$ (dotted line) is for a normal nonlinear response regime. The remaining parameters are $\omega =4.9$ and $\gamma =0.9$.