Transient chaos induces anomalous transport properties of an underdamped Brownian particle

For an underdamped Brownian particle in a one-dimensional periodic potential we theoretically predict three unusual transport properties: (i) A static bias force (of either sign) generates an average particle motion in the opposite direction. (ii) A small bias leads to a particle transport in the direction of the bias, but upon increasing the bias the particle velocity reverses direction. (iii) For a given bias force, the particle motion follows the direction of the force for low temperatures, but upon increasing the temperature reverses its direction. The considered model is shown to be minimal for the occurrence of these phenomena. A detailed analysis of its deterministic properties and the influence of thermal noise is carried out with numerical simulations that are complemented by analytical approximations. Intuitive explanations of the basic mechanism behind the three effects are provided; their origin is attributed to a subtle interplay between the stability of coexisting attractors, noise induced metastability, and transient chaos. An experimental system for the realization of the predicted effects is given within the Stewart-McCumber model for Josephson junctions. Suitable parameter values for which these effects can be observed are quite realistic experimentally.

Figure 1

(Color) Regions in the $(\eta ,A)$ plane with periodic and phase locked aperiodic attractors of the unbiased, deterministic, dissipative dynamics (1, 2, 5), i.e., $F=0$, $T=0$, $\eta >0$, $\omega =0.6$, obtained by numerical integration sampling many different initial conditions. Colors: average velocity $v$ from Eq. (3) on these attractors for a few dominating rational $v$ values. Black: other rational $v$ values. White: no phase locked attractors have been found. Overlapping of regions with different colors typically indicates coexistence of attractors. The larger black cross represents Eq. (7) and the smaller one Eq. (12).

Figure 2

(Color) (a) Magnification of the lowest red stripe in Fig. 1. The large black cross represents Eq. (7) and the small one represents Eq. (12). [(b)–(d)] Same except for $F>0$. Periodic and phase locked aperiodic attractors with $v=1$ are indicated in orange and those with $v=-1$ are indicated in blue, yielding red in the coexistence regions. Apart from this degeneracy due to symmetry (see also main text), the asymptotic velocities $v=\pm 1$ are unique, i.e., independent of initial conditions [39].

Figure 3

(Color online) Bifurcation diagrams of attractors (red) [36] and concomitant velocities (blue) versus damping strength $\eta$ from numerical solutions of the deterministic dissipative dynamics (1, 2, 3) with $T=0$, $\omega =0.6$, $A=1.24$ and the same values of the static bias $F$ as in Fig. 2. For any given $\eta$ and $F$, the red dots represent $x\left(k\tau \right)$ modulo $L$ (stroboscopic map of the reduced spatial dynamics) for a set of sufficiently large integers $k$ such that initial transients have died out. The dashed line at $\eta =0.465$ corresponds to the larger black cross in Fig. 2. Coexistence of different velocities $v$ implies coexistence of attractors, but not vice versa.

Figure 4

(Color online) Same as Fig. 3a, but keeping $\eta =0.465$ fixed and instead varying $F$. In addition, the numerically determined velocity $v$ from Eq. (3) for various $T$ is indicated in the lower part. The numerical uncertainty is of the order of the symbol size. The connecting lines serve as guides to the eye. Since $F\mapsto -F$ implies $v\mapsto -v$, negative $F$ are omitted.

Figure 5

(Color online) Time evolution of the unique attractor of the deterministic dynamics (1, 2) with $T=0$, $\omega =0.6$, $A=1.24$, $\eta =0.465$ [cf. Eqs. (5, 7)], and $F=0.1$. The black dots and red arrows indicate position and velocity of the stable period-one orbit $x\left(t\right)$ at nine time instances during one driving period. The blue curves represent the instantaneous total potential $V\left(x\right)-x[f\left(t\right)+F]+1$ according to Eqs. (1, 2).

Figure 6

(Color) Same as Fig. 1 but for $T=0.001$ and $F=0.1$. The value of the velocity $v$ from Eq. (3) is independent of initial conditions and indicated by the coloring.

Figure 7

(Color online) Symbols: Average velocity $v$ from Eq. (3) versus temperature $T$ from numerical solutions of Eqs. (1, 2, 5) with parameter values (7) (labeled by the larger black cross in Figs. 1, 2, 6) and with bias $F=0.1$. The numerical uncertainty is smaller than the symbol size. Red solid line: analytic approximation (8, 10, 11); green dashed line: analytic approximation (9, 10, 11), with velocities $v_{+}=0.88$ and $v_{-}=-1$, cf. Fig. 9. Arrow: analytical asymptotics $v\to 0.358$ for $T\to \infty$ according to Eq. (6).

Figure 8

(Color online) Typical numerical solution $x\left(t\right)$ of the dynamics (1, 2, 5), with parameter values (7) (labeled by the larger black cross in Figs. 1, 2, 6) and with $T=0.001$ and $F=0.1$. (a) Global behavior. (b) Magnification of the green window in (a). Dashed: straight lines with slopes $v_{+}=0.88$ and $v_{-}=-1$.

Figure 9

(Color) Stroboscopic representation of the phase locked periodic attractor with $v=-1$ (black crosses), whose time evolution is illustrated in Fig. 5, shown within three spatial periods of the full $(x,v)$ phase space (indicated by the dashed lines) for three consecutive driving periods. The colored regions represent the synchronous basin of attraction of the attractor (see main text) also for these three driving periods. The right cross and blue region correspond to the first, the middle cross and green region correspond to the second, and the left cross and red region correspond to the third driving period. This picture can be periodically repeated to the right and to the left to obtain the attractor and its synchronous basin of attraction for previous and later driving periods, respectively. The synchronous basins of attraction of all these periods have filaments that extend into the shown part of the phase space, and if one included them, they would completely cover the white region.

Figure 10

(Color online) Arrhenius plot of the escape rates $k_{-}$ and $k_{+}$ from the attractor (red squares) and the repeller (blue dots), respectively. Remaining parameter values: same as in Figs. 5, 7, 8, 9. The statistical uncertainties are of the order of the symbol sizes. The solid lines represent the data fits (8); the dashed lines show Eq. (9). All fits are based only on the escape rates for temperatures $T\lesssim 0.0025$, because in this temperature range the description of the noisy trajectories in terms of transitions between the “+” and “−” state with respective escape rates $k_{+}$ and $k_{-}$ is expected to be valid, as is self-consistently concluded from the obtained values in the exponents of Eqs. (8, 9).

Figure 11

(Color) Same as Fig. 6, but for $F=0.05$ and various $T$; blue indicates negative velocities with transport against the bias (see color panel in Fig. 6). (d) and (e) represent magnifications of the dashed box in (c), but for higher temperatures. Negative average velocities can still be observed for $T=0.04$. The black cross represents the parameter values (7).

Figure 12

(Color online) Average velocity $v$ for the given parameter values and for various temperatures, obtained from numerical simulations of Eqs. (1, 2). The indicated values for $\eta$ and $A$ are located at the upper border of a green region in Fig. 1 with deterministic transport velocities $\mid v\mid =3$.

Figure 13

(Color) Same as Fig. 2, but for the case of coexisting $v=0$ and $v=\pm \frac{1}{2}$ attractors. The indicated cross and the chosen frequency $\omega$ and bias force $F$ represent the parameter values used in Figs. 1 and 2 of Ref. 15; a vertical section through the cross corresponds to Fig. 1a therein. (Note that the units used by us and in Ref. 15 differ by some factors of $2\pi$.)

Figure 14

(Color online) Symbols: Numerically determined velocity $v$ from Eq. (3) for Eqs. (1, 2) with parameters as indicated in the plot. The numerical uncertainty is smaller than the symbol size. The red line is a guide to the eye.

Figure 15

(Color online) Same as Fig. 7, but for $F=0.007$. Lines: approximations (10, 11) with $v_{+}=2∕3$, $v_{-}=-1$, and $k_{+}=0.024\exp (-1.35\times {10}^{-7}∕T)$, $k_{-}=0.036$ (red solid line), $k_{-}=0.038-6.6\sqrt{T}$, $k_{+}=63\sqrt{T}\exp (-1.1\times {10}^{-7}∕T)$ (green dashed line).

Figure 16

(Color online) Same data as the green symbols in Fig. 4 but presented according to the Stewart-McCumber model (1, 2, 13, 14) for a Josephson junction with resistance $R\approx 0.2\Omega$, capacity $C\approx 250\mathrm{pF}$, critical current $I_c\approx 180\mu \mathrm{A}$, temperature $T\approx 4.2\mathrm{K}$, driven by an ac current of frequency $28\mathrm{GHz}$ and amplitude $220\mu \mathrm{A}$.