Enhanced diffusion with abnormal temperature dependence in underdamped space-periodic systems subject to time-periodic driving

We present a study of the diffusion enhancement of underdamped Brownian particles in a one-dimensional symmetric space-periodic potential due to external symmetric time-periodic driving with zero mean. We show that the diffusivity can be enhanced by many orders of magnitude at an appropriate choice of the driving amplitude and frequency. The diffusivity demonstrates abnormal (decreasing) temperature dependence at the driving amplitudes exceeding a certain value. At any fixed driving frequency $\mathrm{\Omega }$ normal temperature dependence of the diffusivity is restored at low enough temperatures, $T<T_{\mathrm{TAD}}\left(\mathrm{\Omega }\right)$—in contrast with the problem with constant external driving. At fixed temperature at small driving frequency the diffusivity either slowly decreases with $\mathrm{\Omega }$, or (at stronger driving) goes through a maximum near ${\mathrm{\Omega }}_2$, the reciprocal superdiffusion regime termination time. At high frequencies, between ${\mathrm{\Omega }}_2$ and a fraction of the oscillation frequency at the potential minimum, the diffusivity is shown to decrease with $\mathrm{\Omega }$ according to a power law, with the exponent related to the transient superdiffusion exponent. This behavior is found similar for the cases of sinusoidal in time and piecewise constant periodic (“square”) driving.

Figure 1

Dependence of the dimensionless diffusivity on the external constant driving at different temperatures (top). The region of anomalous temperature dependence (II) is hatched. The friction coefficient ${\gamma }^{\prime }=0.2$. 1: $T^{†}=0.097 [T=90\mathrm{K}$ for the potential characteristic of the H-adatom diffusion on metal surface (6)]; 2: $T^{†}=0.129$; 3: $T^{†}=0.194$; 4: $T^{†}=0.388$; 5: $T^{†}=0.582$; 6: $T^{†}=0.776$; 7: $T^{†}=0.969$.

Bottom: the effective velocity potential in the three force regions. Left, middle, and right diagrams correspond to $F_e/F_0=0.06$, 0.095, and 0.25 respectively.

Figure 2

Time dependence of the dimensionless dispersion for the force value $F_e=0.06F_0$ (region I) for different force frequencies. ${\tau }_2$ denotes the time when superdiffusion transitions to normal diffusion at constant driving ($\omega =0$). $\tau =2\pi /\omega$ is the period of the external periodic driving. $T^{†}=0.194$. 1: $\omega =1$; 2: $\omega ={10}^{-1}$; 3: $\omega ={10}^{-2}$; 4: $\omega =0$ (constant external force). Straight dashed and dotted lines show the power-law fitting behavior of the dispersion.

Figure 3

Dependence of diffusivity $D^{\prime }$ on the driving frequency at $T^{†}=0.129$ (triangles) and 0.194 (circles). The dashed line asymptotes show the diffusivity values at constant driving with the same $F_e/F_0=0.06$. The slanting dotted line shows the power-law drop of $D^{\prime }$ at intermediate $\omega$'s.

Figure 4

Dispersion variation with the temperature, at constant driving ($\omega =0$). 1: $T^{†}=0.097$; 2: $T^{†}=0.129$; 3: $T^{†}=0.194$; 4: $T^{†}=0.388$. The short-dashed straight lines show power-law fits for the transient superdiffusion regime, and the late-time normal diffusion.

Figure 5

Time dependence of the dimensionless dispersion for the force value $F_e=0.095F_0$ (region II) for different force frequencies. Curves 1–5, bottom to top: (dimensionless) $\omega ={10}^{-1}, {10}^{-2}, {10}^{-3}, {10}^{-4}$, 0 (constant driving). $T^{†}=0.194$.

Figure 6

Dependence of diffusivity $D^{\prime }$ on the external force frequency $\omega$, at $T^{†}=0.129$ (triangles) and 0.194 (circles). The dashed line asymptotes show the diffusivity values for constant driving with the same $F_e/F_0=0.095$. Dotted line shows the power-law drop of $D^{\prime }$ at higher $\omega$'s. The dot-dashed horizontal line asymptote (for high $\omega$) is at the diffusivity value in the lattice periodic potential in the absence of extra driving, $F_t\left(t\right)=0$, for $T^{†}=0.194$.

Figure 7

The dispersion variation with the temperature, at constant driving ($\omega =0$). 1: $T^{†}=0.097$; 2: $T^{†}=0.129$; 3: $T^{†}=0.194$; 4: $T^{†}=0.388$. $F_e/F_0=0.095$.

Figure 8

Time dependence of the dimensionless dispersion at $F_e=0.25F_0$ (region III) for different force frequencies. Curves 1–5: (dimensionless) $\omega =1, {10}^{-1}, {10}^{-2}, {10}^{-3}, {10}^{-4}$. $T^{†}=0.194$. Dotted lines show power-law fitting behavior of the dispersion.

Figure 9

Dependence of the dimensionless diffusivity $D^{\prime }$ on the driving frequency, at $T^{†}=0.129$ (triangles) and 0.258 (circles). The dashed lines show approximately linear growth at $\omega \in ({\omega }_2;{\omega }_3)$.

Figure 10

Evolution of the diffusivity and the kinetic temperature for high and small frequencies of the driving, $\omega =2$ and ${10}^{-2}$. $F_e/F_0=0.06$. Note that time normalized to one driving period $\tau$ (rather than to ${\mathcal{T}}_0$ as in the previous $t$-dependence graphs) is shown along the abscissa axis.

Figure 11

Dimensionless diffusivity vs $1/T$ at left: $\omega ={10}^{-2}$, force amplitudes $F_e/F_0=$ 0 (squares, and linear fit), 0.06 (circles), 0.095 (diamonds), 0.25 (triangles); right: $F_e/F_0=0.095$, top to bottom, $\omega =$ 0 (squares), ${10}^{-3}$ (triangles), ${10}^{-2}$ (diamonds), $3\times {10}^{-2}$ (circles), ${10}^{-1}$ (pluses).

Figure 12

Dispersion of the particles as a function of time at different temperatures. Force amplitudes in regions I–III. $\omega ={10}^{-2}$.

Figure 13

Features of the dependence of the diffusivity on the driving frequency at fixed temperature (top) and on the inverse temperature at fixed frequency (bottom), for $F_e$ in regions I–III.

Figure 14

$D^{\prime }\left(\omega \right)$ for force amplitude in regions I–III. $T^{†}=0.258$. Sinusoidal in time (circles) and PCP (squares) driving.