Markovian embedding of non-Markovian superdiffusion

We consider different Markovian embedding schemes of non-Markovian stochastic processes that are described by generalized Langevin equations and obey thermal detailed balance under equilibrium conditions. At thermal equilibrium, superdiffusive behavior can emerge if the total integral of the memory kernel vanishes. Such a situation of vanishing static friction is caused by a super-Ohmic thermal bath. One of the simplest models of ballistic superdiffusion is determined by a biexponential memory kernel that was proposed by [Bao J. Stat. Phys. 114, 503 (2004)]. We show that this non-Markovian model has infinitely many different four-dimensional Markovian embeddings. Implementing numerically the simplest one, we demonstrate that (i) the presence of a periodic potential with arbitrarily low barriers changes the asymptotic large-time behavior from free ballistic superdiffusion into normal diffusion; (ii) an additional biasing force renders the asymptotic dynamics superdiffusive again. The development of transients that display a qualitatively different behavior compared to the true large-time asymptotics presents a general feature of this non-Markovian dynamics. These transients though may be extremely long. As a consequence, they can be even mistaken as the true asymptotics. We find that such intermediate asymptotics exhibit a giant enhancement of superdiffusion in tilted washboard potentials and it is accompanied by a giant transient superballistic current growing proportional to $t^{{\alpha }_{\text{eff}}}$ with an exponent ${\alpha }_{\text{eff}}$ that can exceed the ballistic value of 2.

Figure 1

(Color online) The mean square displacement of the position (being independent of the bias $F$) as a function of time changes from a $t^4$ law at small times to a ballistic $t^2$ law. Comparison of the analytical solution according to Eq. (25) (solid line) and results from numerical simulations of the first embedding (square symbols) exhibit good agreement. A strongly deviating result is obtained if the auxiliary variables of the first embedding initially assume vanishing values (dashed line). The other parameters $\kappa =2$, $\nu =3$, ${\omega }_0=1$, and $v\left(0\right)=0$ are the same for the displayed curves. The estimate of the mean square displacement is obtained by an average over an ensemble of ${10}^4$ simulated trajectories.

Figure 2

(Color online) Unbiased superdiffusion in a washboard potential. The mean square displacement of a particle that spreads ballistically in the absence of a bias $F$, see Fig. 1, eventually changes its behavior from superdiffusive to normal diffusive behavior under the influence of a periodic potential of strength $V_0$. From bottom right hand side to top, we use $V_0=0.5,0.2,0.1,0.05,0.02,0$. The time of the turn-over shifts to later times with decreasing potential strength. For the simulation of the displayed data, the first embedding was used with $\kappa =2$, $\nu =3$, ${\omega }_0=1$, and $v\left(0\right)=0$.

Figure 3

(Color online) Biased superdiffusion. After diffusion has become normal in the presence of a periodic potential, see Fig. 2, it is again changing to ballistic diffusion under the influence of an additional finite bias $F$. Long transients exhibiting hyperdiffusion emerge before the ballistic diffusion regime is approached. A fixed barrier height $V_0=1$ was used for the simulations and the tilt $F$ is variable. The other parameters are again $\kappa =2$, $\nu =3$, ${\omega }_0=1$, and $v\left(0\right)=0$.

Figure 4

(Color online) Anomalous drift behavior. A finite bias $F$ induces anomalous drift. From bottom right hand side to top, we use $F=0.1,0.2,0.4,1.5$. Ballistic currents appear asymptotically $⟨\Delta x\left(t\right)⟩\sim t^2$. Like in Fig. 3, transient regimes appear with enhanced particle transport stronger than ballistic. The used parameters are: $V_0=1$, $\kappa =2$, $\nu =3$, ${\omega }_0=1$, and $v\left(0\right)=0$.

Figure 5

(Color online) Mean square displacement of position for different initially distributed velocities. The solid lines mark thermally distributed initial velocities $⟨v^2\left(0\right)⟩=v_T^2$, whereas dashed lines mark initially zero velocity $v\left(0\right)=0$. The differences in time evolution vanish in the asymptotic long time limit in the presence of a biased periodic washboard potential with potential height $V_0$ and bias $F$, implying wide sense ergodicity for the mean square displacement. However, in case of free ballistic diffusion, see lines labeled by $V_0=0,F=0$, a constant deviation remains according to Eq. (35). Parameters are chosen as $\kappa =2$, $\nu =3$, and ${\omega }_0=1$.

Figure 6

(Color online) Role of deviation from stationary fluctuation-dissipation relation in Eqs. (2, A9) on the time evolution of the mean square displacement $⟨\Delta x^2\left(t\right)⟩$. In contrast to the choice with a stationary fluctuation-dissipation relation, see solid lines marking $⟨u_i\left(0\right)u_j\left(0\right)⟩={\kappa }^2{\delta }_{i,j}$, the initial choice $u_1\left(0\right)=u_2\left(0\right)=0$, see dashed lines, yields a Gaussian noise $\zeta \left(t\right)$ that initially is nonstationary, see Eq. (A8). The noise $\zeta \left(t\right)$ assumes, however, stationary noise at asymptotic long times. The initial velocity was set to zero, i.e., $v\left(0\right)=0$, the potential strength to $V_0=1$, and the remaining parameters are chosen as in Fig. 5, $\kappa =2$, $\nu =3$, and ${\omega }_0=1$.