Internal noise and system size effects induce nondiffusive kink dynamics

We investigate the effects of inherent fluctuations and system size in the dynamics of domain between uniform symmetric states. In the case of monotonous kinks, this dynamics is characterized by exhibiting nonsymmetric random walks, being attracted to the system borders. For nonmonotonous interface, the dynamics is replaced by a hopping dynamic. Based on bistable universal models, we characterize the origin of these unexpected dynamics through use of the stochastic kinematic laws for the interface position and the survival probability. Numerical simulations show a quite good agreement with the theoretical predictions.

Figure 1

Temporal evolution of a granular kink position, $x_0$, in a shallow one-dimensional fluidized granular layer subjected to a periodic air flow [45]. The inset snapshot corresponds to a granular kink observed at a given moment.

Figure 2

Domain solution between two symmetric states, kink solution (top), $\{\Delta ,l\}$ account for the position and the core of the kink. Bottom shows a kink and an antikink solution as consequence of the specular boundary condition.

Figure 3

Stochastic dynamics of a kink state. (a) Numerical simulations of a single kink of Eq. (4) in a finite size system with specular boundary conditions, with $\mu =1$ and $\eta =0.0018$ and $L=25$, starting at the position $\Delta =0.3L$. (b) The profile link at an initial moment. (c) Evolution of survival probability of a single kink for different instants of time, $t_1=0$ (black), $t_2=20000$ (blue), and $t_3=40000$ (green).

Figure 4

Kinematic law of the kink position as a result of size effect. Points are obtained numerically by considering a uniform distribution of initial conditions for a single kink, then numerically the system evolves during a brief moment of time, and finally the temporal variation of the kink position is calculated. The solid line is the analytical expression (13) with $\eta =0$, and $f\left(\Delta \right)=f_1+f_2+f_3$.

Figure 5

Temporal evolution of Fokker-Plank Eq. (20) with absorbing boundary conditions, using a drift force $h\left(\Delta \right)=-C[{\Delta }^{-a}+{(L-\Delta )}^{-a}]$ with $C=0.033,a=3.08,{\Delta }_0=0.3L$, and $\eta =2.69\times {10}^{-3}$.

Figure 6

Domain solution between two symmetric states with damping spatial oscillations (top), $\{\Delta ,l\}$ account for the position and the core of the kink. Bottom: kink and an antikink solution as consequence of the specular boundary condition.

Figure 7

Bifurcation diagram of Swift-Hohenberg Eq. (21), where the darkened area accounts for the amplitude of the patterns. The continuous and dashed lines account for stable and unstable uniform states, respectively. $\{{\mu }_0,{\mu }_p\}$ correspond to the spatial and the pitchfork bifurcations of the state $u=0$, respectively. ${\mu }_s$ stands for the spatial bifurcation of the states $u_{\pm }=\pm \sqrt{\mu -q^4}$.

Figure 8

Kink solution of the Swift-Hohenberg equation with the parameters $\mu =q=1$ and $\eta =0.0018$. (a) kink solution profile and (b) spatiotemporal evolution of the kink.

Figure 9

Stochastic dynamics of a kink state of the stochastic Swift-Hohenberg Eq. (21). (a) Numerical simulations of a single kink of the model (21) in a finite size system with specular boundary conditions, with $\mu =1,q=1$ and $\eta =0.0018$ starting at the position $\Delta ={\Delta }_0\approx 0.15L$. (b) The kink at an initial moment. (c) Evolution of survival probability of a single kink at different instants of time, $t_1=0$ (gray), $t_2={10}^6$ (green), and $t_3=2\times {10}^6$ (blue).

Figure 10

Kinematic law of the kink position as a result of size effect. Points are obtained numerically by considering a uniform distribution of initial conditions for a single kink, then numerically the system evolves in a brief moment of time, and finally the temporal variation of the kink position is calculated. The solid curve is obtained by using the integral form of the kinematic law of the kink position $f\left(\Delta \right)=f_1^{\prime }+f_2^{\prime }+f_3^{\prime }$, using the expressions $(27-29)$. The lower panel shows the potential $U\left(\Delta \right)$.

Figure 11

Temporal evolution of Fokker-Plank Eq. (32) with absorbing boundary conditions, using the potential (30) with ${\alpha }^{\prime }=1.157,a=0.4940$, and $b=1.026$ as fitting parameters and ${\eta }^{\prime }=2\times {10}^{-3}$.

Figure 12

Stochastic dynamics of a kink state of the forcing real Ginzburg-Landau Eq. (33) with $k=\pi /3,a=0.02$, and $\eta =0.02$. Different colors represent different trajectories of the kink solution. The bottom panel shows the profile of the kink at the initial time.