Anisotropic velocity statistics of topological defects under shear flow

We report numerical results on the velocity statistics of topological defects during the dynamics of phase ordering and nonrelaxational evolution assisted by an external shear flow. We propose a numerically efficient tracking method for finding the position and velocity of defects and apply it to vortices in a uniform field and dislocations in anisotropic stripe patterns. During relaxational dynamics, the distribution function of the velocity fluctuations is characterized by a dynamical scaling with a scaling function that has a robust algebraic tail with an inverse cube power law. This is characteristic of defects of codimension 2, e.g., point defects in two dimensions and filaments in three dimensions, regardless of whether the motion is isotropic (as for vortices) or highly anisotropic (as for dislocations). However, the anisotropic dislocation motion leads to anisotropic statistical properties when the interaction between defects and their motion is influenced by the presence of an external shear flow transverse to the stripe orientation.

Figure 1

Snapshot of a measure of the charge-density vector field for a configuration of vortex filaments in 3D simulation of phase ordering. The figure shows the $x$ component ${\mathcal{D}}_x$ of the Jacobian determinant defined in Sec. 2. The system size in this simulation is ${128}^3$.

Figure 2

(a) Snapshot of the charge-density field corresponding to a configuration of point vortices in 2D simulations. In panel (b), we show the dislocations in an underlying anisotropic stripe configuration. The lighter blobs correspond to defects that have a positive charge, while the darker blobs are the defects of the opposite charge. The system size in both cases is ${1024}^2$, while the snapshots are drawn from a subset.

Figure 3

Collapsed probability distribution function (scaling function) of the absolute velocity of point vortices in 2D during phase ordering. (Inset) PDF of the defect velocity for different core sizes (open circles correspond to larger core size and crosses correspond to smaller core size) to show that the Gaussian cutoff depends effectively on the vortex core size. The model parameters for the inset figure are $A=2.05$ (open circles), $A=1.05$ (crosses), and $C=3/20(1+A)$. In the main graph, $A=1.5$. Here $v\equiv \left|\mathbf{v}\right|$.

Figure 4

Scaling function of the probability distribution function of the absolute velocity of vortex filaments in 3D during phase ordering. Here $v\equiv \left|\mathbf{v}\right|$.

Figure 5

(a) Density of defects ${\rho }_d$ as a function of time $t$ for different values of the imposed shear velocity $v_0$. (b) Data collapse of the rescaled density with the mean density in the statistical steady state ${\rho }_0\sim t_0^{-2/3}$, where $t_0$ is the crossover time to the steady state. The inset figure shows how $t_0$ scales with $v_0$ as $t_0\sim 1/v_0$.

Figure 6

(a) Evolution with time of the ensemble average of the velocity component $v_x\equiv \langle |v_x\left(t\right)|\rangle$ for different values of the shear rate. (b) Data collapse of the rescaled $v_x$ as a function of the rescaled time $t/t_0$.

Figure 7

(a) Evolution with time of the ensemble average of the velocity component $v_y\equiv \langle |v_y\left(t\right)|\rangle$ for different values of the shear rate. (b) Rescaled $v_y$ plotted as a function of the rescaled time $t/t_0$.

Figure 8

Time-independent scaling function of the probability distribution corresponding to climb (crosses), respectively glide (open circles) velocities during the phase ordering in the absence of an external field. The rescaled velocity variable is given as ${\tilde{v}}_j\equiv v_j/\langle |v_j|\rangle$ for $j=x,y$. The inset figure shows the log-log plot of the PDF calculated with logarithmic binning.

Figure 9

(a) PDF of the gliding velocity for positive (open diamonds and circles) and negative (open triangles and crosses) dislocations in the statistically stationary state at different shear rates. (b) Probability distribution function of the climb velocity for positive (open circles and diamonds) and negative (open crosses and triangles) dislocations in the statistically stationary state. In both cases, the rescaled velocity variable is ${\tilde{v}}_j\equiv v_j/\langle |v_j|\rangle$ for $j=x,y$. The inset figure shows the log-log plot of the tail PDF with logarithmic binning.