Life and Death at the Edge of a Windy Cliff

The survival probability of a particle diffusing in the two dimensional domain $x>0$ near a ``windy cliff'' at $x=0$ is investigated. The particle dies upon reaching the edge of the cliff. In addition to diffusion, the particle is influenced by a steady ``wind shear'' with velocity $\vec v(x,y)=v\,{\rm sign}(y)\,\hat x$, \ie, no average bias either toward or away from the cliff. For this semi-infinite system, the particle survival probability decays with time as $t^{-1/4}$, compared to $t^{-1/2}$ in the absence of wind. Scaling descriptions are developed to elucidate this behavior, as well as the survival probability within a semi-infinite strip of finite width $|y|<w$ with particle absorption at $x=0$. The behavior in the strip geometry can be described in terms of Taylor diffusion, an approach which accounts for the crossover to the $t^{-1/4}$ decay when the width of the strip diverges. Supporting numerical simulations of our analytical results are presented.Comment: 13 pages, plain TeX, 5 figures available upon request to SR (submitted to J. Stat. Phys.

Reduction and reconstruction of stochastic differential equations via symmetries

An algorithmic method to exploit a general class of infinitesimal symmetries for reducing stochastic differential equations is presented and a natural definition of reconstruction, inspired by the classical reconstruction by quadratures, is proposed. As a side result the well-known solution formula for linear one-dimensional stochastic differential equations is obtained within this symmetry approach. The complete procedure is applied to several examples with both theoretical and applied relevance

Doppler Effect of Nonlinear Waves and Superspirals in Oscillatory Media

Nonlinear waves emitted from a moving source are studied. A meandering spiral in a reaction-diffusion medium provides an example, where waves originate from a source exhibiting a back-and-forth movement in radial direction. The periodic motion of the source induces a Doppler effect that causes a modulation in wavelength and amplitude of the waves (``superspiral''). Using the complex Ginzburg-Landau equation, we show that waves subject to a convective Eckhaus instability can exhibit monotonous growth or decay as well as saturation of these modulations away from the source depending on the perturbation frequency. Our findings allow a consistent interpretation of recent experimental observations concerning superspirals and their decay to spatio-temporal chaos.Comment: 4 pages, 4 figure

Spatially Resolved Mapping of Local Polarization Dynamics in an Ergodic Phase of Ferroelectric Relaxor

Spatial variability of polarization relaxation kinetics in relaxor ferroelectric 0.9Pb(Mg1/3Nb2/3)O3-0.1PbTiO3 is studied using time-resolved Piezoresponse Force Microscopy. Local relaxation attributed to the reorientation of polar nanoregions is shown to follow stretched exponential dependence, exp(-(t/tau)^beta), with beta~~0.4, much larger than the macroscopic value determined from dielectric spectra (beta~~0.09). The spatial inhomogeneity of relaxation time distributions with the presence of 100-200 nm "fast" and "slow" regions is observed. The results are analyzed to map the Vogel-Fulcher temperatures on the nanoscale.Comment: 23 pages, 4 figures, supplementary materials attached; to be submitted to Phys. Rev. Let

Interaction of Vortices in Complex Vector Field and Stability of a ``Vortex Molecule''

We consider interaction of vortices in the vector complex Ginzburg--Landau equation (CVGLE). In the limit of small field coupling, it is found analytically that the interaction between well-separated defects in two different fields is long-range, in contrast to interaction between defects in the same field which falls off exponentially. In a certain region of parameters of CVGLE, we find stable rotating bound states of two defects -- a ``vortex molecule".Comment: 4 pages, 5 figures, submitted to Phys. Rev. Let

When are active Brownian particles and run-and-tumble particles equivalent? Consequences for motility-induced phase separation

Active Brownian particles (ABPs, such as self-phoretic colloids) swim at fixed speed $v$ along a body-axis ${\bf u}$ that rotates by slow angular diffusion. Run-and-tumble particles (RTPs, such as motile bacteria) swim with constant \u until a random tumble event suddenly decorrelates the orientation. We show that when the motility parameters depend on density $\rho$ but not on ${\bf u}$, the coarse-grained fluctuating hydrodynamics of interacting ABPs and RTPs can be mapped onto each other and are thus strictly equivalent. In both cases, a steeply enough decreasing $v(\rho)$ causes phase separation in dimensions $d=2,3$, even when no attractive forces act between the particles. This points to a generic role for motility-induced phase separation in active matter. However, we show that the ABP/RTP equivalence does not automatically extend to the more general case of \u-dependent motilities

Spiral Motion in a Noisy Complex Ginzburg-Landau Equation

The response of spiral waves to external perturbations in a stable regime of the two-dimensional complex Ginzburg-Landau equation (CGLE) is investigated. It is shown that the spiral core has a finite mobility and performs Brownian motion when driven by white noise. Combined with simulation results, this suggests that defect-free and quasi-frozen states in the noiseless CGLE are unstable against free vortex excitation at any non-zero noise strength.Comment: RevTex, 4 pages, 3 figures, submitted to Phys. Rev. Let

Vortex Glass and Vortex Liquid in Oscillatory Media

We study the disordered, multi-spiral solutions of two-dimensional homogeneous oscillatory media for parameter values at which the single spiral/vortex solution is fully stable. In the framework of the complex Ginzburg-Landau (CGLE) equation, we show that these states, heretofore believed to be static, actually evolve on ultra-slow timescales. This is achieved via a reduction of the CGLE to the evolution of the sole vortex position and phase coordinates. This true defect-mediated turbulence occurs in two distinct phases, a vortex liquid characterized by normal diffusion of individual spirals, and a slowly relaxing, intermittent, ``vortex glass''.Comment: 4 pages, 2 figures, submitted to Physical Review Letter

Noncoaxial multivortices in the complex sine-Gordon theory on the plane

We construct explicit multivortex solutions for the complex sine-Gordon equation (the Lund-Regge model) in two Euclidean dimensions. Unlike the previously found (coaxial) multivortices, the new solutions comprise $n$ single vortices placed at arbitrary positions (but confined within a finite part of the plane.) All multivortices, including the single vortex, have an infinite number of parameters. We also show that, in contrast to the coaxial complex sine-Gordon multivortices, the axially-symmetric solutions of the Ginzburg-Landau model (the stationary Gross-Pitaevskii equation) {\it do not} belong to a broader family of noncoaxial multivortex configurations.Comment: 40 pages, 7 figures in colou

Nonequilibrium dislocation dynamics and instability of driven vortex lattices in two dimensions

We consider dislocations in a vortex lattice that is driven in a two-dimensional superconductor with random impurities. The structure and dynamics of dislocations is studied in this genuine nonequilibrium situation on the basis of a coarse-grained equation of motion for the displacement field. The presence of dislocations leads to a characteristic anisotropic distortion of the vortex density that is controlled by a Kardar-Parisi-Zhang nonlinearity in the coarse-grained equation of motion. This nonlinearity also implies a screening of the interaction between dislocations and thereby an instability of the vortex lattice to the proliferation of free dislocations.Comment: published version with minor correction