Abstract:
A selection problem that appears in the Lifshitz-Slyozov (LS) theory of Ostwald ripening is reexamined. The problem concerns selection of a self-similar distribution function (DF) of the minority domains with respect to their sizes from a whole one-parameter family of solutions. A strong selection rule is found via an account of fluctuations. Fluctuations produce an infinite tail in the DF and drive the DF towards the "limiting solution" of LS or its analogs for other growth mechanisms.

Abstract:
Suppose that an infinite lattice gas of constant density $n_0$, whose dynamics are described by the symmetric simple exclusion process, is brought in contact with a spherical absorber of radius $R$. Employing the macroscopic fluctuation theory and assuming the additivity principle, we evaluate the probability distribution ${\mathcal P}(N)$ that $N$ particles are absorbed during a long time $T$. The limit of $N=0$ corresponds to the survival problem, whereas $N\gg \bar{N}$ describes the opposite extreme. Here $\bar{N}=4\pi R D_0 n_0 T$ is the \emph{average} number of absorbed particles (in three dimensions), and $D_0$ is the gas diffusivity. For $n_0\ll 1$ the exclusion effects are negligible, and ${\mathcal P}(N)$ can be approximated, for not too large $N$, by the Poisson distribution with mean $\bar{N}$. For finite $n_0$, ${\mathcal P}(N)$ is non-Poissonian. We show that $-\ln{\mathcal P}(N) \simeq n_0 N^2/\bar{N}$ at $N\gg \bar{N}$. At sufficiently large $N$ and $n_0<1/2$ the most likely density profile of the gas, conditional on the absorption of $N$ particles, is non-monotonic in space. We also establish a close connection between this problem and that of statistics of current in finite open systems.

Abstract:
Suppose that a point-like steady source at $x=0$ injects particles into a half-infinite line. The particles diffuse and die. At long times a non-equilibrium steady state sets in, and we assume that it involves many particles. If the particles are non-interacting, their total number $N$ in the steady state is Poisson-distributed with mean $\bar{N}$ predicted from a deterministic reaction-diffusion equation. Here we determine the most likely density history of this driven system conditional on observing a given $N$. We also consider two prototypical examples of \emph{interacting} diffusing particles: (i) a family of mortal diffusive lattice gases with constant diffusivity (as illustrated by the simple symmetric exclusion process with mortal particles), and (ii) random walkers that can annihilate in pairs. In both examples we calculate the variances of the (non-Poissonian) stationary distributions of $N$.

Abstract:
We develop a numerical method for solving a free boundary problem which describes shape relaxation, by surface tension, of a long and thin bubble of an inviscid fluid trapped inside a viscous fluid in a Hele-Shaw cell. The method of solution of the exterior Dirichlet problem employs a classical boundary integral formulation. Our version of the numerical method is especially advantageous for following the dynamics of a very long and thin bubble, for which an asymptotic scaling theory has been recently developed. Because of the very large aspect ratio of the bubble, a direct implementation of the boundary integral algorithm would be impractical. We modify the algorithm by introducing a new approximation of the integrals which appear in the Fredholm integral equation and in the integral expression for the normal derivative of the pressure at the bubble interface. The new approximation allows one to considerably reduce the number of nodes at the almost flat part of the bubble interface, while keeping a good accuracy. An additional benefit from the new approximation is in that it eliminates numerical divergence of the integral for the tangential derivative of the harmonic conjugate. The interface's position is advanced in time by using explicit node tracking, whereas the larger node spacing enables one to use larger time steps. The algorithm is tested on two model problems, for which approximate analytical solutions are available.

Abstract:
Symmetry-breaking instability of a laterally uniform granular cluster (strip state) in a prototypical driven granular gas is investigated. The system consists of smooth hard disks in a two-dimensional box, colliding inelastically with each other and driven, at zero gravity, by a "thermal" wall. The limit of nearly elastic particle collisions is considered, and granular hydrodynamics with the Jenkins-Richman constitutive relations is employed. The hydrodynamic problem is completely described by two scaled parameters and the aspect ratio of the box. Marginal stability analysis predicts a spontaneous symmetry breaking instability of the strip state, similar to that predicted recently for a different set of constitutive relations. If the system is big enough, the marginal stability curve becomes independent of the details of the boundary condition at the driving wall. In this regime, the density perturbation is exponentially localized at the elastic wall opposite to the thermal wall. The short- and long-wavelength asymptotics of the marginal stability curves are obtained analytically in the dilute limit. The physics of the symmetry-breaking instability is discussed.

Abstract:
The Navier-Stokes granular hydrodynamics is employed for determining the threshold of thermal convection in an infinite horizontal layer of granular gas. The dependence of the convection threshold, in terms of the inelasticity of particle collisions, on the Froude and Knudsen numbers is found. A simple necessary condition for convection is formulated in terms of the Schwarzschild's criterion, well-known in thermal convection of (compressible) classical fluids. The morphology of convection cells at the onset is determined. At large Froude numbers, the Froude number drops out of the problem. As the Froude number goes to zero, the convection instability turns into a recently discovered phase separation instability.

Abstract:
We discovered an oscillatory instability in a system of inelastically colliding hard spheres, driven by two opposite "thermal" walls at zero gravity. The instability, predicted by a linear stability analysis of the equations of granular hydrodynamics, occurs when the inelasticity of particle collisions exceeds a critical value. Molecular dynamic simulations support the theory and show a stripe-shaped cluster moving back and forth in the middle of the box away from the driving walls. The oscillations are irregular but have a single dominating frequency that is close to the frequency at the instability onset, predicted from hydrodynamics.

Abstract:
We develop a theory of parametric excitation of weakly nonlinear standing gravity waves in a tank, which is under vertical vibrations with a slowly time-dependent ("chirped") vibration frequency. We show that, by using a negative chirp, one can excite a steadily growing wave via parametric autoresonance. The method of averaging is employed to derive the governing equations for the primary mode. These equations are solved analytically and numerically, for typical initial conditions, for both inviscid and weakly viscous fluids. It is shown that, when passing through resonance, capture into resonance always occurs when the chirp rate is sufficiently small. The critical chirp rate, above which breakdown of autoresonance occurs, is found for different initial conditions. The autoresonance excitation is expected to terminate at large amplitudes, when the underlying constant-frequency system ceases to exhibit a non-trivial stable fixed point.

Abstract:
We use the macroscopic fluctuation theory to determine the statistics of large currents in the Kipnis-Marchioro-Presutti (KMP) model in a ring geometry. About 10 years ago this simple setting was instrumental in identifying a breakdown of the additivity principle in a class of lattice gases at currents exceeding a critical value. Building on earlier work, we assume that, for supercritical currents, the optimal density profile, conditioned on the given current, has the form of a traveling wave (TW). For the KMP model we find this TW analytically, in terms of elliptic functions, for any supercritical current $I$. Using this TW solution, we evaluate, up to a pre-exponential factor, the probability distribution $P(I)$. We obtain simple asymptotics of the TW and of $P(I)$ for currents close to the critical current, and for currents much larger than the critical current. In the latter case we show that $-\ln P (I) \sim I\ln I$, whereas the optimal density profile acquires a soliton shape. Our analytic results are in a very good agreement with Monte-Carlo simulations and numerical solutions of Hurtado and Garrido (2011).

Abstract:
We suggest a general spectral method for calculating statistics of multi-step birth-death processes and chemical reactions of the type mA->nA (m and n are positive integers) which possess an absorbing state. The method yields accurate results for the extinction statistics, and for the quasi-stationary probability distribution, including large deviations, of the metastable state. The power of the method is demonstrated on the example of binary annihilation and triple branching 2A->0 and A->3A, representative of the rather general class of dissociation-recombination reactions.