Oscillatory instability in a driven granular gas

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.Comment: 7 pages, 4 figures, to appear in Europhysics Letter

Time-resolved extinction rates of stochastic populations

Extinction of a long-lived isolated stochastic population can be described as an exponentially slow decay of quasi-stationary probability distribution of the population size. We address extinction of a population in a two-population system in the case when the population turnover -- renewal and removal -- is much slower than all other processes. In this case there is a time scale separation in the system which enables one to introduce a short-time quasi-stationary extinction rate W_1 and a long-time quasi-stationary extinction rate W_2, and develop a time-dependent theory of the transition between the two rates. It is shown that W_1 and W_2 coincide with the extinction rates when the population turnover is absent, and present but very slow, respectively. The exponentially large disparity between the two rates reflects fragility of the extinction rate in the population dynamics without turnover.Comment: 8 pages, 4 figure

Logarithmically Slow Expansion of Hot Bubbles in Gases

We report logarithmically slow expansion of hot bubbles in gases in the process of cooling. A model problem first solved, when the temperature has compact support. Then temperature profile decaying exponentially at large distances is considered. The periphery of the bubble is shown to remain essentially static ("glassy") in the process of cooling until it is taken over by a logarithmically slowly expanding "core". An analytical solution to the problem is obtained by matched asymptotic expansion. This problem gives an example of how logarithmic corrections enter dynamic scaling.Comment: 4 pages, 1 figur

Wavelength limits on isobaricity of perturbations in a thermally unstable radiatively cooling medium

Nonlinear evolution of one-dimensional planar perturbations in an optically thin radiatively cooling medium in the long-wavelength limit is studied numerically. The accepted cooling function generates in thermal equilibrium a bistable equation of state $P(\rho)$. The unperturbed state is taken close to the upper (low-density) unstable state with infinite compressibility ($dP/d\rho= 0$). The evolution is shown to proceed in three different stages. At first stage, pressure and density set in the equilibrium equation of state, and velocity profile steepens gradually as in case of pressure-free flows. At second stage, those regions of the flow where anomalous pressure (i.e. with negative compressibility) holds, create velocity profile more sharp than in pressure-free case, which in turn results in formation of a very narrow (short-wavelength) region where gas separates the equilibrium equation of state and pressure equilibrium sets in rapidly. On this stage, variation in pressure between narrow dense region and extended environment does not exceed more than 0.01 of the unperturbed value. On third stage, gas in the short-wavelength region reaches the second (high-density) stable state, and pressure balance establishes through the flow with pressure equal to the one in the unperturbed state. In external (long-wavelength) regions, gas forms slow isobaric inflow toward the short-wavelength layer. The duration of these stages decreases when the ratio of the acoustic time to the radiative cooling time increases. Limits in which nonlinear evolution of thermally unstable long-wavelength perturbations develops in isobaric regime are obtained.Comment: 21 pages with 7 figures, Revtex, accepted in Physics of Plasma

Reducing multiphoton ionization in a linearly polarized microwave field by local control

We present a control procedure to reduce the stochastic ionization of hydrogen atom in a strong microwave field by adding to the original Hamiltonian a comparatively small control term which might consist of an additional set of microwave fields. This modification restores select invariant tori in the dynamics and prevents ionization. We demonstrate the procedure on the one-dimensional model of microwave ionization.Comment: 8 page

Emergence of stability in a stochastically driven pendulum: beyond the Kapitsa effect

We consider a prototypical nonlinear system which can be stabilized by multiplicative noise: an underdamped non-linear pendulum with a stochastically vibrating pivot. A numerical solution of the pertinent Fokker-Planck equation shows that the upper equilibrium point of the pendulum can become stable even when the noise is white, and the "Kapitsa pendulum" effect is not at work. The stabilization occurs in a strong-noise regime where WKB approximation does not hold.Comment: 4 pages, 7 figure

Extinction of metastable stochastic populations

We investigate extinction of a long-lived self-regulating stochastic population, caused by intrinsic (demographic) noise. Extinction typically occurs via one of two scenarios depending on whether the absorbing state n=0 is a repelling (scenario A) or attracting (scenario B) point of the deterministic rate equation. In scenario A the metastable stochastic population resides in the vicinity of an attracting fixed point next to the repelling point n=0. In scenario B there is an intermediate repelling point n=n_1 between the attracting point n=0 and another attracting point n=n_2 in the vicinity of which the metastable population resides. The crux of the theory is WKB method which assumes that the typical population size in the metastable state is large. Starting from the master equation, we calculate the quasi-stationary probability distribution of the population sizes and the (exponentially long) mean time to extinction for each of the two scenarios. When necessary, the WKB approximation is complemented (i) by a recursive solution of the quasi-stationary master equation at small n and (ii) by the van Kampen system-size expansion, valid near the fixed points of the deterministic rate equation. The theory yields both entropic barriers to extinction and pre-exponential factors, and holds for a general set of multi-step processes when detailed balance is broken. The results simplify considerably for single-step processes and near the characteristic bifurcations of scenarios A and B.Comment: 19 pages, 7 figure

Attempted density blowup in a freely cooling dilute granular gas: hydrodynamics versus molecular dynamics

It has been recently shown (Fouxon et al. 2007) that, in the framework of ideal granular hydrodynamics (IGHD), an initially smooth hydrodynamic flow of a granular gas can produce an infinite gas density in a finite time. Exact solutions that exhibit this property have been derived. Close to the singularity, the granular gas pressure is finite and almost constant. This work reports molecular dynamics (MD) simulations of a freely cooling gas of nearly elastically colliding hard disks, aimed at identifying the "attempted" density blowup regime. The initial conditions of the simulated flow mimic those of one particular solution of the IGHD equations that exhibits the density blowup. We measure the hydrodynamic fields in the MD simulations and compare them with predictions from the ideal theory. We find a remarkable quantitative agreement between the two over an extended time interval, proving the existence of the attempted blowup regime. As the attempted singularity is approached, the hydrodynamic fields, as observed in the MD simulations, deviate from the predictions of the ideal solution. To investigate the mechanism of breakdown of the ideal theory near the singularity, we extend the hydrodynamic theory by accounting separately for the gradient-dependent transport and for finite density corrections.Comment: 11 pages, 9 figures, accepted for publication on Physical Review

Area-preserving dynamics of a long slender finger by curvature: a test case for the globally conserved phase ordering

A long and slender finger can serve as a simple ``test bed'' for different phase ordering models. In this work, the globally-conserved, interface-controlled dynamics of a long finger is investigated, analytically and numerically, in two dimensions. An important limit is considered when the finger dynamics are reducible to the area-preserving motion by curvature. A free boundary problem for the finger shape is formulated. An asymptotic perturbation theory is developed that uses the finger aspect ratio as a small parameter. The leading-order approximation is a modification of ``the Mullins finger" (a well-known analytic solution) which width is allowed to slowly vary with time. This time dependence is described, in the leading order, by an exponential law with the characteristic time proportional to the (constant) finger area. The subleading terms of the asymptotic theory are also calculated. Finally, the finger dynamics is investigated numerically, employing the Ginzburg-Landau equation with a global conservation law. The theory is in a very good agreement with the numerical solution.Comment: 8 pages, 4 figures, Latex; corrected typo