Exact statistical properties of the Burgers equation

The one dimensional Burgers equation in the inviscid limit with white noise initial condition is revisited. The one- and two-point distributions of the Burgers field as well as the related distributions of shocks are obtained in closed analytical forms. In particular, the large distance behavior of spatial correlations of the field is determined. Since higher order distributions factorize in terms of the one and two points functions, our analysis provides an explicit and complete statistical description of this problem.Comment: 21 pages, 6 figures include

Ballistic aggregation: a solvable model of irreversible many particles dynamics

The adhesive dynamics of a one-dimensional aggregating gas of point particles is rigorously described. The infinite hierarchy of kinetic equations for the distributions of clusters of nearest neighbours is shown to be equivalent to a system of two coupled equations for a large class of initial conditions. The solution to these nonlinear equations is found by a direct construction of the relevant probability distributions in the limit of a continuous initial mass distribution. We show that those limiting distributions are identical to those of the statistics of shocks in the Burgers turbulence. The analysis relies on a mapping on a Brownian motion problem with parabolic constraints.Comment: 23 pages, 6 figures include

On the adiabatic properties of a stochastic adiabatic wall: Evolution, stationary non-equilibrium, and equilibrium states

The time evolution of the adiabatic piston problem and the consequences of its stochastic motion are investigated. The model is a one dimensional piston of mass $M$ separating two ideal fluids made of point particles with mass $m\ll M$. For infinite systems it is shown that the piston evolves very rapidly toward a stationary nonequilibrium state with non zero average velocity even if the pressures are equal but the temperatures different on both sides of the piston. For finite system it is shown that the evolution takes place in two stages: first the system evolves rather rapidly and adiabatically toward a metastable state where the pressures are equal but the temperatures different; then the evolution proceeds extremely slowly toward the equilibrium state where both the pressures and the temperatures are equal. Numerical simulations of the model are presented. The results of the microscopical approach, the thermodynamical equations and the simulations are shown to be qualitatively in good agreement.Comment: 28 pages, 10 figures include

Ballistic Annihilation

Ballistic annihilation with continuous initial velocity distributions is investigated in the framework of Boltzmann equation. The particle density and the rms velocity decay as $c=t^{-\alpha}$ and $=t^{-\beta}$, with the exponents depending on the initial velocity distribution and the spatial dimension. For instance, in one dimension for the uniform initial velocity distribution we find $\beta=0.230472...$. We also solve the Boltzmann equation for Maxwell particles and very hard particles in arbitrary spatial dimension. These solvable cases provide bounds for the decay exponents of the hard sphere gas.Comment: 4 RevTeX pages and 1 Eps figure; submitted to Phys. Rev. Let

Statistics of Largest Loops in a Random Walk

We report further findings on the size distribution of the largest neutral segments in a sequence of N randomly charged monomers [D. Ertas and Y. Kantor, Phys. Rev. E53, 846 (1996); cond-mat/9507005]. Upon mapping to one--dimensional random walks (RWs), this corresponds to finding the probability distribution for the size L of the largest segment that returns to its starting position in an N--step RW. We primarily focus on the large N, \ell = L/N << 1 limit, which exhibits an essential singularity. We establish analytical upper and lower bounds on the probability distribution, and numerically probe the distribution down to \ell \approx 0.04 (corresponding to probabilities as low as 10^{-15}) using a recursive Monte Carlo algorithm. We also investigate the possibility of singularities at \ell=1/k for integer k.Comment: 5 pages and 4 eps figures, requires RevTeX, epsf and multicol. Postscript file also available at http://cmtw.harvard.edu/~deniz/publications.htm

Phase transition in a spatial Lotka-Volterra model

Spatial evolution is investigated in a simulated system of nine competing and mutating bacterium strains, which mimics the biochemical war among bacteria capable of producing two different bacteriocins (toxins) at most. Random sequential dynamics on a square lattice is governed by very symmetrical transition rules for neighborhood invasion of sensitive strains by killers, killers by resistants, and resistants by by sensitives. The community of the nine possible toxicity/resistance types undergoes a critical phase transition as the uniform transmutation rates between the types decreases below a critical value $P_c$ above which all the nine types of strain coexist with equal frequencies. Passing the critical mutation rate from above, the system collapses into one of the three topologically identical states, each consisting of three strain types. Of the three final states each accrues with equal probability and all three maintain themselves in a self-organizing polydomain structure via cyclic invasions. Our Monte Carlo simulations support that this symmetry breaking transition belongs to the universality class of the three-state Potts model.Comment: 4 page

Stochastic Aggregation: Scaling Properties

We study scaling properties of stochastic aggregation processes in one dimension. Numerical simulations for both diffusive and ballistic transport show that the mass distribution is characterized by two independent nontrivial exponents corresponding to the survival probability of particles and monomers. The overall behavior agrees qualitatively with the mean-field theory. This theory also provides a useful approximation for the decay exponents, as well as the limiting mass distribution.Comment: 6 pages, 7 figure

Influence of correlations on the velocity statistics of scalar granular gases

The free evolution of inelastic particles in one dimension is studied by means of Molecular Dynamics (MD), of an inelastic pseudo-Maxwell model and of a lattice model, with emphasis on the role of spatial correlations. We present an exact solution of the 1d granular pseudo-Maxwell model for the scaling distribution of velocities and discuss how this model fails to describe correctly the homogeneous cooling stage of the 1d granular gas. Embedding the pseudo-Maxwell gas on a lattice (hence allowing for the onset of spatial correlations), we find a much better agreement with the MD simulations even in the inhomogeneous regime. This is seen by comparing the velocity distributions, the velocity profiles and the structure factors of the velocity field.Comment: Latex file: 6 pages, 5 figures (.eps). See also http://axtnt3.phys.uniroma1.it/Maxwel