142 research outputs found
Ballistic aggregation for one-sided Brownian initial velocity
We study the one-dimensional ballistic aggregation process in the continuum
limit for one-sided Brownian initial velocity (i.e. particles merge when they
collide and move freely between collisions, and in the continuum limit the
initial velocity on the right side is a Brownian motion that starts from the
origin ). We consider the cases where the left side is either at rest or
empty at . We derive explicit expressions for the velocity distribution
and the mean density and current profiles built by this out-of-equilibrium
system. We find that on the right side the mean density remains constant
whereas the mean current is uniform and grows linearly with time. All
quantities show an exponential decay on the far left. We also obtain the
properties of the leftmost cluster that travels towards the left. We find that
in both cases relevant lengths and masses scale as and the evolution is
self-similar.Comment: 18 pages, published in Physica
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 separating two ideal fluids made of point particles with mass . 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 and , 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 . 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
Ballistic annihilation kinetics for a multi-velocity one-dimensional ideal gas
Ballistic annihilation kinetics for a multi-velocity one-dimensional ideal
gas is studied in the framework of an exact analytic approach. For an initial
symmetric three-velocity distribution, the problem can be solved exactly and it
is shown that different regimes exist depending on the initial fraction of
particles at rest. Extension to the case of a n-velocity distribution is
discussed.Comment: 19 pages, latex, uses Revtex macro
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 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
Stable distribution in fragmentation processes
We introduce three models of fragmentation in which the largest fragment in the system can be broken at each time step with a fixed probability, p. We solve these models exactly in the long time limit to reveal stable time invariant (scaling) solutions which depend on p and the precise details of the fragmentation process. Various features of these models are compared with those of conventional fragmentation models.
To get Figures e-mail to G.J. [email protected]
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