38 research outputs found
Long Range Hops and the Pair Annihilation Reaction A+A->0: Renormalization Group and Simulation
A simple example of a non-equilibrium system for which fluctuations are
important is a system of particles which diffuse and may annihilate in pairs on
contact. The renormalization group can be used to calculate the time dependence
of the density of particles, and provides both an exact value for the exponent
governing the decay of particles and an epsilon-expansion for the amplitude of
this power law. When the diffusion is anomalous, as when the particles perform
Levy flights, the critical dimension depends continuously on the control
parameter for the Levy distribution. The epsilon-expansion can then become an
expansion in a small parameter. We present a renormalization group calculation
and compare these results with those of a simulation.Comment: As-published version; two significant errors fixed, two references
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Scaling Relations for Collision-less Dark Matter Turbulence
Many scaling relations are observed for self-gravitating systems in the
universe. We explore the consistent understanding of them from a simple
principle based on the proposal that the collision-less dark matter fluid terns
into a turbulent state, i.e. dark turbulence, after crossing the caustic
surface in the non-linear stage. The dark turbulence will not eddy dominant
reflecting the collision-less property. After deriving Kolmogorov scaling laws
from Navier-Stokes equation by the method similar to the one for Smoluchowski
coagulation equation, we apply this to several observations such as the
scale-dependent velocity dispersion, mass-luminosity ratio, magnetic fields,
and mass-angular momentum relation, power spectrum of density fluctuations.
They all point the concordant value for the constant energy flow per mass: , which may be understood as the speed of the hierarchical
coalescence process in the cosmic structure formation.Comment: 26 pages, 6 figure
The scaling attractor and ultimate dynamics for Smoluchowski's coagulation equations
We describe a basic framework for studying dynamic scaling that has roots in
dynamical systems and probability theory. Within this framework, we study
Smoluchowski's coagulation equation for the three simplest rate kernels
, and . In another work, we classified all self-similar
solutions and all universality classes (domains of attraction) for scaling
limits under weak convergence (Comm. Pure Appl. Math 57 (2004)1197-1232). Here
we add to this a complete description of the set of all limit points of
solutions modulo scaling (the scaling attractor) and the dynamics on this limit
set (the ultimate dynamics). The main tool is Bertoin's L\'{e}vy-Khintchine
representation formula for eternal solutions of Smoluchowski's equation (Adv.
Appl. Prob. 12 (2002) 547--64). This representation linearizes the dynamics on
the scaling attractor, revealing these dynamics to be conjugate to a continuous
dilation, and chaotic in a classical sense. Furthermore, our study of scaling
limits explains how Smoluchowski dynamics ``compactifies'' in a natural way
that accounts for clusters of zero and infinite size (dust and gel)
Brownian Dynamics Simulation of Polydisperse Hard Spheres
Standard algorithms for the numerical integration of the Langevin equation
require that interactions are slowly varying during to the integration
timestep. This in not the case for hard-body systems, where there is no
clearcut between the correlation time of the noise and the timescale of the
interactions. Starting from a short time approximation of the Smoluchowsky
equation, we introduce an algorithm for the simulation of the overdamped
Brownian dynamics of polydisperse hard-spheres in absence of hydrodynamics
interactions and briefly discuss the extension to the case of external drifts
Lattice theory of trapping reactions with mobile species
We present a stochastic lattice theory describing the kinetic behavior of
trapping reactions , in which both the and particles
perform an independent stochastic motion on a regular hypercubic lattice. Upon
an encounter of an particle with any of the particles, is
annihilated with a finite probability; finite reaction rate is taken into
account by introducing a set of two-state random variables - "gates", imposed
on each particle, such that an open (closed) gate corresponds to a reactive
(passive) state. We evaluate here a formal expression describing the time
evolution of the particle survival probability, which generalizes our
previous results. We prove that for quite a general class of random motion of
the species involved in the reaction process, for infinite or finite number of
traps, and for any time , the particle survival probability is always
larger in case when stays immobile, than in situations when it moves.Comment: 12 pages, appearing in PR
Mirror symmetry breaking through an internal degree of freedom leading to directional motion
We analyze here the minimal conditions for directional motion (net flow in
phase space) of a molecular motor placed on a mirror-symmetric environment and
driven by a center-symmetric and time-periodic force field. The complete
characterization of the deterministic limit of the dissipative dynamics of
several realizations of this minimal model, reveals a complex structure in the
phase diagram in parameter space, with intertwined regions of pinning (closed
orbits) and directional motion. This demonstrates that the mirror-symmetry
breaking which is needed for directional motion to occur, can operate through
an internal degree of freedom coupled to the translational one.Comment: Accepted for publication in Phys. Rev.
A Dissipative-Particle-Dynamics Model for Simulating Dynamics of Charged Colloid
A mesoscopic colloid model is developed in which a spherical colloid is
represented by many interacting sites on its surface. The hydrodynamic
interactions with thermal fluctuations are taken accounts in full using
Dissipative Particle Dynamics, and the electrostatic interactions are simulated
using Particle-Particle-Particle Mesh method. This new model is applied to
investigate the electrophoretic mobility of a charged colloid under an external
electric field, and the influence of salt concentration and colloid charge are
systematically studied. The simulation results show good agreement with
predictions from the electrokinetic theory.Comment: 17 pages, 8 figures, submitted to the proceedings of High Performance
Computing in Science & Engineering '1