5,358 research outputs found
Maximum Distance Between the Leader and the Laggard for Three Brownian Walkers
We consider three independent Brownian walkers moving on a line. The process
terminates when the left-most walker (the `Leader') meets either of the other
two walkers. For arbitrary values of the diffusion constants D_1 (the Leader),
D_2 and D_3 of the three walkers, we compute the probability distribution
P(m|y_2,y_3) of the maximum distance m between the Leader and the current
right-most particle (the `Laggard') during the process, where y_2 and y_3 are
the initial distances between the leader and the other two walkers. The result
has, for large m, the form P(m|y_2,y_3) \sim A(y_2,y_3) m^{-\delta}, where
\delta = (2\pi-\theta)/(\pi-\theta) and \theta =
cos^{-1}(D_1/\sqrt{(D_1+D_2)(D_1+D_3)}. The amplitude A(y_2,y_3) is also
determined exactly
Persistence of a Brownian particle in a Time Dependent Potential
We investigate the persistence probability of a Brownian particle in a
harmonic potential, which decays to zero at long times -- leading to an
unbounded motion of the Brownian particle. We consider two functional forms for
the decay of the confinement, an exponential and an algebraic decay. Analytical
calculations and numerical simulations show, that for the case of the
exponential relaxation, the dynamics of Brownian particle at short and long
times are independent of the parameters of the relaxation. On the contrary, for
the algebraic decay of the confinement, the dynamics at long times is
determined by the exponent of the decay. Finally, using the two-time
correlation function for the position of the Brownian particle, we construct
the persistence probability for the Brownian walker in such a scenario.Comment: 7 pages, 5 figures, Accepted for publication in Phys. Rev.
Blocking and Persistence in the Zero-Temperature Dynamics of Homogeneous and Disordered Ising Models
A ``persistence'' exponent theta has been extensively used to describe the
nonequilibrium dynamics of spin systems following a deep quench: for
zero-temperature homogeneous Ising models on the d-dimensional cubic lattice,
the fraction p(t) of spins not flipped by time t decays to zero like
t^[-theta(d)] for low d; for high d, p(t) may decay to p(infinity)>0, because
of ``blocking'' (but perhaps still like a power). What are the effects of
disorder or changes of lattice? We show that these can quite generally lead to
blocking (and convergence to a metastable configuration) even for low d, and
then present two examples --- one disordered and one homogeneous --- where p(t)
decays exponentially to p(infinity).Comment: 8 pages (LaTeX); to appear in Physical Review Letter
Persistence of Manifolds in Nonequilibrium Critical Dynamics
We study the persistence P(t) of the magnetization of a d' dimensional
manifold (i.e., the probability that the manifold magnetization does not flip
up to time t, starting from a random initial condition) in a d-dimensional spin
system at its critical point. We show analytically that there are three
distinct late time decay forms for P(t) : exponential, stretched exponential
and power law, depending on a single parameter \zeta=(D-2+\eta)/z where D=d-d'
and \eta, z are standard critical exponents. In particular, our theory predicts
that the persistence of a line magnetization decays as a power law in the d=2
Ising model at its critical point. For the d=3 critical Ising model, the
persistence of the plane magnetization decays as a power law, while that of a
line magnetization decays as a stretched exponential. Numerical results are
consistent with these analytical predictions.Comment: 4 pages revtex, 1 eps figure include
Area Distribution of Elastic Brownian Motion
We calculate the excursion and meander area distributions of the elastic
Brownian motion by using the self adjoint extension of the Hamiltonian of the
free quantum particle on the half line. We also give some comments on the area
of the Brownian motion bridge on the real line with the origin removed. We will
stress on the power of self adjoint extension to investigate different possible
boundary conditions for the stochastic processes.Comment: 18 pages, published versio
Global Persistence Exponent in Critical Dynamics: Finite Size induced Crossover
We extend the definition of a global order parameter to the case of a
critical system confined between two infinite parallel plates separated by a
finite distance . For a quench to the critical point we study the
persistence property of the global order parameter and show that there is a
crossover behaviour characterized by a non universal exponent which depends on
the ratio of the system size to a dynamic length scale
Exact solution of a model of time-dependent evolutionary dynamics in a rugged fitness landscape
A simplified form of the time dependent evolutionary dynamics of a
quasispecies model with a rugged fitness landscape is solved via a mapping onto
a random flux model whose asymptotic behavior can be described in terms of a
random walk. The statistics of the number of changes of the dominant genotype
from a finite set of genotypes are exactly obtained confirming existing
conjectures based on numerics.Comment: 5 pages RevTex 2 figures .ep
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