408 research outputs found
Exact Occupation Time Distribution in a Non-Markovian Sequence and Its Relation to Spin Glass Models
We compute exactly the distribution of the occupation time in a discrete {\em
non-Markovian} toy sequence which appears in various physical contexts such as
the diffusion processes and Ising spin glass chains. The non-Markovian property
makes the results nontrivial even for this toy sequence. The distribution is
shown to have non-Gaussian tails characterized by a nontrivial large deviation
function which is computed explicitly. An exact mapping of this sequence to an
Ising spin glass chain via a gauge transformation raises an interesting new
question for a generic finite sized spin glass model: at a given temperature,
what is the distribution (over disorder) of the thermally averaged number of
spins that are aligned to their local fields? We show that this distribution
remains nontrivial even at infinite temperature and can be computed explicitly
in few cases such as in the Sherrington-Kirkpatrick model with Gaussian
disorder.Comment: 10 pages Revtex (two-column), 1 eps figure (included
Persistence in higher dimensions : a finite size scaling study
We show that the persistence probability , in a coarsening system of
linear size at a time , has the finite size scaling form where is the persistence exponent and
is the coarsening exponent. The scaling function for
and is constant for large . The scaling form implies a fractal
distribution of persistent sites with power-law spatial correlations. We study
the scaling numerically for Glauber-Ising model at dimension to 4 and
extend the study to the diffusion problem. Our finite size scaling ansatz is
satisfied in all these cases providing a good estimate of the exponent
.Comment: 4 pages in RevTeX with 6 figures. To appear in Phys. Rev.
Survival Probability of a Ballistic Tracer Particle in the Presence of Diffusing Traps
We calculate the survival probability P_S(t) up to time t of a tracer
particle moving along a deterministic trajectory in a continuous d-dimensional
space in the presence of diffusing but mutually noninteracting traps. In
particular, for a tracer particle moving ballistically with a constant velocity
c, we obtain an exact expression for P_S(t), valid for all t, for d<2. For d
\geq 2, we obtain the leading asymptotic behavior of P_S(t) for large t. In all
cases, P_S(t) decays exponentially for large t, P_S(t) \sim \exp(-\theta t). We
provide an explicit exact expression for the exponent \theta in dimensions d
\leq 2, and for the physically relevant case, d=3, as a function of the system
parameters.Comment: RevTeX, 4 page
Persistence in a Stationary Time-series
We study the persistence in a class of continuous stochastic processes that
are stationary only under integer shifts of time. We show that under certain
conditions, the persistence of such a continuous process reduces to the
persistence of a corresponding discrete sequence obtained from the measurement
of the process only at integer times. We then construct a specific sequence for
which the persistence can be computed even though the sequence is
non-Markovian. We show that this may be considered as a limiting case of
persistence in the diffusion process on a hierarchical lattice.Comment: 8 pages revte
Persistence of a Continuous Stochastic Process with Discrete-Time Sampling: Non-Markov Processes
We consider the problem of `discrete-time persistence', which deals with the
zero-crossings of a continuous stochastic process, X(T), measured at discrete
times, T = n(\Delta T). For a Gaussian Stationary Process the persistence (no
crossing) probability decays as exp(-\theta_D T) = [\rho(a)]^n for large n,
where a = \exp[-(\Delta T)/2], and the discrete persistence exponent, \theta_D,
is given by \theta_D = \ln(\rho)/2\ln(a). Using the `Independent Interval
Approximation', we show how \theta_D varies with (\Delta T) for small (\Delta
T) and conclude that experimental measurements of persistence for smooth
processes, such as diffusion, are less sensitive to the effects of discrete
sampling than measurements of a randomly accelerated particle or random walker.
We extend the matrix method developed by us previously [Phys. Rev. E 64,
015151(R) (2001)] to determine \rho(a) for a two-dimensional random walk and
the one-dimensional random acceleration problem. We also consider `alternating
persistence', which corresponds to a < 0, and calculate \rho(a) for this case.Comment: 14 pages plus 8 figure
Random Walks in Logarithmic and Power-Law Potentials, Nonuniversal Persistence, and Vortex Dynamics in the Two-Dimensional XY Model
The Langevin equation for a particle (`random walker') moving in
d-dimensional space under an attractive central force, and driven by a Gaussian
white noise, is considered for the case of a power-law force, F(r) = -
Ar^{-sigma}. The `persistence probability', P_0(t), that the particle has not
visited the origin up to time t, is calculated. For sigma > 1, the force is
asymptotically irrelevant (with respect to the noise), and the asymptotics of
P_0(t) are those of a free random walker. For sigma < 1, the noise is
(dangerously) irrelevant and the asymptotics of P_0(t) can be extracted from a
weak noise limit within a path-integral formalism. For the case sigma=1,
corresponding to a logarithmic potential, the noise is exactly marginal. In
this case, P_0(t) decays as a power-law, P_0(t) \sim t^{-theta}, with an
exponent theta that depends continuously on the ratio of the strength of the
potential to the strength of the noise. This case, with d=2, is relevant to the
annihilation dynamics of a vortex-antivortex pair in the two-dimensional XY
model. Although the noise is multiplicative in the latter case, the relevant
Langevin equation can be transformed to the standard form discussed in the
first part of the paper. The mean annihilation time for a pair initially
separated by r is given by t(r) \sim r^2 ln(r/a) where a is a microscopic
cut-off (the vortex core size). Implications for the nonequilibrium critical
dynamics of the system are discussed and compared to numerical simulation
results.Comment: 10 pages, 1 figur
Scaling of loop-erased walks in 2 to 4 dimensions
We simulate loop-erased random walks on simple (hyper-)cubic lattices of
dimensions 2,3, and 4. These simulations were mainly motivated to test recent
two loop renormalization group predictions for logarithmic corrections in
, simulations in lower dimensions were done for completeness and in order
to test the algorithm. In , we verify with high precision the prediction
, where the number of steps after erasure scales with the number
of steps before erasure as . In we again find a power law,
but with an exponent different from the one found in the most precise previous
simulations: . Finally, we see clear deviations from the
naive scaling in . While they agree only qualitatively with the
leading logarithmic corrections predicted by several authors, their agreement
with the two-loop prediction is nearly perfect.Comment: 3 pages, including 3 figure
Slow Relaxation in a Constrained Ising Spin Chain: a Toy Model for Granular Compaction
We present detailed analytical studies on the zero temperature coarsening
dynamics in an Ising spin chain in presence of a dynamically induced field that
favors locally the `-' phase compared to the `+' phase. We show that the
presence of such a local kinetic bias drives the system into a late time state
with average magnetization m=-1. However the magnetization relaxes into this
final value extremely slowly in an inverse logarithmic fashion. We further map
this spin model exactly onto a simple lattice model of granular compaction that
includes the minimal microscopic moves needed for compaction. This toy model
then predicts analytically an inverse logarithmic law for the growth of density
of granular particles, as seen in recent experiments and thereby provides a new
mechanism for the inverse logarithmic relaxation. Our analysis utilizes an
independent interval approximation for the particle and the hole clusters and
is argued to be exact at late times (supported also by numerical simulations).Comment: 9 pages RevTeX, 1 figures (.eps
Fraction of uninfected walkers in the one-dimensional Potts model
The dynamics of the one-dimensional q-state Potts model, in the zero
temperature limit, can be formulated through the motion of random walkers which
either annihilate (A + A -> 0) or coalesce (A + A -> A) with a q-dependent
probability. We consider all of the walkers in this model to be mutually
infectious. Whenever two walkers meet, they experience mutual contamination.
Walkers which avoid an encounter with another random walker up to time t remain
uninfected. The fraction of uninfected walkers is investigated numerically and
found to decay algebraically, U(t) \sim t^{-\phi(q)}, with a nontrivial
exponent \phi(q). Our study is extended to include the coupled
diffusion-limited reaction A+A -> B, B+B -> A in one dimension with equal
initial densities of A and B particles. We find that the density of walkers
decays in this model as \rho(t) \sim t^{-1/2}. The fraction of sites unvisited
by either an A or a B particle is found to obey a power law, P(t) \sim
t^{-\theta} with \theta \simeq 1.33. We discuss these exponents within the
context of the q-state Potts model and present numerical evidence that the
fraction of walkers which remain uninfected decays as U(t) \sim t^{-\phi},
where \phi \simeq 1.13 when infection occurs between like particles only, and
\phi \simeq 1.93 when we also include cross-species contamination.Comment: Expanded introduction with more discussion of related wor
Global persistence exponent of the two-dimensional Blume-Capel model
The global persistence exponent is calculated for the
two-dimensional Blume-Capel model following a quench to the critical point from
both disordered states and such with small initial magnetizations.
Estimates are obtained for the nonequilibrium critical dynamics on the
critical line and at the tricritical point.
Ising-like universality is observed along the critical line and a different
value is found at the tricritical point.Comment: 7 pages with 3 figure
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