159 research outputs found
First passages for a search by a swarm of independent random searchers
In this paper we study some aspects of search for an immobile target by a
swarm of N non-communicating, randomly moving searchers (numbered by the index
k, k = 1, 2,..., N), which all start their random motion simultaneously at the
same point in space. For each realization of the search process, we record the
unordered set of time moments \{\tau_k\}, where \tau_k is the time of the first
passage of the k-th searcher to the location of the target. Clearly, \tau_k's
are independent, identically distributed random variables with the same
distribution function \Psi(\tau). We evaluate then the distribution P(\omega)
of the random variable \omega \sim \tau_1/bar{\tau}, where bar{\tau} = N^{-1}
\sum_{k=1}^N \tau_k is the ensemble-averaged realization-dependent first
passage time. We show that P(\omega) exhibits quite a non-trivial and sometimes
a counterintuitive behaviour. We demonstrate that in some well-studied cases
e.g., Brownian motion in finite d-dimensional domains) the \textit{mean} first
passage time is not a robust measure of the search efficiency, despite the fact
that \Psi(\tau) has moments of arbitrary order. This implies, in particular,
that even in this simplest case (not saying about complex systems and/or
anomalous diffusion) first passage data extracted from a single particle
tracking should be regarded with an appropriate caution because of the
significant sample-to-sample fluctuations.Comment: 35 pages, 18 figures, to appear in JSTA
Persistence and First-Passage Properties in Non-equilibrium Systems
In this review we discuss the persistence and the related first-passage
properties in extended many-body nonequilibrium systems. Starting with simple
systems with one or few degrees of freedom, such as random walk and random
acceleration problems, we progressively discuss the persistence properties in
systems with many degrees of freedom. These systems include spins models
undergoing phase ordering dynamics, diffusion equation, fluctuating interfaces
etc. Persistence properties are nontrivial in these systems as the effective
underlying stochastic process is non-Markovian. Several exact and approximate
methods have been developed to compute the persistence of such non-Markov
processes over the last two decades, as reviewed in this article. We also
discuss various generalisations of the local site persistence probability.
Persistence in systems with quenched disorder is discussed briefly. Although
the main emphasis of this review is on the theoretical developments on
persistence, we briefly touch upon various experimental systems as well.Comment: Review article submitted to Advances in Physics: 149 pages, 21
Figure
Condensation of the roots of real random polynomials on the real axis
We introduce a family of real random polynomials of degree n whose
coefficients a_k are symmetric independent Gaussian variables with variance
= e^{-k^\alpha}, indexed by a real \alpha \geq 0. We compute exactly
the mean number of real roots for large n. As \alpha is varied, one finds
three different phases. First, for 0 \leq \alpha \sim
(\frac{2}{\pi}) \log{n}. For 1 < \alpha < 2, there is an intermediate phase
where grows algebraically with a continuously varying exponent,
\sim \frac{2}{\pi} \sqrt{\frac{\alpha-1}{\alpha}} n^{\alpha/2}. And finally for
\alpha > 2, one finds a third phase where \sim n. This family of real
random polynomials thus exhibits a condensation of their roots on the real line
in the sense that, for large n, a finite fraction of their roots /n are
real. This condensation occurs via a localization of the real roots around the
values \pm \exp{[\frac{\alpha}{2}(k+{1/2})^{\alpha-1} ]}, 1 \ll k \leq n.Comment: 13 pages, 2 figure
Domain walls and chaos in the disordered SOS model
Domain walls, optimal droplets and disorder chaos at zero temperature are
studied numerically for the solid-on-solid model on a random substrate. It is
shown that the ensemble of random curves represented by the domain walls obeys
Schramm's left passage formula with kappa=4 whereas their fractal dimension is
d_s=1.25, and therefore is NOT described by "Stochastic-Loewner-Evolution"
(SLE). Optimal droplets with a lateral size between L and 2L have the same
fractal dimension as domain walls but an energy that saturates at a value of
order O(1) for L->infinity such that arbitrarily large excitations exist which
cost only a small amount of energy. Finally it is demonstrated that the
sensitivity of the ground state to small changes of order delta in the disorder
is subtle: beyond a cross-over length scale L_delta ~ 1/delta the correlations
of the perturbed ground state with the unperturbed ground state, rescaled by
the roughness, are suppressed and approach zero logarithmically.Comment: 23 pages, 11 figure
Specific Heat of Quantum Elastic Systems Pinned by Disorder
We present the detailed study of the thermodynamics of vibrational modes in
disordered elastic systems such as the Bragg glass phase of lattices pinned by
quenched impurities. Our study and our results are valid within the (mean
field) replica Gaussian variational method. We obtain an expression for the
internal energy in the quantum regime as a function of the saddle point
solution, which is then expanded in powers of at low temperature .
In the calculation of the specific heat a non trivial cancellation of the
term linear in occurs, explicitly checked to second order in . The
final result is at low temperatures in dimension three and
two. The prefactor is controlled by the pinning length. This result is
discussed in connection with other analytical or numerical studies.Comment: 14 page
Finite temperature behavior of strongly disordered quantum magnets coupled to a dissipative bath
We study the effect of dissipation on the infinite randomness fixed point and
the Griffiths-McCoy singularities of random transverse Ising systems in chains,
ladders and in two-dimensions. A strong disorder renormalization group scheme
is presented that allows the computation of the finite temperature behavior of
the magnetic susceptibility and the spin specific heat. In the case of Ohmic
dissipation the susceptibility displays a crossover from Griffiths-McCoy
behavior (with a continuously varying dynamical exponent) to classical Curie
behavior at some temperature . The specific heat displays Griffiths-McCoy
singularities over the whole temperature range. For super-Ohmic dissipation we
find an infinite randomness fixed point within the same universality class as
the transverse Ising system without dissipation. In this case the phase diagram
and the parameter dependence of the dynamical exponent in the Griffiths-McCoy
phase can be determined analytically.Comment: 23 pages, 12 figure
Specific heat of the quantum Bragg Glass
We study the thermodynamics of the vibrational modes of a lattice pinned by
impurity disorder in the absence of topological defects (Bragg glass phase).
Using a replica variational method we compute the specific heat in the
quantum regime and find at low temperatures in dimension
three and two. The prefactor is controlled by the pinning length. The non
trivial cancellation of the linear term in arises from the so-called
marginality condition and has important consequences for other mean field
models.Comment: 5 pages, RevTex, strongly revised versio
Symmetry breaking between statistically equivalent, independent channels in a few-channel chaotic scattering
We study the distribution function of the random variable , where 's are the partial Wigner
delay times for chaotic scattering in a disordered system with independent,
statistically equivalent channels. In this case, 's are i.i.d. random
variables with a distribution characterized by a "fat" power-law
intermediate tail , truncated by an exponential (or a
log-normal) function of . For and N=3, we observe a surprisingly
rich behavior of revealing a breakdown of the symmetry between
identical independent channels. For N=2, numerical simulations of the quasi
one-dimensional Anderson model confirm our findings.Comment: 4 pages, 5 figure
Large time zero temperature dynamics of the spherical p=2-spin glass model of finite size
18 pages, 3 figures. Published version18 pages, 3 figures. Published versionWe revisit the long time dynamics of the spherical fully connected -spin glass model when the number of spins is large but {\it finite}. At where the system is in a (trivial) spin-glass phase, and on long time scale we show that the behavior of physical observables, like the energy, correlation and response functions, is controlled by the density of near-extreme eigenvalues at the edge of the spectrum of the coupling matrix , and are thus non self-averaging. We show that the late time decay of these observables, once averaged over the disorder, is controlled by new universal exponents which we compute exactly
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