144 research outputs found
Dynamics at the angle of repose: jamming, bistability, and collapse
When a sandpile relaxes under vibration, it is known that its measured angle
of repose is bistable in a range of values bounded by a material-dependent
maximal angle of stability; thus, at the same angle of repose, a sandpile can
be stationary or avalanching, depending on its history. In the nearly jammed
slow dynamical regime, sandpile collapse to a zero angle of repose can also
occur, as a rare event. We claim here that fluctuations of {\it dilatancy} (or
local density) are the key ingredient that can explain such varied phenomena.
In this work, we model the dynamics of the angle of repose and of the density
fluctuations, in the presence of external noise, by means of coupled stochastic
equations. Among other things, we are able to describe sandpile collapse in
terms of an activated process, where an effective temperature (related to the
density as well as to the external vibration intensity) competes against the
configurational barriers created by the density fluctuations.Comment: 15 pages, 1 figure. Minor changes and update
Approach to Asymptotic Behaviour in the Dynamics of the Trapping Reaction
We consider the trapping reaction A + B -> B in space dimension d=1, where
the A and B particles have diffusion constants D_A, D_B respectively. We
calculate the probability, Q(t), that a given A particle has not yet reacted at
time t. Exploiting a recent formulation in which the B particles are eliminated
from the problem we find, for t -> \infty, , where
is the density of B particles and for .Comment: 8 pages, 2 figures; minor change
Onsager reciprocity relations without microscopic reversibility
In this paper we show that Onsager--Machlup time reversal properties of
thermodynamic fluctuations and Onsager reciprocity relations for transport
coefficients can hold also if the microscopic dynamics is not reversible. This
result is based on the explicit construction of a class of conservative models
which can be analysed rigorously.Comment: revtex, no figure
Diffusion with critically correlated traps and the slow relaxation of the longest wavelength mode
We study diffusion on a substrate with permanent traps distributed with
critical positional correlation, modeled by their placement on the perimeters
of a critical percolation cluster. We perform a numerical analysis of the
vibrational density of states and the largest eigenvalue of the equivalent
scalar elasticity problem using the method of Arnoldi and Saad. We show that
the critical trap correlation increases the exponent appearing in the stretched
exponential behavior of the low frequency density of states by approximately a
factor of two as compared to the case of no correlations. A finite size scaling
hypothesis of the largest eigenvalue is proposed and its relation to the
density of states is given. The numerical analysis of this scaling postulate
leads to the estimation of the stretch exponent in good agreement with the
density of states result.Comment: 15 pages, LaTeX (RevTeX
Trapping of a random walk by diffusing traps
We present a systematic analytical approach to the trapping of a random walk
by a finite density rho of diffusing traps in arbitrary dimension d. We confirm
the phenomenologically predicted e^{-c_d rho t^{d/2}} time decay of the
survival probability, and compute the dimension dependent constant c_d to
leading order within an eps=2-d expansion.Comment: 16 pages, to appear in J. Phys.
Self-intersection local times of random walks: Exponential moments in subcritical dimensions
Fix , not necessarily integer, with . We study the -fold
self-intersection local time of a simple random walk on the lattice up
to time . This is the -norm of the vector of the walker's local times,
. We derive precise logarithmic asymptotics of the expectation of
for scales that are bounded from
above, possibly tending to zero. The speed is identified in terms of mixed
powers of and , and the precise rate is characterized in terms of
a variational formula, which is in close connection to the {\it
Gagliardo-Nirenberg inequality}. As a corollary, we obtain a large-deviation
principle for for deviation functions satisfying
t r_t\gg\E[\|\ell_t\|_p]. Informally, it turns out that the random walk
homogeneously squeezes in a -dependent box with diameter of order to produce the required amount of self-intersections. Our main tool is
an upper bound for the joint density of the local times of the walk.Comment: 15 pages. To appear in Probability Theory and Related Fields. The
final publication is available at springerlink.co
Fluctuations in Stationary non Equilibrium States
In this paper we formulate a dynamical fluctuation theory for stationary non
equilibrium states (SNS) which covers situations in a nonlinear hydrodynamic
regime and is verified explicitly in stochastic models of interacting
particles. In our theory a crucial role is played by the time reversed
dynamics. Our results include the modification of the Onsager-Machlup theory in
the SNS, a general Hamilton-Jacobi equation for the macroscopic entropy and a
non equilibrium, non linear fluctuation dissipation relation valid for a wide
class of systems
Metastability and small eigenvalues in Markov chains
In this letter we announce rigorous results that elucidate the relation
between metastable states and low-lying eigenvalues in Markov chains in a much
more general setting and with considerable greater precision as was so far
available. This includes a sharp uncertainty principle relating all low-lying
eigenvalues to mean times of metastable transitions, a relation between the
support of eigenfunctions and the attractor of a metastable state, and sharp
estimates on the convergence of probability distribution of the metastable
transition times to the exponential distribution.Comment: 5pp, AMSTe
Simulations for trapping reactions with subdiffusive traps and subdiffusive particles
While there are many well-known and extensively tested results involving
diffusion-limited binary reactions, reactions involving subdiffusive reactant
species are far less understood. Subdiffusive motion is characterized by a mean
square displacement with . Recently we
calculated the asymptotic survival probability of a (sub)diffusive
particle () surrounded by (sub)diffusive traps () in one
dimension. These are among the few known results for reactions involving
species characterized by different anomalous exponents. Our results were
obtained by bounding, above and below, the exact survival probability by two
other probabilities that are asymptotically identical (except when
and ). Using this approach, we were not able to
estimate the time of validity of the asymptotic result, nor the way in which
the survival probability approaches this regime. Toward this goal, here we
present a detailed comparison of the asymptotic results with numerical
simulations. In some parameter ranges the asymptotic theory describes the
simulation results very well even for relatively short times. However, in other
regimes more time is required for the simulation results to approach asymptotic
behavior, and we arrive at situations where we are not able to reach asymptotia
within our computational means. This is regrettably the case for
and , where we are therefore not able to prove
or disprove even conjectures about the asymptotic survival probability of the
particle.Comment: 15 pages, 10 figures, submitted to Journal of Physics: Condensed
Matter; special issue on Chemical Kinetics Beyond the Textbook: Fluctuations,
Many-Particle Effects and Anomalous Dynamics, eds. K.Lindenberg, G.Oshanin
and M.Tachiy
Stretched exponential relaxation in the mode-coupling theory for the Kardar-Parisi-Zhang equation
We study the mode-coupling theory for the Kardar-Parisi-Zhang equation in the
strong-coupling regime, focusing on the long time properties. By a saddle point
analysis of the mode-coupling equations, we derive exact results for the
correlation function in the long time limit - a limit which is hard to study
using simulations. The correlation function at wavevector k in dimension d is
found to behave asymptotically at time t as C(k,t)\simeq 1/k^{d+4-2z}
(Btk^z)^{\gamma/z} e^{-(Btk^z)^{1/z}}, with \gamma=(d-1)/2, A a determined
constant and B a scale factor.Comment: RevTex, 4 pages, 1 figur
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