4,798 research outputs found
Survival of a diffusing particle in an expanding cage
We consider a Brownian particle, with diffusion constant D, moving inside an
expanding d-dimensional sphere whose surface is an absorbing boundary for the
particle. The sphere has initial radius L_0 and expands at a constant rate c.
We calculate the joint probability density, p(r,t|r_0), that the particle
survives until time t, and is at a distance r from the centre of the sphere,
given that it started at a distance r_0 from the centre.Comment: 5 page
On the Use of Finite-Size Scaling to Measure Spin-Glass Exponents
Finite-size scaling (FSS) is a standard technique for measuring scaling
exponents in spin glasses. Here we present a critique of this approach,
emphasizing the need for all length scales to be large compared to microscopic
scales. In particular we show that the replacement, in FSS analyses, of the
correlation length by its asymptotic scaling form can lead to apparently good
scaling collapses with the wrong values of the scaling exponents.Comment: RevTeX, 5 page
Global Persistence Exponent for Critical Dynamics
A `persistence exponent' is defined for nonequilibrium critical
phenomena. It describes the probability, , that the
global order parameter has not changed sign in the time interval following
a quench to the critical point from a disordered state. This exponent is
calculated in mean-field theory, in the limit of the model,
to first order in , and for the 1-d Ising model. Numerical
results are obtained for the 2-d Ising model. We argue that is a new
independent exponent.Comment: 4 pages, revtex, one figur
Evidence for the droplet/scaling picture of spin glasses
We have studied the Parisi overlap distribution for the three dimensional
Ising spin glass in the Migdal-Kadanoff approximation. For temperatures T
around 0.7Tc and system sizes upto L=32, we found a P(q) as expected for the
full Parisi replica symmetry breaking, just as was also observed in recent
Monte Carlo simulations on a cubic lattice. However, for lower temperatures our
data agree with predictions from the droplet or scaling picture. The failure to
see droplet model behaviour in Monte Carlo simulations is due to the fact that
all existing simulations have been done at temperatures too close to the
transition temperature so that sytem sizes larger than the correlation length
have not been achieved.Comment: 4 pages, 6 figure
Self-propelled particles with fluctuating speed and direction of motion
We study general aspects of active motion with fluctuations in the speed and
the direction of motion in two dimensions. We consider the case in which
fluctuations in the speed are not correlated to fluctuations in the direction
of motion, and assume that both processes can be described by independent
characteristic time-scales. We show the occurrence of a complex transient that
can exhibit a series of alternating regimes of motion, for two different
angular dynamics which correspond to persistent and directed random walks. We
also show additive corrections to the diffusion coefficient. The characteristic
time-scales are also exposed in the velocity autocorrelation, which is a sum of
exponential forms.Comment: to appear in Phys. Rev. Let
Dynamic regimes of hydrodynamically coupled self-propelling particles
We analyze the collective dynamics of self-propelling particles (spps) which
move at small Reynolds numbers including the hydrodynamic coupling to the
suspending solvent through numerical simulations. The velocity distribution
functions show marked deviations from Gaussian behavior at short times, and the
mean-square displacement at long times shows a transition from diffusive to
ballistic motion for appropriate driving mechanism at low concentrations. We
discuss the structures the spps form at long times and how they correlate to
their dynamic behavior.Comment: 7 pages, 4 figure
Free energy landscapes, dynamics and the edge of chaos in mean-field models of spin glasses
Metastable states in Ising spin-glass models are studied by finding iterative
solutions of mean-field equations for the local magnetizations. Two different
equations are studied: the TAP equations which are exact for the SK model, and
the simpler `naive-mean-field' (NMF) equations. The free-energy landscapes that
emerge are very different. For the TAP equations, the numerical studies confirm
the analytical results of Aspelmeier et al., which predict that TAP states
consist of close pairs of minima and index-one (one unstable direction) saddle
points, while for the NMF equations saddle points with large indices are found.
For TAP the barrier height between a minimum and its nearby saddle point scales
as (f-f_0)^{-1/3} where f is the free energy per spin of the solution and f_0
is the equilibrium free energy per spin. This means that for `pure states', for
which f-f_0 is of order 1/N, the barriers scale as N^{1/3}, but between states
for which f-f_0 is of order one the barriers are finite and also small so such
metastable states will be of limited physical significance. For the NMF
equations there are saddles of index K and we can demonstrate that their
complexity Sigma_K scales as a function of K/N. We have also employed an
iterative scheme with a free parameter that can be adjusted to bring the system
of equations close to the `edge of chaos'. Both for the TAP and NME equations
it is possible with this approach to find metastable states whose free energy
per spin is close to f_0. As N increases, it becomes harder and harder to find
solutions near the edge of chaos, but nevertheless the results which can be
obtained are competitive with those achieved by more time-consuming computing
methods and suggest that this method may be of general utility.Comment: 13 page
One-dimensional infinite component vector spin glass with long-range interactions
We investigate zero and finite temperature properties of the one-dimensional
spin-glass model for vector spins in the limit of an infinite number m of spin
components where the interactions decay with a power, \sigma, of the distance.
A diluted version of this model is also studied, but found to deviate
significantly from the fully connected model. At zero temperature, defect
energies are determined from the difference in ground-state energies between
systems with periodic and antiperiodic boundary conditions to determine the
dependence of the defect-energy exponent \theta on \sigma. A good fit to this
dependence is \theta =3/4-\sigma. This implies that the upper critical value of
\sigma is 3/4, corresponding to the lower critical dimension in the
d-dimensional short-range version of the model. For finite temperatures the
large m saddle-point equations are solved self-consistently which gives access
to the correlation function, the order parameter and the spin-glass
susceptibility. Special attention is paid to the different forms of finite-size
scaling effects below and above the lower critical value, \sigma =5/8, which
corresponds to the upper critical dimension 8 of the hypercubic short-range
model.Comment: 27 pages, 27 figures, 4 table
Non-equilibrium Dynamics Following a Quench to the Critical Point in a Semi-infinite System
We study the non-equilibrium dynamics (purely dissipative and relaxational)
in a semi-infinite system following a quench from the high temperature
disordered phase to its critical temperature. We show that the local
autocorrelation near the surface of a semi-infinite system decays algebraically
in time with a new exponent which is different from the bulk. We calculate this
new non-equilibrium surface exponent in several cases, both analytically and
numerically.Comment: revtex, 9 pages, 2 figures available from the author
Unusual Dynamical Scaling in the Spatial Distribution of Persistent Sites in 1D Potts Models
The distribution, n(k,t), of the interval sizes, k, between clusters of
persistent sites in the dynamical evolution of the one-dimensional q-state
Potts model is studied using a combination of numerical simulations, scaling
arguments, and exact analysis. It is shown to have the scaling form n(k,t) =
t^{-2z} f(k/t^z), with z= max(1/2,theta), where theta(q) is the persistence
exponent which characterizes the fraction of sites which have not changed their
state up to time t. When theta > 1/2, the scaling length, t^theta, for the
interval-size distribution is larger than the coarsening length scale, t^{1/2},
that characterizes spatial correlations of the Potts variables.Comment: RevTex, 11 page
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