630 research outputs found
Persistence exponents of non-Gaussian processes in statistical mechanics
Motivated by certain problems of statistical physics we consider a stationary
stochastic process in which deterministic evolution is interrupted at random
times by upward jumps of a fixed size. If the evolution consists of linear
decay, the sample functions are of the "random sawtooth" type and the level
dependent persistence exponent \theta can be calculated exactly. We then
develop an expansion method valid for small curvature of the deterministic
curve. The curvature parameter g plays the role of the coupling constant of an
interacting particle system. The leading order curvature correction to \theta
is proportional to g^{2/3}. The expansion applies in particular to exponential
decay in the limit of large level, where the curvature correction considerably
improves the linear approximation. The Langevin equation, with Gaussian white
noise, is recovered as a singular limiting case.Comment: 20 pages, 3 figure
Exact analytical solution of the collapse of self-gravitating Brownian particles and bacterial populations at zero temperature
We provide an exact analytical solution of the collapse dynamics of
self-gravitating Brownian particles and bacterial populations at zero
temperature. These systems are described by the Smoluchowski-Poisson system or
Keller-Segel model in which the diffusion term is neglected. As a result, the
dynamics is purely deterministic. A cold system undergoes a gravitational
collapse leading to a finite time singularity: the central density increases
and becomes infinite in a finite time t_coll. The evolution continues in the
post collapse regime. A Dirac peak emerges, grows and finally captures all the
mass in a finite time t_end, while the central density excluding the Dirac peak
progressively decreases. Close to the collapse time, the pre and post collapse
evolution is self-similar. Interestingly, if one starts from a parabolic
density profile, one obtains an exact analytical solution that describes the
whole collapse dynamics, from the initial time to the end, and accounts for non
self-similar corrections that were neglected in previous works. Our results
have possible application in different areas including astrophysics,
chemotaxis, colloids and nanoscience
Weak Disorder in Fibonacci Sequences
We study how weak disorder affects the growth of the Fibonacci series. We
introduce a family of stochastic sequences that grow by the normal Fibonacci
recursion with probability 1-epsilon, but follow a different recursion rule
with a small probability epsilon. We focus on the weak disorder limit and
obtain the Lyapunov exponent, that characterizes the typical growth of the
sequence elements, using perturbation theory. The limiting distribution for the
ratio of consecutive sequence elements is obtained as well. A number of
variations to the basic Fibonacci recursion including shift, doubling, and
copying are considered.Comment: 4 pages, 2 figure
Random Geometric Series
Integer sequences where each element is determined by a previous randomly
chosen element are investigated analytically. In particular, the random
geometric series x_n=2x_p with 0<=p<=n-1 is studied. At large n, the moments
grow algebraically, n^beta(s) with beta(s)=2^s-1, while the typical
behavior is x_n n^ln 2. The probability distribution is obtained explicitly in
terms of the Stirling numbers of the first kind and it approaches a log-normal
distribution asymptotically.Comment: 6 pages, 2 figure
Non-equilibrium Phase-Ordering with a Global Conservation Law
In all dimensions, infinite-range Kawasaki spin exchange in a quenched Ising
model leads to an asymptotic length-scale
at because the kinetic coefficient is renormalized by the broken-bond
density, . For , activated kinetics recovers the
standard asymptotic growth-law, . However, at all temperatures,
infinite-range energy-transport is allowed by the spin-exchange dynamics. A
better implementation of global conservation, the microcanonical Creutz
algorithm, is well behaved and exhibits the standard non-conserved growth law,
, at all temperatures.Comment: 2 pages and 2 figures, uses epsf.st
Growth and Structure of Stochastic Sequences
We introduce a class of stochastic integer sequences. In these sequences,
every element is a sum of two previous elements, at least one of which is
chosen randomly. The interplay between randomness and memory underlying these
sequences leads to a wide variety of behaviors ranging from stretched
exponential to log-normal to algebraic growth. Interestingly, the set of all
possible sequence values has an intricate structure.Comment: 4 pages, 4 figure
On War: The Dynamics of Vicious Civilizations
The dynamics of ``vicious'', continuously growing civilizations (domains),
which engage in ``war'' whenever two domains meet, is investigated. In the war
event, the smaller domain is annihilated, while the larger domain is reduced in
size by a fraction \e of the casualties of the loser. Here \e quantifies
the fairness of the war, with \e=1 corresponding to a fair war with equal
casualties on both side, and \e=0 corresponding to a completely unfair war
where the winner suffers no casualties. In the heterogeneous version of the
model, evolution begins from a specified initial distribution of domains, while
in the homogeneous system, there is a continuous and spatially uniform input of
point domains, in addition to the growth and warfare. For the heterogeneous
case, the rate equations are derived and solved, and comparisons with numerical
simulations are made. An exact solution is also derived for the case of equal
size domains in one dimension. The heterogeneous system is found to coarsen,
with the typical cluster size growing linearly in time and the number
density of domains decreases as . For the homogeneous system, two
different long-time behaviors arise as a function of \e. When 1/2<\e\leq 1
(relatively fair wars), a steady state arises which is characterized by
egalitarian competition between domains of comparable size. In the limiting
case of \e=1, rate equations which simultaneously account for the
distribution of domains and that of the intervening gaps are derived and
solved. The steady state is characterized by domains whose age is typically
much larger than their size. When 0\leq\e<1/2 (unfair wars), a few
``superpowers'' ultimately dominate. Simulations indicate that this coarsening
process is characterized by power-law temporal behavior, with non-universalComment: 43 pages, plain TeX, 12 figures included, gzipped and uuencode
Analytical results for random walk persistence
In this paper, we present the detailed calculation of the persistence
exponent for a nearly-Markovian Gaussian process , a problem
initially introduced in [Phys. Rev. Lett. 77, 1420 (1996)], describing the
probability that the walker never crosses the origin. New resummed perturbative
and non-perturbative expressions for are obtained, which suggest a
connection with the result of the alternative independent interval
approximation (IIA). The perturbation theory is extended to the calculation of
for non-Gaussian processes, by making a strong connection between the
problem of persistence and the calculation of the energy eigenfunctions of a
quantum mechanical problem. Finally, we give perturbative and non-perturbative
expressions for the persistence exponent , describing the
probability that the process remains bigger than .Comment: 23 pages; accepted for publication to Phys. Rev. E (Dec. 98
Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion
For a specific choice of the diffusion, the parabolic-elliptic
Patlak-Keller-Segel system with non-linear diffusion (also referred to as the
quasi-linear Smoluchowski-Poisson equation) exhibits an interesting threshold
phenomenon: there is a critical mass such that all the solutions with
initial data of mass smaller or equal to exist globally while the
solution blows up in finite time for a large class of initial data with mass
greater than . Unlike in space dimension 2, finite mass self-similar
blowing-up solutions are shown to exist in space dimension
Random Fibonacci Sequences
Solutions to the random Fibonacci recurrence x_{n+1}=x_{n} + or - Bx_{n-1}
decrease (increase) exponentially, x_{n} = exp(lambda n), for sufficiently
small (large) B. In the limits B --> 0 and B --> infinity, we expand the
Lyapunov exponent lambda(B) in powers of B and B^{-1}, respectively. For the
classical case of we obtain exact non-perturbative results. In
particular, an invariant measure associated with Ricatti variable
r_n=x_{n+1}/x_{n} is shown to exhibit plateaux around all rational.Comment: 11 Pages (Multi-Column); 3 EPS Figures ; Submitted to J. Phys.
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