72 research outputs found
Time-Dependent Random Walks and the Theory of Complex Adaptive Systems
Motivated by novel results in the theory of complex adaptive systems, we
analyze the dynamics of random walks in which the jumping probabilities are
{\it time-dependent}. We determine the survival probability in the presence of
an absorbing boundary. For an unbiased walk the survival probability is
maximized in the case of large temporal oscillations in the jumping
probabilities. On the other hand, a random walker who is drifted towards the
absorbing boundary performs best with a constant jumping probability. We use
the results to reveal the underlying dynamics responsible for the phenomenon of
self-segregation and clustering observed in the evolutionary minority game.Comment: 5 pages, 2 figure
Perturbation Theory for Fractional Brownian Motion in Presence of Absorbing Boundaries
Fractional Brownian motion is a Gaussian process x(t) with zero mean and
two-time correlations ~ t^{2H} + s^{2H} - |t-s|^{2H}, where H, with
0<H<1 is called the Hurst exponent. For H = 1/2, x(t) is a Brownian motion,
while for H unequal 1/2, x(t) is a non-Markovian process. Here we study x(t) in
presence of an absorbing boundary at the origin and focus on the probability
density P(x,t) for the process to arrive at x at time t, starting near the
origin at time 0, given that it has never crossed the origin. It has a scaling
form P(x,t) ~ R(x/t^H)/t^H. Our objective is to compute the scaling function
R(y), which up to now was only known for the Markov case H=1/2. We develop a
systematic perturbation theory around this limit, setting H = 1/2 + epsilon, to
calculate the scaling function R(y) to first order in epsilon. We find that
R(y) behaves as R(y) ~ y^phi as y -> 0 (near the absorbing boundary), while
R(y) ~ y^gamma exp(-y^2/2) as y -> oo, with phi = 1 - 4 epsilon + O(epsilon^2)
and gamma = 1 - 2 epsilon + O(epsilon^2). Our epsilon-expansion result confirms
the scaling relation phi = (1-H)/H proposed in Ref. [28]. We verify our
findings via numerical simulations for H = 2/3. The tools developed here are
versatile, powerful, and adaptable to different situations.Comment: 16 pages, 8 figures; revised version 2 adds discussion on spatial
small-distance cutof
Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records
We review how the renormalized force correlator Delta(u), the function
computed in the functional RG field theory, can be measured directly in
numerics and experiments on the dynamics of elastic manifolds in presence of
pinning disorder. We show how this function can be computed analytically for a
particle dragged through a 1-dimensional random-force landscape. The limit of
small velocity allows to access the critical behavior at the depinning
transition. For uncorrelated forces one finds three universality classes,
corresponding to the three extreme value statistics, Gumbel, Weibull, and
Frechet. For each class we obtain analytically the universal function Delta(u),
the corrections to the critical force, and the joint probability distribution
of avalanche sizes s and waiting times w. We find P(s)=P(w) for all three
cases. All results are checked numerically. For a Brownian force landscape,
known as the ABBM model, avalanche distributions and Delta(u) can be computed
for any velocity. For 2-dimensional disorder, we perform large-scale numerical
simulations to calculate the renormalized force correlator tensor
Delta_{ij}(u), and to extract the anisotropic scaling exponents zeta_x >
zeta_y. We also show how the Middleton theorem is violated. Our results are
relevant for the record statistics of random sequences with linear trends, as
encountered e.g. in some models of global warming. We give the joint
distribution of the time s between two successive records and their difference
in value w.Comment: 41 pages, 35 figure
On the Inelastic Collapse of a Ball Bouncing on a Randomly Vibrating Platform
We study analytically the dynamics of a ball bouncing inelastically on a
randomly vibrating platform, as a simple toy model of inelastic collapse. Of
principal interest are the distributions of the number of flights n_f till the
collapse and the total time \tau_c elapsed before the collapse. In the strictly
elastic case, both distributions have power law tails characterised by
exponents which are universal, i.e., independent of the details of the platform
noise distribution. In the inelastic case, both distributions have exponential
tails: P(n_f) ~ exp[-\theta_1 n_f] and P(\tau_c) ~ exp[-\theta_2 \tau_c]. The
decay exponents \theta_1 and \theta_2 depend continuously on the coefficient of
restitution and are nonuniversal; however as one approches the elastic limit,
they vanish in a universal manner that we compute exactly. An explicit
expression for \theta_1 is provided for a particular case of the platform noise
distribution.Comment: 32 page
Comment on "Mean First Passage Time for Anomalous Diffusion"
We correct a previously erroneous calculation [Phys. Rev. E 62, 6065 (2000)]
of the mean first passage time of a subdiffusive process to reach either end of
a finite interval in one dimension. The mean first passage time is in fact
infinite.Comment: To appear in Phys. Rev.
Asymptotic behavior of self-affine processes in semi-infinite domains
We propose to model the stochastic dynamics of a polymer passing through a
pore (translocation) by means of a fractional Brownian motion, and study its
behavior in presence of an absorbing boundary. Based on scaling arguments and
numerical simulations, we present a conjecture that provides a link between the
persistence exponent and the Hurst exponent of the process, thus
sheding light on the spatial and temporal features of translocation.
Furthermore, we show that this conjecture applies more generally to a broad
class of self affine processes undergoing anomalous diffusion in bounded
domains, and we discuss some significant examples.Comment: 4 pages, 3 figures; to be published in Phys. Rev. Let
Survival of a Diffusing Particle in a Transverse Shear Flow: A First-Passage Problem with Continuously Varying Persistence Exponent
We consider a particle diffusing in the y-direction, dy/dt=\eta(t), subject
to a transverse shear flow in the x-direction, dx/dt=f(y), where x \ge 0 and
x=0 is an absorbing boundary. We treat the class of models defined by f(y) =
\pm v_{\pm}(\pm y)^\alpha where the upper (lower) sign refers to y>0 (y<0). We
show that the particle survives with probability Q(t) \sim t^{-\theta} with
\theta = 1/4, independent of \alpha, if v_{+}=v_{-}. If v_{+} \ne v_{-},
however, we show that \theta depends on both \alpha and the ratio v_{+}/v_{-},
and we determine this dependence.Comment: 4 page
Anomalous diffusion and generalized Sparre-Andersen scaling
We are discussing long-time, scaling limit for the anomalous diffusion
composed of the subordinated L\'evy-Wiener process. The limiting anomalous
diffusion is in general non-Markov, even in the regime, where ensemble averages
of a mean-square displacement or quantiles representing the group spread of the
distribution follow the scaling characteristic for an ordinary stochastic
diffusion. To discriminate between truly memory-less process and the non-Markov
one, we are analyzing deviation of the survival probability from the (standard)
Sparre-Andersen scaling.Comment: 5 pages, 3 figure
The problem of analytical calculation of barrier crossing characteristics for Levy flights
By using the backward fractional Fokker-Planck equation we investigate the
barrier crossing event in the presence of Levy noise. After shortly review
recent results obtained with different approaches on the time characteristics
of the barrier crossing, we derive a general differential equation useful to
calculate the nonlinear relaxation time. We obtain analytically the nonlinear
relaxation time for free Levy flights and a closed expression in quadrature of
the same characteristics for cubic potential.Comment: 12 pages, 2 figures, presented at 5th International Conference on
Unsolved Problems on Noise, Lyon, France, 2008, to appear in J. Stat. Mech.:
Theory and Experimen
Universal Record Statistics of Random Walks and L\'evy Flights
It is shown that statistics of records for time series generated by random
walks are independent of the details of the jump distribution, as long as the
latter is continuous and symmetric. In N steps, the mean of the record
distribution grows as the sqrt(4N/pi) while the standard deviation grows as
sqrt((2-4/pi) N), so the distribution is non-self-averaging. The mean shortest
and longest duration records grow as sqrt(N/pi) and 0.626508... N,
respectively. The case of a discrete random walker is also studied, and similar
asymptotic behavior is found.Comment: 4 pages, 3 figures. Added journal ref. and made small changes.
Compatible with published versio
- …