102 research outputs found
Coherent-state path integral versus coarse-grained effective stochastic equation of motion: From reaction diffusion to stochastic sandpiles
We derive and study two different formalisms used for non-equilibrium
processes: The coherent-state path integral, and an effective, coarse-grained
stochastic equation of motion. We first study the coherent-state path integral
and the corresponding field theory, using the annihilation process
as an example. The field theory contains counter-intuitive quartic vertices. We
show how they can be interpreted in terms of a first-passage problem.
Reformulating the coherent-state path integral as a stochastic equation of
motion, the noise generically becomes imaginary. This renders it not only
difficult to interpret, but leads to convergence problems at finite times. We
then show how alternatively an effective coarse-grained stochastic equation of
motion with real noise can be constructed. The procedure is similar in spirit
to the derivation of the mean-field approximation for the Ising model, and the
ensuing construction of its effective field theory. We finally apply our
findings to stochastic Manna sandpiles. We show that the coherent-state path
integral is inappropriate, or at least inconvenient. As an alternative, we
derive and solve its mean-field approximation, which we then use to construct a
coarse-grained stochastic equation of motion with real noise.Comment: 29 pages, 33 figures. This is a pedagogic introduction to stochastic
processes, their modeling, and effective field theory. Version 2: writing
improved + a new appendi
Perturbative Expansion for the Maximum of Fractional Brownian Motion
Brownian motion is the only random process which is Gaussian, stationary and
Markovian. Dropping the Markovian property, i.e. allowing for memory, one
obtains a class of processes called fractional Brownian motion, indexed by the
Hurst exponent . For , Brownian motion is recovered. We develop a
perturbative approach to treat the non-locality in time in an expansion in
. This allows us to derive analytic results beyond scaling
exponents for various observables related to extreme value statistics: The
maximum of the process and the time at which this maximum
is reached, as well as their joint distribution. We test our analytical
predictions with extensive numerical simulations for different values of .
They show excellent agreement, even for far from .Comment: 28 pages, 9 figure
Elasticity of a contact-line and avalanche-size distribution at depinning
Motivated by recent experiments, we extend the Joanny-deGennes calculation of
the elasticity of a contact line to an arbitrary contact angle and an arbitrary
plate inclination in presence of gravity. This requires a diagonalization of
the elastic modes around the non-linear equilibrium profile, which is carried
out exactly. We then make detailed predictions for the avalanche-size
distribution at quasi-static depinning: we study how the universal (i.e.
short-scale independent) rescaled size distribution and the ratio of moments of
local to global avalanches depend on the precise form of the elastic kernel.Comment: 15 pages, 11 figure
Classification of Perturbations for Membranes with Bending Rigidity
A complete classification of the renormalization-group flow is given for
impurity-like marginal operators of membranes whose elastic stress scales like
(\Delta r)^2 around the external critical dimension d_c=2. These operators are
classified by characteristic functions on R^2 x R^2.Comment: latex, 3 .eps-file
Spatial shape of avalanches in the Brownian force model
We study the Brownian force model (BFM), a solvable model of avalanche
statistics for an interface, in a general discrete setting. The BFM describes
the overdamped motion of elastically coupled particles driven by a parabolic
well in independent Brownian force landscapes. Avalanches are defined as the
collective jump of the particles in response to an arbitrary monotonous change
in the well position (i.e. in the applied force). We derive an exact formula
for the joint probability distribution of these jumps. From it we obtain the
joint density of local avalanche sizes for stationary driving in the
quasi-static limit near the depinning threshold. A saddle-point analysis
predicts the spatial shape of avalanches in the limit of large aspect ratios
for the continuum version of the model. We then study fluctuations around this
saddle point, and obtain the leading corrections to the mean shape, the
fluctuations around the mean shape and the shape asymmetry, for finite aspect
ratios. Our results are finally confronted to numerical simulations.Comment: 41 pages, 16 figure
Avalanche shape and exponents beyond mean-field theory
Elastic systems, such as magnetic domain walls, density waves, contact lines,
and cracks, are all pinned by substrate disorder. When driven, they move via
successive jumps called avalanches, with power law distributions of size,
duration and velocity. Their exponents, and the shape of an avalanche, defined
as its mean velocity as function of time, have recently been studied. They are
known approximatively from experiments and simulations, and were predicted from
mean-field models, such as the Brownian force model (BFM), where each point of
the elastic interface sees a force field which itself is a random walk. As we
showed in EPL 97 (2012) 46004, the BFM is the starting point for an expansion around the upper critical dimension, with
for short-ranged elasticity, and for long-ranged elasticity. Here
we calculate analytically the , i.e. 1-loop, correction to
the avalanche shape at fixed duration , for both types of elasticity. The
exact expression is well approximated by \left_T\simeq [
Tx(1-x)]^{\gamma-1} \exp\left( {\cal A}\left[\frac12-x\right]\right), .
The asymmetry is negative for
close to , skewing the avalanche towards its end, as observed in
numerical simulations in and . The exponent is
given by the two independent exponents at depinning, the roughness and
the dynamical exponent . We propose a general procedure to predict other
avalanche exponents in terms of and . We finally introduce and
calculate the shape at fixed avalanche size, not yet measured in experiments or
simulations.Comment: 6 pages, 2 figure
Statistics of Avalanches with Relaxation, and Barkhausen Noise: A Solvable Model
We study a generalization of the Alessandro-Beatrice-Bertotti-Montorsi (ABBM)
model of a particle in a Brownian force landscape, including retardation
effects. We show that under monotonous driving the particle moves forward at
all times, as it does in absence of retardation (Middleton's theorem). This
remarkable property allows us to develop an analytical treatment. The model
with an exponentially decaying memory kernel is realized in Barkhausen
experiments with eddy-current relaxation, and has previously been shown
numerically to account for the experimentally observed asymmetry of
Barkhausen-pulse shapes. We elucidate another qualitatively new feature: the
breakup of each avalanche of the standard ABBM model into a cluster of
sub-avalanches, sharply delimited for slow relaxation under quasi-static
driving. These conditions are typical for earthquake dynamics. With relaxation
and aftershock clustering, the present model includes important ingredients for
an effective description of earthquakes. We analyze quantitatively the limits
of slow and fast relaxation for stationary driving with velocity v>0. The
v-dependent power-law exponent for small velocities, and the critical driving
velocity at which the particle velocity never vanishes, are modified. We also
analyze non-stationary avalanches following a step in the driving magnetic
field. Analytically, we obtain the mean avalanche shape at fixed size, the
duration distribution of the first sub-avalanche, and the time dependence of
the mean velocity. We propose to study these observables in experiments,
allowing to directly measure the shape of the memory kernel, and to trace eddy
current relaxation in Barkhausen noise.Comment: 39 pages, 26 figure
The Freezing of Random RNA
We study secondary structures of random RNA molecules by means of a
renormalized field theory based on an expansion in the sequence disorder. We
show that there is a continuous phase transition from a molten phase at higher
temperatures to a low-temperature glass phase. The primary freezing occurs
above the critical temperature, with local islands of stable folds forming
within the molten phase. The size of these islands defines the correlation
length of the transition. Our results include critical exponents at the
transition and in the glass phase.Comment: 4 pages, 8 figures. v2: presentation improve
Hausdorff Dimension of the Record Set of a Fractional Brownian Motion
We prove that the Hausdorff dimension of the record set of a fractional
Brownian motion with Hurst parameter equals
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