151 research outputs found

### The number statistics and optimal history of non-equilibrium steady states of mortal diffusing particles

Suppose that a point-like steady source at $x=0$ injects particles into a
half-infinite line. The particles diffuse and die. At long times a
non-equilibrium steady state sets in, and we assume that it involves many
particles. If the particles are non-interacting, their total number $N$ in the
steady state is Poisson-distributed with mean $\bar{N}$ predicted from a
deterministic reaction-diffusion equation. Here we determine the most likely
density history of this driven system conditional on observing a given $N$. We
also consider two prototypical examples of \emph{interacting} diffusing
particles: (i) a family of mortal diffusive lattice gases with constant
diffusivity (as illustrated by the simple symmetric exclusion process with
mortal particles), and (ii) random walkers that can annihilate in pairs. In
both examples we calculate the variances of the (non-Poissonian) stationary
distributions of $N$.Comment: 11 pages in one-column format, 1 figure, revised and extended versio

### Mortal Brownian motion: three short stories

Mortality introduces an intrinsic time scale into the scale-invariant
Brownian motion. This fact has important consequences for different statistics
of Brownian motion. Here we are telling three short stories, where spontaneous
death, such as radioactive decay, puts a natural limit to "lifetime
achievements" of a Brownian particle. In story 1 we determine the probability
distribution of a mortal Brownian particle (MBP) reaching a specified point in
space at the time of its death. In story 2 we determine the probability
distribution of the area $A=\int_0^{T} x(t) dt$ of a MBP on the line. Story 3
addresses the distribution of the winding angle of a MBP wandering around a
reflecting disk in the plane. In stories 1 and 2 the probability distributions
exhibit integrable singularities at zero values of the position and the area,
respectively. In story 3 a singularity at zero winding angle appears only in
the limit of very high mortality. A different integrable singularity appears at
a nonzero winding angle. It is inherited from the recently uncovered
singularity of the short-time large-deviation function of the winding angle for
immortal Brownian motion.Comment: 7 one-column pages, 4 figures, small change

### Fluctuations provide strong selection in Ostwald ripening

A selection problem that appears in the Lifshitz-Slyozov (LS) theory of
Ostwald ripening is reexamined. The problem concerns selection of a
self-similar distribution function (DF) of the minority domains with respect to
their sizes from a whole one-parameter family of solutions. A strong selection
rule is found via an account of fluctuations. Fluctuations produce an infinite
tail in the DF and drive the DF towards the "limiting solution" of LS or its
analogs for other growth mechanisms.Comment: 4 pages, RevTex, more details and reference adde

### Large fluctuations of the area under a constrained Brownian excursion

We study large fluctuations of the area $\mathcal{A}$ under a Brownian
excursion $x(t)$ on the time interval $|t|\leq T$, constrained to stay away
from a moving wall $x_0(t)$ such that $x_0(-T)=x_0(T)=0$ and $x_0(|t|0$. We
focus on wall functions described by a family of generalized parabolas
$x_0(t)=T^{\gamma} [1-(t/T)^{2k}]$, where $k\geq 1$. Using the optimal
fluctuation method (OFM), we calculate the large deviation function (LDF) of
the area at long times. The OFM provides a simple description of the area
fluctuations in terms of optimal paths, or rays, of the Brownian motion. We
show that the LDF has a jump in the third derivative with respect to
$\mathcal{A}$ at a critical value of $\mathcal{A}$. This singularity results
from a qualitative change of the optimal path, and it can be interpreted as a
third-order dynamical phase transition. Although the OFM is not applicable for
typical (small) area fluctuations, we argue that it correctly captures their
power-law scaling of $\mathcal{A}$ with $T$ with an exponent that depends
continuously on $\gamma$ and on $k$. We also consider the cosine wall
$x_0(t)=T^{\gamma} \cos[\pi t/(2T)]$ to illustrate a different possible
behavior of the optimal path and of the scaling of typical fluctuations. For
some wall functions additional phase transitions, which result from a
coexistence of multiple OFM solutions, should be possible.Comment: 10 pages, 5 figure

### Spectral theory of metastability and extinction in birth-death systems

We suggest a general spectral method for calculating statistics of multi-step
birth-death processes and chemical reactions of the type mA->nA (m and n are
positive integers) which possess an absorbing state. The method yields accurate
results for the extinction statistics, and for the quasi-stationary probability
distribution, including large deviations, of the metastable state. The power of
the method is demonstrated on the example of binary annihilation and triple
branching 2A->0 and A->3A, representative of the rather general class of
dissociation-recombination reactions.Comment: 4 pages, 3 figure

### Noise enhanced persistence in a biochemical regulatory network with feedback control

We find that discrete noise of inhibiting (signal) molecules can greatly
delay the extinction of plasmids in a plasmid replication system: a
prototypical biochemical regulatory network. We calculate the probability
distribution of the metastable state of the plasmids and show on this example
that the reaction rate equations may fail in predicting the average number of
regulated molecules even when this number is large, and the time is much
shorter than the mean extinction time.Comment: 4 pages, 4 figures, submitted for publicatio

### Statistics of large currents in the Kipnis-Marchioro-Presutti model in a ring geometry

We use the macroscopic fluctuation theory to determine the statistics of
large currents in the Kipnis-Marchioro-Presutti (KMP) model in a ring geometry.
About 10 years ago this simple setting was instrumental in identifying a
breakdown of the additivity principle in a class of lattice gases at currents
exceeding a critical value. Building on earlier work, we assume that, for
supercritical currents, the optimal density profile, conditioned on the given
current, has the form of a traveling wave (TW). For the KMP model we find this
TW analytically, in terms of elliptic functions, for any supercritical current
$I$. Using this TW solution, we evaluate, up to a pre-exponential factor, the
probability distribution $P(I)$. We obtain simple asymptotics of the TW and of
$P(I)$ for currents close to the critical current, and for currents much larger
than the critical current. In the latter case we show that $-\ln P (I) \sim
I\ln I$, whereas the optimal density profile acquires a soliton-like shape. Our
analytic results are in a very good agreement with Monte-Carlo simulations and
numerical solutions of Hurtado and Garrido (2011).Comment: 18 pages in one-column format, 9 figure

### How a Long Bubble Shrinks: a Numerical Method for an Unforced Hele-Shaw Flow

We develop a numerical method for solving a free boundary problem which
describes shape relaxation, by surface tension, of a long and thin bubble of an
inviscid fluid trapped inside a viscous fluid in a Hele-Shaw cell. The method
of solution of the exterior Dirichlet problem employs a classical boundary
integral formulation. Our version of the numerical method is especially
advantageous for following the dynamics of a very long and thin bubble, for
which an asymptotic scaling theory has been recently developed. Because of the
very large aspect ratio of the bubble, a direct implementation of the boundary
integral algorithm would be impractical. We modify the algorithm by introducing
a new approximation of the integrals which appear in the Fredholm integral
equation and in the integral expression for the normal derivative of the
pressure at the bubble interface. The new approximation allows one to
considerably reduce the number of nodes at the almost flat part of the bubble
interface, while keeping a good accuracy. An additional benefit from the new
approximation is in that it eliminates numerical divergence of the integral for
the tangential derivative of the harmonic conjugate. The interface's position
is advanced in time by using explicit node tracking, whereas the larger node
spacing enables one to use larger time steps. The algorithm is tested on two
model problems, for which approximate analytical solutions are available.Comment: 16 pages, 6 eps figure

### Dynamics of fractal dimension during phase ordering of a geometrical multifractal

A simple multifractal coarsening model is suggested that can explain the
observed dynamical behavior of the fractal dimension in a wide range of
coarsening fractal systems. It is assumed that the minority phase (an ensemble
of droplets) at $t=0$ represents a non-uniform recursive fractal set, and that
this set is a geometrical multifractal characterized by a $f(\alpha)$-curve. It
is assumed that the droplets shrink according to their size and preserving
their ordering. It is shown that at early times the Hausdorff dimension does
not change with time, whereas at late times its dynamics follow the $f(\alpha)$
curve. This is illustrated by a special case of a two-scale Cantor dust. The
results are then generalized to a wider range of coarsening mechanisms.Comment: 5 pages, 1 Postscript figur

### Macroscopic fluctuation theory and first-passage properties of surface diffusion

We investigate non-equilibrium fluctuations of a solid surface governed by
the stochastic Mullins-Herring equation with conserved noise. This equation
describes surface diffusion of adatoms accompanied by their exchange between
the surface and the bulk of the solid, when desorption of adatoms is
negligible. Previous works dealt with dynamic scaling behavior of the
fluctuating interface. Here we determine the probability that the interface
first reaches a large given height at a specified time. We also find the
optimal time history of the interface conditional on this non-equilibrium
fluctuation. We obtain these results by developing a macroscopic fluctuation
theory of surface diffusion.Comment: 6 pages including Supplemental Material, 3 figure

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