28,143 research outputs found
The asymmetric simple exclusion process: an integrable model for non-equilibrium statistical mechanics
The asymmetric simple exclusion process (ASEP) plays the role of a paradigm
in non-equilibrium statistical mechanics. We review exact results for the ASEP
obtained by Bethe ansatz and put emphasis on the algebraic properties of this
model. The Bethe equations for the eigenvalues of the Markov matrix of the ASEP
are derived from the algebraic Bethe ansatz. Using these equations we explain
how to calculate the spectral gap of the model and how global spectral
properties such as the existence of multiplets can be predicted. An extension
of the Bethe ansatz leads to an analytic expression for the large deviation
function of the current in the ASEP that satisfies the Gallavotti-Cohen
relation. Finally, we describe some variants of the ASEP that are also solvable
by Bethe ansatz.
Keywords: ASEP, integrable models, Bethe ansatz, large deviations.Comment: 24 pages, 5 figures, published in the "special issue on recent
advances in low-dimensional quantum field theories", P. Dorey, G. Dunne and
J. Feinberg editor
Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution
A birth-death process is a continuous-time Markov chain that counts the
number of particles in a system over time. In the general process with
current particles, a new particle is born with instantaneous rate
and a particle dies with instantaneous rate . Currently no robust and
efficient method exists to evaluate the finite-time transition probabilities in
a general birth-death process with arbitrary birth and death rates. In this
paper, we first revisit the theory of continued fractions to obtain expressions
for the Laplace transforms of these transition probabilities and make explicit
an important derivation connecting transition probabilities and continued
fractions. We then develop an efficient algorithm for computing these
probabilities that analyzes the error associated with approximations in the
method. We demonstrate that this error-controlled method agrees with known
solutions and outperforms previous approaches to computing these probabilities.
Finally, we apply our novel method to several important problems in ecology,
evolution, and genetics
Synchronization of stochastic hybrid oscillators driven by a common switching environment
Many systems in biology, physics and chemistry can be modeled through
ordinary differential equations, which are piecewise smooth, but switch between
different states according to a Markov jump process. In the fast switching
limit, the dynamics converges to a deterministic ODE. In this paper we suppose
that this limit ODE supports a stable limit cycle. We demonstrate that a set of
such oscillators can synchronize when they are uncoupled, but they share the
same switching Markov jump process. The latter is taken to represent the effect
of a common randomly switching environment. We determine the leading order of
the Lyapunov coefficient governing the rate of decay of the phase difference in
the fast switching limit. The analysis bears some similarities to the classical
analysis of synchronization of stochastic oscillators subject to common white
noise. However the discrete nature of the Markov jump process raises some
difficulties: in fact we find that the Lyapunov coefficient from the
quasi-steady-state approximation differs from the Lyapunov coefficient one
obtains from a second order perturbation expansion in the waiting time between
jumps. Finally, we demonstrate synchronization numerically in the radial
isochron clock model and show that the latter Lyapinov exponent is more
accurate
Schwinger-Dyson equations in large-N quantum field theories and nonlinear random processes
We propose a stochastic method for solving Schwinger-Dyson equations in
large-N quantum field theories. Expectation values of single-trace operators
are sampled by stationary probability distributions of the so-called nonlinear
random processes. The set of all histories of such processes corresponds to the
set of all planar diagrams in the perturbative expansions of the expectation
values of singlet operators. We illustrate the method on the examples of the
matrix-valued scalar field theory and the Weingarten model of random planar
surfaces on the lattice. For theories with compact field variables, such as
sigma-models or non-Abelian lattice gauge theories, the method does not
converge in the physically most interesting weak-coupling limit. In this case
one can absorb the divergences into a self-consistent redefinition of expansion
parameters. Stochastic solution of the self-consistency conditions can be
implemented as a "memory" of the random process, so that some parameters of the
process are estimated from its previous history. We illustrate this idea on the
example of two-dimensional O(N) sigma-model. Extension to non-Abelian lattice
gauge theories is discussed.Comment: 16 pages RevTeX, 14 figures; v2: Algorithm for the Weingarten model
corrected; v3: published versio
Many-server queues with customer abandonment: numerical analysis of their diffusion models
We use multidimensional diffusion processes to approximate the dynamics of a
queue served by many parallel servers. The queue is served in the
first-in-first-out (FIFO) order and the customers waiting in queue may abandon
the system without service. Two diffusion models are proposed in this paper.
They differ in how the patience time distribution is built into them. The first
diffusion model uses the patience time density at zero and the second one uses
the entire patience time distribution. To analyze these diffusion models, we
develop a numerical algorithm for computing the stationary distribution of such
a diffusion process. A crucial part of the algorithm is to choose an
appropriate reference density. Using a conjecture on the tail behavior of a
limit queue length process, we propose a systematic approach to constructing a
reference density. With the proposed reference density, the algorithm is shown
to converge quickly in numerical experiments. These experiments also show that
the diffusion models are good approximations for many-server queues, sometimes
for queues with as few as twenty servers
Stability Region of a Slotted Aloha Network with K-Exponential Backoff
Stability region of random access wireless networks is known for only simple
network scenarios. The main problem in this respect is due to interaction among
queues. When transmission probabilities during successive transmissions change,
e.g., when exponential backoff mechanism is exploited, the interactions in the
network are stimulated. In this paper, we derive the stability region of a
buffered slotted Aloha network with K-exponential backoff mechanism,
approximately, when a finite number of nodes exist. To this end, we propose a
new approach in modeling the interaction among wireless nodes. In this
approach, we model the network with inter-related quasi-birth-death (QBD)
processes such that at each QBD corresponding to each node, a finite number of
phases consider the status of the other nodes. Then, by exploiting the
available theorems on stability of QBDs, we find the stability region. We show
that exponential backoff mechanism is able to increase the area of the
stability region of a simple slotted Aloha network with two nodes, more than
40\%. We also show that a slotted Aloha network with exponential backoff may
perform very near to ideal scheduling. The accuracy of our modeling approach is
verified by simulation in different conditions.Comment: 30 pages, 6 figure
Stochastic pump effect and geometric phases in dissipative and stochastic systems
The success of Berry phases in quantum mechanics stimulated the study of
similar phenomena in other areas of physics, including the theory of living
cell locomotion and motion of patterns in nonlinear media. More recently,
geometric phases have been applied to systems operating in a strongly
stochastic environment, such as molecular motors. We discuss such geometric
effects in purely classical dissipative stochastic systems and their role in
the theory of the stochastic pump effect (SPE).Comment: Review. 35 pages. J. Phys. A: Math, Theor. (in press
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