533 research outputs found
Property-driven State-Space Coarsening for Continuous Time Markov Chains
Dynamical systems with large state-spaces are often expensive to thoroughly
explore experimentally. Coarse-graining methods aim to define simpler systems
which are more amenable to analysis and exploration; most current methods,
however, focus on a priori state aggregation based on similarities in
transition rates, which is not necessarily reflected in similar behaviours at
the level of trajectories. We propose a way to coarsen the state-space of a
system which optimally preserves the satisfaction of a set of logical
specifications about the system's trajectories. Our approach is based on
Gaussian Process emulation and Multi-Dimensional Scaling, a dimensionality
reduction technique which optimally preserves distances in non-Euclidean
spaces. We show how to obtain low-dimensional visualisations of the system's
state-space from the perspective of properties' satisfaction, and how to define
macro-states which behave coherently with respect to the specifications. Our
approach is illustrated on a non-trivial running example, showing promising
performance and high computational efficiency.Comment: 16 pages, 6 figures, 1 tabl
Derivation of mean-field equations for stochastic particle systems
We study stochastic particle systems on a complete graph and derive effective
mean-field rate equations in the limit of diverging system size, which are also
known from cluster aggregation models. We establish the propagation of chaos
under generic growth conditions on particle jump rates, and the limit provides
a master equation for the single site dynamics of the particle system, which is
a non-linear birth death chain. Conservation of mass in the particle system
leads to conservation of the first moment for the limit dynamics, and to
non-uniqueness of stationary distributions. Our findings are consistent with
recent results on exchange driven growth, and provide a connection between the
well studied phenomena of gelation and condensation.Comment: 26 page
Persistence and First-Passage Properties in Non-equilibrium Systems
In this review we discuss the persistence and the related first-passage
properties in extended many-body nonequilibrium systems. Starting with simple
systems with one or few degrees of freedom, such as random walk and random
acceleration problems, we progressively discuss the persistence properties in
systems with many degrees of freedom. These systems include spins models
undergoing phase ordering dynamics, diffusion equation, fluctuating interfaces
etc. Persistence properties are nontrivial in these systems as the effective
underlying stochastic process is non-Markovian. Several exact and approximate
methods have been developed to compute the persistence of such non-Markov
processes over the last two decades, as reviewed in this article. We also
discuss various generalisations of the local site persistence probability.
Persistence in systems with quenched disorder is discussed briefly. Although
the main emphasis of this review is on the theoretical developments on
persistence, we briefly touch upon various experimental systems as well.Comment: Review article submitted to Advances in Physics: 149 pages, 21
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Dynamics of condensation in the totally asymmetric inclusion process
We study the dynamics of condensation of the inclusion process on a
one-dimensional periodic lattice in the thermodynamic limit, generalising
recent results on finite lattices for symmetric dynamics. Our main focus is on
totally asymmetric dynamics which have not been studied before, and which we
also compare to exact solutions for symmetric systems. We identify all relevant
dynamical regimes and corresponding time scales as a function of the system
size, including a coarsening regime where clusters move on the lattice and
exchange particles, leading to a growing average cluster size. Suitable
observables exhibit a power law scaling in this regime before they saturate to
stationarity following an exponential decay depending on the system size. Our
results are based on heuristic derivations and exact computations for symmetric
systems, and are supported by detailed simulation data.Comment: 23 pages, 6 figures, updated references and introductio
Metastability of finite state Markov chains: a recursive procedure to identify slow variables for model reduction
Consider a sequence of continuous-time, irreducible
Markov chains evolving on a fixed finite set , indexed by a parameter .
Denote by the jump rates of the Markov chain , and
assume that for any pair of bonds , converges as . Under a
hypothesis slightly more restrictive (cf. \eqref{mhyp} below), we present a
recursive procedure which provides a sequence of increasing time-scales
\theta^1_N, \dots, \theta^{\mf p}_N, , and of
coarsening partitions \{\ms E^j_1, \dots, \ms E^j_{\mf n_j}, \Delta^j\},
1\le j\le \mf p, of the set . Let \phi_j: E \to \{0,1, \dots, \mf n_j\}
be the projection defined by \phi_j(\eta) = \sum_{x=1}^{\mf n_j} x \, \mb
1\{\eta \in \ms E^j_x\}. For each 1\le j\le \mf p, we prove that the hidden
Markov chain converges to a Markov
chain on \{1, \dots, \mf n_j\}
Condensation in stochastic particle systems with stationary product measures
We study stochastic particle systems with stationary product measures that
exhibit a condensation transition due to particle interactions or spatial
inhomogeneities. We review previous work on the stationary behaviour and put it
in the context of the equivalence of ensembles, providing a general
characterization of the condensation transition for homogeneous and
inhomogeneous systems in the thermodynamic limit. This leads to strengthened
results on weak convergence for subcritical systems, and establishes the
equivalence of ensembles for spatially inhomogeneous systems under very general
conditions, extending previous results which were focused on attractive and
finite systems. We use relative entropy techniques which provide simple proofs,
making use of general versions of local limit theorems for independent random
variables.Comment: 44 pages, 4 figures; improved figures and corrected typographical
error
Metastability in a condensing zero-range process in the thermodynamic limit
Zero-range processes with decreasing jump rates are known to exhibit
condensation, where a finite fraction of all particles concentrates on a single
lattice site when the total density exceeds a critical value. We study such a
process on a one-dimensional lattice with periodic boundary conditions in the
thermodynamic limit with fixed, super-critical particle density. We show that
the process exhibits metastability with respect to the condensate location,
i.e. the suitably accelerated process of the rescaled location converges to a
limiting Markov process on the unit torus. This process has stationary,
independent increments and the rates are characterized by the scaling limit of
capacities of a single random walker on the lattice. Our result extends
previous work for fixed lattices and diverging density in [J. Beltran, C.
Landim, Probab. Theory Related Fields, 152(3-4):781-807, 2012], and we follow
the martingale approach developed there and in subsequent publications. Besides
additional technical difficulties in estimating error bounds for transition
rates, the thermodynamic limit requires new estimates for equilibration towards
a suitably defined distribution in metastable wells, corresponding to a typical
set of configurations with a particular condensate location. The total exit
rates from individual wells turn out to diverge in the limit, which requires an
intermediate regularization step using the symmetries of the process and the
regularity of the limit generator. Another important novel contribution is a
coupling construction to provide a uniform bound on the exit rates from
metastable wells, which is of a general nature and can be adapted to other
models.Comment: 55 pages, 1 figur
Time scale separation and dynamic heterogeneity in the low temperature East model
We consider the non-equilibrium dynamics of the East model, a linear chain of
0-1 spins evolving under a simple Glauber dynamics in the presence of a kinetic
constraint which forbids flips of those spins whose left neighbor is 1. We
focus on the glassy effects caused by the kinetic constraint as , where is the equilibrium density of the 0's. In the physical literature
this limit is equivalent to the zero temperature limit. We first prove that,
for any given , the divergence as of three basic
characteristic time scales of the East process of length is the same. Then
we examine the problem of dynamic heterogeneity, i.e. non-trivial
spatio-temporal fluctuations of the local relaxation to equilibrium, one of the
central aspects of glassy dynamics. For any mesoscopic length scale
, , we show that the characteristic time scale of
two East processes of length and respectively are indeed
separated by a factor , , provided that
is large enough (independent of , for ). In
particular, the evolution of mesoscopic domains, i.e. maximal blocks of the
form , occurs on a time scale which depends sharply on the size of the
domain, a clear signature of dynamic heterogeneity. A key result for this part
is a very precise computation of the relaxation time of the chain as a function
of , well beyond the current knowledge, which uses induction on length
scales on one hand and a novel algorithmic lower bound on the other. Finally we
show that no form of time scale separation occurs for , i.e. at the
equilibrium scale , contrary to what was assumed in the physical
literature based on numerical simulations.Comment: 40 pages, 4 figures; minor typographical corrections and improvement
Multiscale Markov Decision Problems: Compression, Solution, and Transfer Learning
Many problems in sequential decision making and stochastic control often have
natural multiscale structure: sub-tasks are assembled together to accomplish
complex goals. Systematically inferring and leveraging hierarchical structure,
particularly beyond a single level of abstraction, has remained a longstanding
challenge. We describe a fast multiscale procedure for repeatedly compressing,
or homogenizing, Markov decision processes (MDPs), wherein a hierarchy of
sub-problems at different scales is automatically determined. Coarsened MDPs
are themselves independent, deterministic MDPs, and may be solved using
existing algorithms. The multiscale representation delivered by this procedure
decouples sub-tasks from each other and can lead to substantial improvements in
convergence rates both locally within sub-problems and globally across
sub-problems, yielding significant computational savings. A second fundamental
aspect of this work is that these multiscale decompositions yield new transfer
opportunities across different problems, where solutions of sub-tasks at
different levels of the hierarchy may be amenable to transfer to new problems.
Localized transfer of policies and potential operators at arbitrary scales is
emphasized. Finally, we demonstrate compression and transfer in a collection of
illustrative domains, including examples involving discrete and continuous
statespaces.Comment: 86 pages, 15 figure
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