4,000 research outputs found
Metastable convergence theorems
The dominated convergence theorem implies that if (f_n) is a sequence of
functions on a probability space taking values in the interval [0,1], and (f_n)
converges pointwise a.e., then the sequence of integrals converges to the
integral of the pointwise limit. Tao has proved a quantitative version of this
theorem: given a uniform bound on the rates of metastable convergence in the
hypothesis, there is a bound on the rate of metastable convergence in the
conclusion that is independent of the sequence (f_n) and the underlying space.
We prove a slight strengthening of Tao's theorem which, moreover, provides an
explicit description of the second bound in terms of the first. Specifically,
we show that when the first bound is given by a continuous functional, the
bound in the conclusion can be computed by a recursion along the tree of
unsecured sequences. We also establish a quantitative version of Egorov's
theorem, and introduce a new mode of convergence related to these notions
On soft capacities, quasi-stationary distributions and the pathwise approach to metastability
Motivated by the study of the metastable stochastic Ising model at
subcritical temperature and in the limit of a vanishing magnetic field, we
extend the notion of (, )-capacities between sets, as well as
the associated notion of soft-measures, to the case of overlapping sets. We
recover their essential properties, sometimes in a stronger form or in a
simpler way, relying on weaker hypotheses. These properties allow to write the
main quantities associated with reversible metastable dynamics, e.g. asymptotic
transition and relaxation times, in terms of objects that are associated with
two-sided variational principles. We also clarify the connection with the
classical "pathwise approach" by referring to temporal means on the appropriate
time scale.Comment: 29 pages, 1 figur
Stationary States and Asymptotic Behaviour of Aggregation Models with Nonlinear Local Repulsion
We consider a continuum aggregation model with nonlinear local repulsion
given by a degenerate power-law diffusion with general exponent. The steady
states and their properties in one dimension are studied both analytically and
numerically, suggesting that the quadratic diffusion is a critical case. The
focus is on finite-size, monotone and compactly supported equilibria. We also
investigate numerically the long time asymptotics of the model by simulations
of the evolution equation. Issues such as metastability and local/ global
stability are studied in connection to the gradient flow formulation of the
model
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
A weak characterization of slow variables in stochastic dynamical systems
We present a novel characterization of slow variables for continuous Markov
processes that provably preserve the slow timescales. These slow variables are
known as reaction coordinates in molecular dynamical applications, where they
play a key role in system analysis and coarse graining. The defining
characteristics of these slow variables is that they parametrize a so-called
transition manifold, a low-dimensional manifold in a certain density function
space that emerges with progressive equilibration of the system's fast
variables. The existence of said manifold was previously predicted for certain
classes of metastable and slow-fast systems. However, in the original work, the
existence of the manifold hinges on the pointwise convergence of the system's
transition density functions towards it. We show in this work that a
convergence in average with respect to the system's stationary measure is
sufficient to yield reaction coordinates with the same key qualities. This
allows one to accurately predict the timescale preservation in systems where
the old theory is not applicable or would give overly pessimistic results.
Moreover, the new characterization is still constructive, in that it allows for
the algorithmic identification of a good slow variable. The improved
characterization, the error prediction and the variable construction are
demonstrated by a small metastable system
- …