780 research outputs found
Dynamical Phase Transitions for Fluxes of Mass on Finite Graphs
We study the time-averaged flux in a model of particles that randomly hop on
a finite directed graph. In the limit as the number of particles and the time
window go to infinity but the graph remains finite, the large-deviation rate
functional of the average flux is given by a variational formulation involving
paths of the density and flux. We give sufficient conditions under which the
large deviations of a given time averaged flux is determined by paths that are
constant in time. We then consider a class of models on a discrete ring for
which it is possible to show that a better strategy is obtained producing a
time-dependent path. This phenomenon, called a dynamical phase transition, is
known to occur for some particle systems in the hydrodynamic scaling limit,
which is thus extended to the setting of a finite graph
From large deviations to semidistances of transport and mixing: coherence analysis for finite Lagrangian data
One way to analyze complicated non-autonomous flows is through trying to
understand their transport behavior. In a quantitative, set-oriented approach
to transport and mixing, finite time coherent sets play an important role.
These are time-parametrized families of sets with unlikely transport to and
from their surroundings under small or vanishing random perturbations of the
dynamics. Here we propose, as a measure of transport and mixing for purely
advective (i.e., deterministic) flows, (semi)distances that arise under
vanishing perturbations in the sense of large deviations. Analogously, for
given finite Lagrangian trajectory data we derive a discrete-time and space
semidistance that comes from the "best" approximation of the randomly perturbed
process conditioned on this limited information of the deterministic flow. It
can be computed as shortest path in a graph with time-dependent weights.
Furthermore, we argue that coherent sets are regions of maximal farness in
terms of transport and mixing, hence they occur as extremal regions on a
spanning structure of the state space under this semidistance---in fact, under
any distance measure arising from the physical notion of transport. Based on
this notion we develop a tool to analyze the state space (or the finite
trajectory data at hand) and identify coherent regions. We validate our
approach on idealized prototypical examples and well-studied standard cases.Comment: J Nonlinear Sci, 201
On microscopic origins of generalized gradient structures
Classical gradient systems have a linear relation between rates and driving
forces. In generalized gradient systems we allow for arbitrary relations
derived from general non-quadratic dissipation potentials. This paper describes
two natural origins for these structures.
A first microscopic origin of generalized gradient structures is given by the
theory of large-deviation principles. While Markovian diffusion processes lead
to classical gradient structures, Poissonian jump processes give rise to
cosh-type dissipation potentials.
A second origin arises via a new form of convergence, that we call
EDP-convergence. Even when starting with classical gradient systems, where the
dissipation potential is a quadratic functional of the rate, we may obtain a
generalized gradient system in the evolutionary -limit. As examples we
treat (i) the limit of a diffusion equation having a thin layer of low
diffusivity, which leads to a membrane model, and (ii) the limit of diffusion
over a high barrier, which gives a reaction-diffusion system.Comment: Keywords: Generalized gradient structure, gradient system,
evolutionary \Gamma-convergence, energy-dissipation principle, variational
evolution, relative entropy, large-deviation principl
Gradient and Generic systems in the space of fluxes, applied to reacting particle systems
In a previous work we devised a framework to derive generalised gradient
systems for an evolution equation from the large deviations of an underlying
microscopic system, in the spirit of the Onsager-Machlup relations. Of
particular interest is the case where the microscopic system consists of random
particles, and the macroscopic quantity is the empirical measure or
concentration. In this work we take the particle flux as the macroscopic
quantity, which is related to the concentration via a continuity equation. By a
similar argument the large deviations can induce a generalised gradient or
Generic system in the space of fluxes. In a general setting we study how flux
gradient or generic systems are related to gradient systems of concentrations.
The arguments are explained by the example of reacting particle systems, which
is later expanded to include spatial diffusion as well
Anisothermal chemical reactions: Onsager--Machlup and macroscopic fluctuation theory
We study a micro and macroscopic model for chemical reactions with feedback between reactions and temperature of the solute. The first result concerns the quasipotential as the large-deviation rate of the microscopic invariant measure. The second result is an application of modern Onsager-Machlup theory to the pathwise large deviations, in case the system is in detailed balance. The third result is an application of macroscopic fluctuation theory to the reaction flux large deviations, in case the system is in complex balance
Gradient and Generic systems in the space of fluxes, applied to reacting particle systems
In a previous work we devised a framework to derive generalised gradient systems for an evolution equation from the large deviations of an underlying microscopic system, in the spirit of the Onsager-Machlup relations. Of particular interest is the case where the microscopic system consists of random particles, and the macroscopic quantity is the empirical measure or concentration. In this work we take the particle flux as the macroscopic quantity, which is related to the concentration via a continuity equation. By a similar argument the large deviations can induce a generalised gradient or Generic system in the space of fluxes. In a general setting we study how flux gradient or generic systems are related to gradient systems of concentrations. The arguments are explained by the example of reacting particle systems, which is later expanded to include spatial diffusion as well
Flux large deviations of independent and reacting particle systems, with implications for macroscopic fluctuation theory
We consider a system of independent particles on a finite state space, and prove a dynamic large-deviation principle for the empirical measure-empirical flux pair, taking the specific fluxes rather than net fluxes into account. We prove the large deviations under deterministic initial conditions, and under random initial conditions satisfying a large-deviation principle. We then show how to use this result to generalise a number of principles from Macroscopic Fluctuation Theory to the finite-space setting
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Gradient and GENERIC Systems in the Space of Fluxes, Applied to Reacting Particle Systems
In a previous work we devised a framework to derive generalised gradient systems for an evolution equation from the large deviations of an underlying microscopic system, in the spirit of the Onsager–Machlup relations. Of particular interest is the case where the microscopic system consists of random particles, and the macroscopic quantity is the empirical measure or concentration. In this work we take the particle flux as the macroscopic quantity, which is related to the concentration via a continuity equation. By a similar argument the large deviations can induce a generalised gradient or GENERIC system in the space of fluxes. In a general setting we study how flux gradient or GENERIC systems are related to gradient systems of concentrations. This shows that many gradient or GENERIC systems arise from an underlying gradient or GENERIC system where fluxes rather than densities are being driven by (free) energies. The arguments are explained by the example of reacting particle systems, which is later expanded to include spatial diffusion as well
Coherence Analysis for Finite Lagrangian Data
One way to analyze complicated non-autonomous flows is through trying to understand their transport behavior. In a quantitative, set-oriented approach to transport and mixing, finite time coherent sets play an important role. These are time-parametrized families of sets with unlikely transport to and from their surroundings under small or vanishing random perturbations of the dynamics. Here we propose, as a measure of transport and mixing for purely advective (i.e., deterministic) flows, (semi)distances that arise under vanishing perturbations in the sense of large deviations. Analogously, for given finite Lagrangian trajectory data we derive a discrete-time-and-space semidistance that comes from the “best” approximation of the randomly perturbed process conditioned on this limited information of the deterministic flow. It can be computed as shortest path in a graph with time-dependent weights. Furthermore, we argue that coherent sets are regions of maximal farness in terms of transport and mixing, and hence they occur as extremal regions on a spanning structure of the state space under this semidistance—in fact, under any distance measure arising from the physical notion of transport. Based on this notion, we develop a tool to analyze the state space (or the finite trajectory data at hand) and identify coherent regions. We validate our approach on idealized prototypical examples and well-studied standard cases
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Dynamical phase transitions for flows on finite graphs
We study the time-averaged flow in a model of particles that randomly hop on a finite directed graph. In the limit as the number of particles and the time window go to infinity but the graph remains finite, the large-deviation rate functional of the average flow is given by a variational formulation involving paths of the density and flow. We give sufficient conditions under which the large deviations of a given time averaged flow is determined by paths that are constant in time. We then consider a class of models on a discrete ring for which it is possible to show that a better strategy is obtained producing a time-dependent path. This phenomenon, called a dynamical phase transition, is known to occur for some particle systems in the hydrodynamic scaling limit, which is thus extended to the setting of a finite graph
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