Large deviations of jump process fluxes

We study a general class of systems of interacting particles that randomly interact to form new or different particles. In addition to the distribution of particles we consider the fluxes, defined as the rescaled number of jumps of each type that take place in a time interval. We prove a dynamic large deviations principle for the fluxes under general assumptions that include mass-action chemical kinetics. This result immediately implies a dynamic large deviations principle for the particle distribution

The Dwarf Nova PQ Andromedae

We report a photometric study of the WZ Sagittae-type dwarf nova PQ Andromedae. The light curve shows strong (0.05 mag full amplitude) signals with periods of 1263(1) and 634(1) s, and a likely double-humped signal with P=80.6(2) min. We interpret the first two as nonradial pulsation periods of the underlying white dwarf, and the last as the orbital period of the underlying binary. We estimate a distance of 150(50) pc from proper motions and the two standard candles available: the white dwarf and the dwarf-nova outburst. At this distance, the K magnitude implies that the secondary is probably fainter than any star on the main sequence -- indicating a mass below the Kumar limit at 0.075 M_sol. PQ And may be another "period bouncer", where evolution now drives the binary out to longer period.Comment: PDF, 13 pages, 2 figures; accepted, in press, to appear September 2005, PASP; more info at http://cba.phys.columbia.edu

A large-deviations principle for all the cluster sizes of a sparse Erdős–Rényi graph

Let (Formula presented.) be the Erdős–Rényi graph with connection probability (Formula presented.) as N → ∞ for a fixed t ∈ (0, ∞). We derive a large-deviations principle for the empirical measure of the sizes of all the connected components of (Formula presented.), registered according to microscopic sizes (i.e., of finite order), macroscopic ones (i.e., of order N), and mesoscopic ones (everything in between). The rate function explicitly describes the microscopic and macroscopic components and the fraction of vertices in components of mesoscopic sizes. Moreover, it clearly captures the well known phase transition at t = 1 as part of a comprehensive picture. The proofs rely on elementary combinatorics and on known estimates and asymptotics for the probability that subgraphs are connected. We also draw conclusions for the strongly related model of the multiplicative coalescent, the Marcus–Lushnikov coagulation model with monodisperse initial condition, and its gelation phase transition

Dynamical large deviations of countable reaction networks under a weak reversibility condition

A dynamic large deviations principle for a countable reaction network including coagulation--fragmentation models is proved. The rate function is represented as the infimal cost of the reaction fluxes and a minimiser for this variational problem is shown to exist. A weak reversibility condition is used to control the boundary behaviour and to guarantee a representation for the optimal fluxes via a Lagrange multiplier that can be used to construct the changes of measure used in standard tilting arguments. Reflecting the pure jump nature of the approximating processes, their paths are treated as elements of a BV function space

Topologies and measures on the space of functions of bounded variation taking values in a Banach or metric space

We study functions of bounded variation with values in a Banach or in a metric space. In finite dimensions, there are three well-known topologies; we argue that in infinite dimensions there is a natural fourth topology. We provide some insight into the structure of these four topologies. In particular, we study the meaning of convergence, duality and regularity for these topologies and provide some useful compactness criteria, also related to the classical Aubin–Lions theorem. After this we study the Bore

Large deviations of reaction fluxes

We study a system of interacting particles that randomly react to form new particles. The reaction flux is the rescaled number of reactions that take place in a time interval. We prove a dynamic large-deviation principle for the reaction fluxes under general assumptions that include mass-action kinetics. This result immediately implies the dynamic large deviations for the empirical concentration

A large-deviations approach to gelation

A large-deviations principle (LDP) is derived for the state at fixed time, of the multiplicative coalescent in the large particle number limit. The rate function is explicit and describes each of the three parts of the state: microscopic, mesoscopic and macroscopic. In particular, it clearly captures the well known gelation phase transition given by the formation of a particle containing a positive fraction of the system mass. Via a standard map of the multiplicative coalescent onto a time-dependent version of the Erdős-Rényi random graph, our results can also be rephrased as an LDP for the component sizes in that graph. The proofs rely on estimates and asymptotics for the probability that smaller Erdős-Rényi graphs are connected

Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics

We study stochastic interacting particle systems that model chemical reaction networks on the micro scale, converging to the macroscopic Reaction Rate Equation. One abstraction level higher, we study the ensemble of such particle systems, converging to the corresponding Liouville transport equation. For both systems, we calculate the corresponding large deviations and show that under the condition of detailed balance, the large deviations induce a non-linear relation between thermodynamic fluxes and free energy driving force

Large deviations for Markov jump processes with uniformly diminishing rates

We prove a large-deviation principle (LDP) for the sample paths of jump Markov processes in the small noise limit when, possibly, all the jump rates vanish uniformly, but slowly enough, in a region of the state space. We further discuss the optimality of our assumptions on the decay of the jump rates. As a direct application of this work we relax the assumptions needed for the application of LDPs to, e.g., Chemical Reaction Network dynamics, where vanishing reaction rates arise naturally particularly the context of mass action kinetics