Coexistence of Stable Limit Cycles in a Generalized Curie–Weiss Model with Dissipation

In this paper, we modify the Langevin dynamics associated to the generalized Curie–Weiss model by introducing noisy and dissipative evolution in the interaction potential. We show that, when a zero-mean Gaussian is taken as single-site distribution, the dynamics in the thermodynamic limit can be described by a finite set of ODEs. Depending on the form of the interaction function, the system can have several phase transitions at different critical temperatures. Because of the dissipation effect, not only the magnetization of the systems displays a self-sustained periodic behavior at sufficiently low temperature, but, in certain regimes, any (finite) number of stable limit cycles can exist. We explore some of these peculiarities with explicit examples

On a class of time-Fractional continuous-State branching processes

We propose a class of non-Markov population models with continuous or discrete state space via a limiting procedure involving sequences of rescaled and randomly time-changed Galton – Watson processes. The class includes as specific cases the classical continuous-state branching processes and Markov branching processes. Several results such as the expressions of moments and the branching inequality governing the evolution of the process are presented and commented. The generalized Feller branching diffusion and the fractional Yule process are analyzed in detail as special cases of the general model

McKean–Vlasov limit for interacting systems with simultaneous jumps

Motivated by several applications, including neuronal models, we consider the McKean–Vlasov limit for a general class of mean-field systems of interacting diffusions characterized by an interaction via simultaneous jumps. We focus our interest on systems where the rate of the jumps is unbounded, which are rarely treated in the mean-field literature, and we prove well-posedness of the McKean–Vlasov limit together with propagation of chaos via a coupling technique. To highlight the role of simultaneous jumps, we introduce an intermediate process which is close to the original particle system but does not display simultaneous jumps. This shows in particular that the simultaneous jumps contribute to the overall rate of convergence of the N-particle empirical measures by a term of order 1/√N

Strong existence and uniqueness of the stationary distribution for a stochastic inviscid dyadic model

We consider an inviscid stochastically forced dyadic model, where the additive noise acts only on the first component. We prove that a strong solution for this problem exists and is unique by means of uniform energy estimates. Moreover, we exploit these results to establish strong existence and uniqueness of the stationary distribution

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

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

A REDUCED KINETIC MECHANISM FOR PROPANE FLAMES

Propane is one of the simplest hydrocarbons that can be a representative of higher hydrocarbons used in many applications. Therefore, this work develops a ten-step reduced kinetic mechanism among 14 reactive species for the propane combustion. The model is based on the solution of the flamelet equations. The equations are discretized using the second-order space finite difference method, using LES (Large-Eddy Simulation). Obtained results compare favorably with data in the literature for a propane jet diffusion flame. The main advantage of this strategy is the decrease of the work needed to solve the system of governing equations

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