Stationarity of multivariate particle systems

A particle system is a family of i.i.d. stochastic processes with values translated by Poisson points. We obtain conditions that ensure the stationarity in time of the particle system in RdRd and in some cases provide a full characterisation of the stationarity property. In particular, a full characterisation of stationary multivariate Brown–Resnick processes is given.

Fluidization and fluid-particle systems

Includes bibliography.

Interacting nonequilibrium systems with two temperatures

We investigate a simplified model of two fully connected magnetic systems maintained at different temperatures by virtue of being connected to two independent thermal baths while simultaneously being interconnected with each other. Using generating functional analysis, commonly used in statistical mechanics, we find exactly soluble expressions for their individual magnetization that define a two-dimensional nonlinear map, the equations of which have the same form as those obtained for densely connected equilibrium systems. Steady states correspond to the fixed points of this map, separating the parameter space into a rich set of nonequilibrium phases that we analyze in asymptotically high and low (nonequilibrium) temperature limits. The theoretical formalism is shown to revert to the classical nonequilibrium steady state problem for two interacting systems with a nonzero heat transfer between them that catalyzes a phase transition between ambient nonequilibrium states. © 2013 American Physical Society.

Exponential Rates of Convergence in the Ergodic Theorem: A Constructive Approach

We prove that, once an algorithm of perfect simulation for a stationary and ergodic random field F taking values in S(Zd), S a bounded subset of R(n), is provided, the speed of convergence in the mean ergodic theorem occurs exponentially fast for F. Applications from (non-equilibrium) statistical mechanics and interacting particle systems are presented.

Random recursion

Thesis (Ph.D.)--University of Washington, 2016-06

Complex evolution of a multi-particle system

This paper studies a discrete dynamical system of interacting particles that evolve by interacting among them. The computational model is an abstraction of the natural world, and real systems can range from the huge cosmological scale down to the scale of biological cell, or even molecules. Different conditions for the system evolution are tested. The emerging patterns are analysed by means of fractal dimension and entropy measures. It is observed that the population of particles evolves towards geometrical objects with a fractal nature. Moreover, the time signature of the entropy can be interpreted at the light of complex dynamical systems.

Energy-level statistics of interacting trapped bosons

It is a well-established fact that statistical properties of energy-level spectra are the most efficient tool to characterize nonintegrable quantum systems. The statistical behavior of different systems such as complex atoms, atomic nuclei, two-dimensional Hamiltonians, quantum billiards, and noninteracting many bosons has been studied. The study of statistical properties and spectral fluctuations in interacting many-boson systems has developed interest in this direction. We are especially interested in weakly interacting trapped bosons in the context of Bose-Einstein condensation (BEC) as the energy spectrum shows a transition from a collective nature to a single-particle nature with an increase in the number of levels. However this has received less attention as it is believed that the system may exhibit Poisson-like fluctuations due to the existence of an external harmonic trap. Here we compute numerically the energy levels of the zero-temperature many-boson systems which are weakly interacting through the van der Waals potential and are confined in the three-dimensional harmonic potential. We study the nearest-neighbor spacing distribution and the spectral rigidity by unfolding the spectrum. It is found that an increase in the number of energy levels for repulsive BEC induces a transition from a Wigner-like form displaying level repulsion to the Poisson distribution for P(s). It does not follow the Gaussian orthogonal ensemble prediction. For repulsive interaction, the lower levels are correlated and manifest level-repulsion. For intermediate levels P(s) shows mixed statistics, which clearly signifies the existence of two energy scales: external trap and interatomic interaction, whereas for very high levels the trapping potential dominates, generating a Poisson distribution. Comparison with mean-field results for lower levels are also presented. For attractive BEC near the critical point we observe the Shnirelman-like peak near s = 0, which signifies the presence of a large number of quasidegenerate states.

Stochastic particle models: mean reversion and burgers dynamics. An application to commodity markets

The aim of this study is to propose a stochastic model for commodity markets linked with the Burgers equation from fluid dynamics. We construct a stochastic particles method for commodity markets, in which particles represent market participants. A discontinuity in the model is included through an interacting kernel equal to the Heaviside function and its link with the Burgers equation is given. The Burgers equation and the connection of this model with stochastic differential equations are also studied. Further, based on the law of large numbers, we prove the convergence, for large N, of a system of stochastic differential equations describing the evolution of the prices of N traders to a deterministic partial differential equation of Burgers type. Numerical experiments highlight the success of the new proposal in modeling some commodity markets, and this is confirmed by the ability of the model to reproduce price spikes when their effects occur in a sufficiently long period of time.