Papers by Zochil Gonzalez Arenas
Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, Dec 20, 2021
Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, Dec 8, 2022
Resumo. Na modelagem epidemiológica, modelos estocásticos permitem estudar propriedades às quais ... more Resumo. Na modelagem epidemiológica, modelos estocásticos permitem estudar propriedades às quais não se tem acesso através de modelos derteministas. Tal é o caso da distribuição de probabilidade para o tamanho final de uma epidemia, uma propriedade muito importante na descrição da dinâmica evolutiva de uma doença. Uma abordagem comum para se obter esta distribuição, é dada pelo cálculo de potências de matrizes que podem ser excessivamente grandes a depender do tamanho da população. Com isso, a modelagem torna-se inviável por conta do custo computacional. Como uma forma alternativa, neste trabalho, deduzimos uma expressão para encontrar a forma fechada do vetor de tamanho final da epidemia no caso de um modelo SIR estocástico definido por meio de Cadeia de Markov de Tempo Contínuo, eliminando a necessidade do cálculo matricial.
Proceeding Series of the Brazilian Society of Computational and Applied Mathematics
Resumo. Na modelagem epidemiológica, modelos estocásticos permitem estudar propriedades às quais ... more Resumo. Na modelagem epidemiológica, modelos estocásticos permitem estudar propriedades às quais não se tem acesso através de modelos derteministas. Tal é o caso da distribuição de probabilidade para o tamanho final de uma epidemia, uma propriedade muito importante na descrição da dinâmica evolutiva de uma doença. Uma abordagem comum para se obter esta distribuição, é dada pelo cálculo de potências de matrizes que podem ser excessivamente grandes a depender do tamanho da população. Com isso, a modelagem torna-se inviável por conta do custo computacional. Como uma forma alternativa, neste trabalho, deduzimos uma expressão para encontrar a forma fechada do vetor de tamanho final da epidemia no caso de um modelo SIR estocástico definido por meio de Cadeia de Markov de Tempo Contínuo, eliminando a necessidade do cálculo matricial.
Anais do Congresso Brasileiro de Automática 2020, Dec 8, 2020
The recent Coronavirus (COVID-19) spread quickly around the world, boosting the research of mathe... more The recent Coronavirus (COVID-19) spread quickly around the world, boosting the research of mathematical modeling applied to epidemiology. One of the commonly used models is the deterministic version of the Susceptible-Infected-Recovered (SIR) model. In this work, we present the stochastic version of the SIR model, using Continuous Time Markov Chain (CTMC). Some numerical simulations were performed in order to compare the two models using public data on COVID-19 and parameter values reported in the literature. Computing some properties of stochastic models demands many computational resources, limiting its usability in large populations. Preliminary results as well as a discussion of future work are presented. Resumo: O novo Coronavirus (COVID-19) espalhou-se rapidamente pelo mundo e dinamizou a pesquisa da modelagem matemática aplicada a epidemiologia. Um dos modelos utilizados na modelagem dessa doençaé o Suscetível-Infectado-Recuperado (SIR) utilizando uma abordagem determinística. Neste trabalho, apresentamos o modelo SIR com uma abordagem estocástica, utilizando Cadeia de Markov de Tempo Contínuo (CTMC). Foram realizadas simulações a fim de comparação entre os dois modelos utilizando dados públicos sobre a COVID-19 e valores de parâmetros reportados na literatura. Algumas das propriedades dos modelos estocásticos possuem um custo computacional elevado que limitam o seu uso em populações grandes. Resultados preliminares são apresentados assim como uma discussão dos trabalhos futuros.
Trends in Computational and Applied Mathematics, Jul 20, 2023
Development of mathematical models and its numerical implementations are essential tools in epide... more Development of mathematical models and its numerical implementations are essential tools in epidemiological modeling. Susceptible-Infected-Recovered (SIR) compartmental model, proposed by Kermack and McKendrick in 1927, is a widely used deterministic model which serves as a basis for more involved mathematical models. In this work, we consider two stochastic versions of the SIR model for analysing a measles outbreak in Ilha Grande, Rio de Janeiro, in 1976; Continuous Time Markov Chain and Stochastic Differential Equations. The SIR Continuous Time Markov Chain model is used to extract specific information from the measles outbreak. The outbreak probability, final size distribution and expected duration of the epidemic were computed, obtaining results in excellent agreement with the reported epidemic values. Numerical simulations are performed in Python.
Physical review, Mar 22, 2019
We address the calculation of transition probabilities in multiplicative noise stochastic differe... more We address the calculation of transition probabilities in multiplicative noise stochastic differential equations using a path integral approach. We show the equivalence between the conditional probability and the propagator of a quantum particle with variable mass. Introducing a time reparametrization, we are able to transform the problem of multiplicative noise fluctuations into an equivalent additive one. We illustrate the method by showing the explicit analytic computation of the conditional probability of a harmonic oscillator in a nonlinear multiplicative environment.
arXiv (Cornell University), Dec 18, 2018
We address the calculation of transition probabilities in multiplicative noise stochastic differe... more We address the calculation of transition probabilities in multiplicative noise stochastic differential equations using a path integral approach. We show the equivalence between the conditional probability and the propagator of a quantum particle with variable mass. Introducing a time reparametrization, we are able to transform the problem of multiplicative noise fluctuations into an equivalent additive one. We illustrate the method by showing the explicit analytic computation of the conditional probability of a harmonic oscillator in a nonlinear multiplicative environment.
Physica D: Nonlinear Phenomena, May 1, 2016
Motivated by experiments in bosonic mixtures composed of a single element in two different hyperf... more Motivated by experiments in bosonic mixtures composed of a single element in two different hyperfine states, we study bosonic binary mixtures in the presence of Josephson interactions between species. We focus on a particular model with O(2) isospin symmetry, lifted by an imbalanced population parametrized by a Rabi frequency, Ω R , and a detuning, ν, which couples the phases of both species. We have studied the model at mean-field approximation plus Gaussian fluctuations. We have found that both species simultaneously condensate below a critical temperature T c and the relative phases are locked by the applied laser phase, α. Moreover, the condensate fractions are strongly dependent on the ratio Ω R /|ν| that is not affected by thermal fluctuations.
EPL, 2016
We present a path integral formalism to compute potentials for nonequilibrium steady states, reac... more We present a path integral formalism to compute potentials for nonequilibrium steady states, reached by a multiplicative stochastic dynamics. We develop a weak-noise expansion, which allows the explicit evaluation of the potential in arbitrary dimensions and for any stochastic prescription. We apply this general formalism to study noise-induced phase transitions. We focus on a class of multiplicative stochastic lattice models and compute the steady state phase diagram in terms of the noise intensity and the lattice coupling. We obtain, under appropriate conditions, an ordered phase induced by noise. By computing entropy production, we show that microscopic irreversibility is a necessary condition to develop noise-induced phase transitions. This property of the nonequilibrium stationary state has no relation with the initial stages of the dynamical evolution, in contrast with previous interpretations, based on the short-time evolution of the order parameter.
Physical Review E, Apr 16, 2012
We present a supersymmetric formulation of Markov processes, represented by a family of Langevin ... more We present a supersymmetric formulation of Markov processes, represented by a family of Langevin equations with multiplicative white-noise. The hidden symmetry encodes equilibrium properties such as fluctuation-dissipation relations. The formulation does not depend on the particular prescription to define the Wiener integral. In this way, different equilibrium distributions, reached at long times for each prescription, can be formally treated on the same footing.
Journal of Statistical Mechanics: Theory and Experiment, Dec 6, 2012
Multiplicative white-noise stochastic processes continuously attract the attention of a wide area... more Multiplicative white-noise stochastic processes continuously attract the attention of a wide area of scientific research. The variety of prescriptions available to define it difficults the development of general tools for its characterization. In this work, we study equilibrium properties of Markovian multiplicative white-noise processes. For this, we define the time reversal transformation for this kind of processes, taking into account that the asymptotic stationary probability distribution depends on the prescription. Representing the stochastic process in a functional Grassman formalism, we avoid the necessity of fixing a particular prescription. In this framework, we analyze equilibrium properties and study hidden symmetries of the process. We show that, using a careful definition of equilibrium distribution and taken into account the appropriate time reversal transformation, usual equilibrium properties are satisfied for any prescription. Finally, we present a detailed deduction of a covariant supersymmetric formulation of a multiplicative Markovian white-noise process and study some of the constraints it imposes on correlation functions using Ward-Takahashi identities.
Physical Review E, Sep 16, 2014
We deduce a nonlinear and inhomogeneous Fokker-Planck equation within a generalized Stratonovich,... more We deduce a nonlinear and inhomogeneous Fokker-Planck equation within a generalized Stratonovich, or stochastic α-, prescription (α = 0, 1/2 and 1 respectively correspond to the Itô, Stratonovich and anti-Itô prescriptions). We obtain its stationary state pst(x) for a class of constitutive relations between drift and diffusion and show that it has a q-exponential form, pst(x) = Nq[1 − (1 − q)βV (x)] 1/(1−q) , with an index q which does not depend on α in the presence of any nonvanishing nonlinearity. This is in contrast with the linear case, for which the index q is α-dependent.
Physical Review B, Aug 8, 2008
We analyze the experimental observation of metastable anisotropy resistance orientation at half f... more We analyze the experimental observation of metastable anisotropy resistance orientation at half filled quantum Hall fluids by means of a model of a quantum nematic liquid in an explicit symmetry breaking potential. We interpret the observed "rotation" of the anisotropy axis as a process of nucleation of nematic domains and compute the nucleation rate within this model. By comparing with experiment, we are able to predict the critical radius of nematic bubbles, Rc ∼ 2.6µm. Each domain contains about 10 4 electrons.
Physical review, Jun 2, 2020
We consider a simple model of a bistable system under the influence of multiplicative noise. We p... more We consider a simple model of a bistable system under the influence of multiplicative noise. We provide a path integral representation of the overdamped Langevin dynamics and compute conditional probabilities and escape rates in the weak noise approximation. The saddle-point solution of the functional integral is given by a diluted gas of instantons and anti-instantons, similarly to the additive noise problem. However, in this case, the integration over fluctuations is more involved. We introduce a local time reparametrization that allows its computation in the form of usual Gaussian integrals. We found corrections to the Kramers' escape rate produced by the diffusion function which governs the state dependent diffusion for arbitrary values of the stochastic prescription parameter. Theoretical results are confirmed through numerical simulations.
Physica D: Nonlinear Phenomena, Nov 1, 2018
We study the stochastic dynamics of a two-dimensional magnetic moment embedded in a threedimensio... more We study the stochastic dynamics of a two-dimensional magnetic moment embedded in a threedimensional environment, described by means of the stochastic Landau-Lifshitz-Gilbert (sLLG) equation. We define a covariant generalization of this equation, valid in the "generalized Stratonovich discretization prescription". We present a path integral formulation that allows to compute any n−point correlation function, independently of the stochastic calculus used. Using this formalism, we show the equivalence between the cartesian formulation with vectorial noise, and the polar formulation with just one scalar fluctuation term. In particular, we show that, for isotropic fluctuations, the system is represented by an additive stochastic process, despite of the multiplicative terms appearing in the original formulation of the sLLG equation, but, for anisotropic fluctuations the noise turns out to be truly multiplicative.
Physical Review E, Apr 6, 2015
We discuss general multi-dimensional stochastic processes driven by a system of Langevin equation... more We discuss general multi-dimensional stochastic processes driven by a system of Langevin equations with multiplicative white noise. In particular, we address the problem of how time reversal diffusion processes are affected by the variety of conventions available to deal with stochastic integrals. We present a functional formalism to built up the generating functional of correlation functions without any type of discretization of the Langevin equations at any intermediate step. The generating functional is characterized by a functional integration over two sets of commuting variables as well as Grassmann variables. In this representation, time reversal transformation became a linear transformation in the extended variables, simplifying in this way the complexity introduced by the mixture of prescriptions and the associated calculus rules. The stochastic calculus is codified in our formalism in the structure of the Grassmann algebra. We study some examples such as higher order derivatives Langevin equations and the functional representation of the micromagnetic stochastic Landau-Lifshitz-Gilbert equation.
Journal of Statistical Mechanics: Theory and Experiment, May 19, 2016
We analyse various properties of stochastic Markov processes with multiplicative white noise. We ... more We analyse various properties of stochastic Markov processes with multiplicative white noise. We take a single-variable problem as a simple example, and we later extend the analysis to the Landau-Lifshitz-Gilbert equation for the stochastic dynamics of a magnetic moment. In particular, we focus on the non-equilibrium transfer of angular momentum to the magnetization from a spin-polarised current of electrons, a technique which is widely used in the context of spintronics to manipulate magnetic moments. We unveil two hidden dynamical symmetries of the generating functionals of these Markovian multiplicative white-noise processes. One symmetry only holds in equilibrium and we use it to prove generic relations such as the fluctuation-dissipation theorems. Out of equilibrium, we take profit of the symmetry-breaking terms to prove fluctuation theorems. The other symmetry yields strong dynamical relations between correlation and response functions which can notably simplify the numerical analysis of these problems. Our construction allows us to clarify some misconceptions on multiplicative white-noise stochastic processes that can be found in the literature. In particular, we show that a first-order differential equation with multiplicative white noise can be transformed into an additive-noise equation, but that the latter keeps a non-trivial memory of the discretisation prescription used to define the former.
Journal of Statistical Mechanics: Theory and Experiment, Sep 8, 2014
We construct a path-integral representation of the generating functional for the dissipative dyna... more We construct a path-integral representation of the generating functional for the dissipative dynamics of a classical magnetic moment as described by the stochastic generalization of the Landau-Lifshitz-Gilbert equation proposed by Brown [1], with the possible addition of spin-torque terms. In the process of constructing this functional in the Cartesian coordinate system, we critically revisit this stochastic equation. We present it in a form that accommodates for any discretization scheme thanks to the inclusion of a drift term. The generalized equation ensures the conservation of the magnetization modulus and the approach to the Gibbs-Boltzmann equilibrium in the absence of nonpotential and time-dependent forces. The drift term vanishes only if the mid-point Stratonovich prescription is used. We next reset the problem in the more natural spherical coordinate system. We show that the noise transforms non-trivially to spherical coordinates acquiring a non-vanishing mean value in this coordinate system, a fact that has been often overlooked in the literature. We next construct the generating functional formalism in this system of coordinates for any discretization prescription. The functional formalism in Cartesian or spherical coordinates should serve as a starting point to study different aspects of the out-of-equilibrium dynamics of magnets. Extensions to colored noise, micro-magnetism and disordered problems are straightforward.
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Papers by Zochil Gonzalez Arenas