Metastable behavior of stochastic dynamics: A pathwise approach

In this paper a new approach to metastability for stochastic dynamics is proposed. The basic idea is to study the statistics of each path, performing time averages along the evolution. Metastability would be characterized by the fact that the process of these time averages converges, under a suitable rescaling, to a measure valued Markov jump process. Here this convergence is shown for the Curie-Weiss mean field dynamics and also for a model with spatial structure: Harris contact process.Partially supported by CNPq, Grant No. 301301/79.     

A stochastic approach to Wilson loops

Wilson loops exp (i A (x) dx) are investigated in two-dimensional Euclidean space-time. The electromagnetic vector potential A is regarded as a generalized random field given by the stochastic partial differential equation A = F where is a first-order differential operator and F is white noise. We give a rigorous definition of Wilson loops and examine the properties of the N-loop Schwinger functions.     

A stochastic approach to open quantum systems

Stochastic methods are ubiquitous to a variety of fields, ranging from physics to economics and mathematics. In many cases, in the investigation of natural processes, stochasticity arises every time one considers the dynamics of a system in contact with a somewhat bigger system, an environment with which it is considered in thermal equilibrium. Any small fluctuation of the environment has some random effect on the system. In physics, stochastic methods have been applied to the investigation of phase transitions, thermal and electrical noise, thermal relaxation, quantum information, Brownian motion and so on. In this review, we will focus on the so-called stochastic Schr?dinger equation. This is useful as a starting point to investigate the dynamics of open quantum systems capable of exchanging energy and momentum with an external environment. We discuss in some detail the general derivation of a stochastic Schr?dinger equation and some of its recent applications to spin thermal transport, thermal relaxation, and Bose-Einstein condensation. We thoroughly discuss the advantages of this formalism with respect to the more common approach in terms of the reduced density matrix. The applications discussed here constitute only a few examples of a much wider range of applicability.     

A variational approach to stochastic nonlinear problems

A variational principle is formulated which enables the mean value and higher moments of the solution of a stochastic nonlinear differential equation to be expressed as stationary values of certain quantities. Approximations are generated by using suitable trial functions in this variational principle and some of these are investigated numerically for the case of a Bernoulli oscillator driven by white noise. Comparison with exact data available for this system shows that the variational approach to such problems can be quite effective.     

Evolutionary dynamics of imatinib-treated leukemic cells by stochastic approach

The evolutionary dynamics of a system of cancerous cells in a model of chronic myeloid leukemia (CML) is investigated by a statistical approach. Cancer progression is explored by applying a Monte Carlo method to simulate the stochastic behavior of cell reproduction and death in a population of blood cells which can experience genetic mutations. In CML front line therapy is represented by the tyrosine kinase inhibitor imatinib which strongly affects the reproduction of leukemic cells only. In this work, we analyze the effects of a targeted therapy on the evolutionary dynamics of normal, first-mutant and cancerous cell populations. Several scenarios of the evolutionary dynamics of imatinib-treated leukemic cells are described as a consequence of the efficacy of the different modelled therapies. We show how the patient response to the therapy changes when a high value of the mutation rate from healthy to cancerous cells is present. Our results are in agreement with clinical observations. Unfortunately, development of resistance to imatinib is observed in a fraction of patients, whose blood cells are characterized by an increasing number of genetic alterations. We find that the occurrence of resistance to the therapy can be related to a progressive increase of deleterious mutations.      

A stochastic approach to simulations of fermionic systems

We introduce a Langevin equation approach to the analysis of fermionic theories. We find a Langevin type equation, depending on a parameter τ, such that its equilibrium distributions are those of the original fermionic system in the limit of τ going to zero. We explicitly treat a simple example, proving the exactness of the method in the limit τ → 0, and we estimate the error induced by the method at first order in τ.     

A stochastic approach to hopping transport in semiconductors

Generalized charge carrier equations for hopping transport in semiconductors are derived which include also the widely used Van Roosbroeck equations. The approach is based on a microscopic stochastic interacting particle system which models the hopping of electrons on a random set of states.     

A sigma-model approach to glassy dynamics

In this contribution we review recent progress in understanding fluctuations in the aging process of macroscopic systems, and we propose further tests of these ideas. We discuss how the emergence of a symmetry in aging systems, global timereparametrization invariance, could be responsible for the observed ‘universal’ behavior of local and mesoscopic non-equilibrium fluctuations. We discuss (i) the two-time scaling and functional form of the distribution of local correlations and responses; (ii) the scaling of multi-time correlations and susceptibilities; (iii) how the above can be derived from a random surface effective action; (iv) the behavior of a diverging two-time dependent correlation length; (v) how these ideas apply to off-lattice particle systems.     

A stochastic approach to the hadron spectrum. I

In this paper the squared mass of the hadron is defined as a random variable, whose average is the measured quantity. This leads to a mass formula, of a unique type for mesons and baryons, with a general law for the spin variation of the coefficients. The central squared masses form an overall geometrical scheme; in the baryon case it contains trajectories which are a fine structure of the Regge trajectories. For the accurately measured masses the difference between the computed and experimental value lies within 5 MeV for mesons and baryons. The geometrical scheme will be the basis for the computation of the decay rates, to be developed in the next paper.     

A stochastic approach to the kinetic theory of gases

A theory of fluctuations in non-equilibrium diluted gases is presented. The velocity distribution function is treated as a stochastic variable and a master equation for its probability is derived. This evolution equation is based on two processes: binary hard sphere collisions and free flow. A mean-field approximation leads to a non-linear master equation containing explicitly a parameter which represents the spatial correlation length of the fluctuations. An infinite hierarchy of equations for the successive moments is found. If the correlation length is sufficiently short a truncation after the first equation is possible and this leads to the Boltzmann kinetic equation. The associated probability distribution is Poissonian. As to the fluctuation of the macroscopic quantities, an approximation scheme permits to recover the Langevin approach of fluctuating hydrodynamics near equilibrium and its fluctuation-dissipation relations.     

A stochastic approach to the hadron spectrum. III

The connection with the quarks of the stochastic model proposed in the two preceding papers is studied; the slopes of the baryon trajectories are calculated with reference to the quarks. Suggestions are made for the interpretation of the model (quadratic or linear addition of the contributions to the mass, dependence of the decay on the quantum numbers of the hadrons involved, etc.) and concerning its link with the quarkonium model, which describes the mesons with charm or beauty. The controversial question of the subquantum level is examined.     

A stochastic approach to the hadron spectrum. II

The definition of mass as a random variable is applied to the study of the decay rates. A decay is assumed possible when the fluctuation of the Gaussian variables involved makes a definite relation satisfied. Computing the probability of this process leads to the determination of the decay amplitude. This calculation, unified for baryons and mesons, is worked out in the lower and medium spectrum (up to2000 MeV for baryons and mesons), and fits to20 MeV the accurate measurements of width and branching ratios.     

A simple approach to the nuclear dynamics

We propose an approach to the nuclear dynamics which enables us to understand the relations between the time-dependent Hartree-Fock method and the quantumstatistical approach to the heavy ion reactions. The same formalism allows us to generalize the random phase approximation in order to include the spreading width of collective states.     

A constructive approach to gene expression dynamics

Recently, experiments on mRNA abundance (gene expression) have revealed that gene expression shows a stationary organization described by a scale-free distribution. Here we propose a constructive approach to gene expression dynamics which restores the scale-free exponent and describes the intermediate state dynamics. This approach requires only one assumption: Markov property.     

A stochastic approach to nonequilibrium thermodynamics of chemical reaction system

Nonequilibrium thermodynamics is formulated by combining the nonlinear Fokker-Planck equation with the so-called Gibbs entropy postulate. The entropy production thus derived consists of two parts: one is of the same form as the usual entropy production and the other is the fluctuating part attendant on it. The evolution criterion can easily be verified in the stochastic framework. For illustration the system governed by the linear Fokker-Planck equation is in detail discussed.