Exact solutions of stochastic fractional Korteweg de–Vries equation with conformable derivatives

We deal with the Wick-type stochastic fractional Korteweg de–Vries(KdV) equation with conformable derivatives.With the aid of the Exp-function method, white noise theory, and Hermite transform, we produce a novel set of exact soliton and periodic wave solutions to the fractional KdV equation with conformable derivatives. With the help of inverse Hermite transform, we get stochastic soliton and periodic wave solutions of the Wick-type stochastic fractional KdV equation with conformable derivatives. Eventually, by an application example, we show how the stochastic solutions can be given as Brownian motion functional solutions.     

Foundations of quantum mechanics: The Langevin equations for QM

Stochastic derivations of the Schrödinger equation are always developed on very general and abstract grounds. Thus, one is never enlightened which specific stochastic process corresponds to some particular quantum mechanical system, that is, given the physical system—expressed by the potential function, which fluctuation structure one should impose on a Langevin equation in order to arrive at results identical to those comming from the solutions of the Schrödinger equation. We show, from first principles, how to write the Langevin stochastic equations for any particular quantum system. We also show the relation between these Langevin equations and those proposed by Bohm in 1952. We present numerical simulations of the Langevin equations for some quantum mechanical problems and compare them with the usual analytic solutions to show the adequacy of our approach. The model also allows us to address important topics on the interpretation of quantum mechanics.     

Stochastic least-action principle for the incompressible Navier-Stokes equation

We formulate a stochastic least-action principle for solutions of the incompressible Navier-Stokes equation, which formally reduces to Hamilton’s principle for the incompressible Euler solutions in the case of zero viscosity. We use this principle to give a new derivation of a stochastic Kelvin Theorem for the Navier-Stokes equation, recently established by Constantin and Iyer, which shows that this stochastic conservation law arises from particle-relabelling symmetry of the action. We discuss issues of irreversibility, energy dissipation, and the inviscid limit of Navier-Stokes solutions in the framework of the stochastic variational principle. In particular, we discuss the connection of the stochastic Kelvin Theorem with our previous “martingale hypothesis” for fluid circulations in turbulent solutions of the incompressible Euler equations.     

A Stochastic Nonlinear Schrödinger Equation¶with Multiplicative Noise

We study a conservative stochastic nonlinear Schr?dinger equation with a multiplicative noise. We show the global existence and uniqueness of square integrable solutions for subcritical nonlinearities, the critical exponent being the same, in dimension 1 or 2, as the critical exponent of the deterministic equation. Received: 21 December 1998 / Accepted: 22 February 1999     

Hydrodynamics in two dimensions and vortex theory

We consider the Navier-Stokes equation for a viscous and incompressible fluid inR ². We show that such an equation may be interpreted as a mean field equation (Vlasov-like limit) for a system of particles, called vortices, interacting via a logarithmic potential, on which, in addition, a stochastic perturbation is acting. More precisely we prove that the solutions of the Navier-Stokes equation may be approximated, in a suitable way, by finite dimensional diffusion processes with the diffusion constant related to the viscosity. As a particular case, when the diffusion constant is zero, the finite dimensional theory reduces to the usual deterministic vortex theory, and the limiting equation reduces to the Euler equation.Partially supported by Italian CNR     

The diffusion equation of random genetic drift – biology’s analogue of the Schrödinger equation?

This work is a self-contained introduction to some basic aspects of the dynamics that occurs in biological populations. It focusses on the proportion (or frequency) of a population that carries a particular gene. We make use of the notion of a force, in the context of genetics and evolution, to describe the dynamics of the frequency in an effectively infinite population. We then show how randomness enters into the dynamics of populations with a finite size, a randomness known as random genetic drift. We derive an equation, involving random numbers, which describes how the frequency behaves in a population of finite size. It is shown that in some situations this equation exhibits irreversible absorption phenomena. These phenomena are associated with the extinction (or loss) of the gene, or the complete takeover by the gene (termed fixation), where 100% of the population carries the gene. Taking the theory further, we show how an approximation leads to a stochastic differential equation for the frequency, where random genetic drift takes the form of an additional contribution to the force, that randomly fluctuates. The stochastic differential equation is, in turn, related to a diffusion equation, which encompasses many fundamental phenomena. Because of this, the diffusion equation plausibly has a similar status in biology to the Schrödinger equation in physics. It is notable that both the Schrödinger equation and the diffusion equation have a somewhat similar mathematical structure: they both involve first order derivatives of time and second order derivatives of space (or the analogue of space). There are, however, some significant mathematical differences. In contrast to the Schrödinger equation, the diffusion equation can routinely have solutions which are singular, in that the solutions contain Dirac delta functions. The delta functions are not, however, problematic, and have an explicit biological significance. We illustrate results with some basic calculations and computer simulations.     

A stochastic lattice gas for Burgers' equation: A practical study

We continue our investigation of stochastic lattice gases as a (highly parallel) means of simulating given PDEs, in this case Burgers' equation in one dimension. The lattice dynamics consists of stochastic unidirectional particle displacement, and our attention is turned toward the reliability of the model, i.e., its ability to reproduce the unique physical solution of Burgers' equation. Lattice gas results are discussed and compared against finite-difference calculations and exact solutions in examples which include shocks and rarefaction waves.     

Stability of Symmetric Solitary Wave Solutions of a Forced Korteweg-de Vries Equation and the Polynomial Chaos

In this paper, we consider the numerical stability of gravity-capillary waves generated by a localized pressure in water of finite depth based on the forced Korteweg-de Vries (FKdV) framework and the polynomial chaos. The stability studies are focused on the symmetric solitary wave for the subcritical flow with the Bond number greater than one third. When its steady symmetric solitary-wave-like solutions are randomly perturbed, the evolutions of some waves show stability in time regardless of the randomness while other waves produce unstable fluctuations. By representing the perturbation with a random variable, the governing FKdV equation is interpreted as a stochastic equation. The polynomial chaos expansion of the random solution has been used for the study of stability in two ways. Firstly, it allows us to identify the stable solution of the stochastic governing equation. Secondly, it is used to construct upper and lower bounding surfaces for unstable solutions, which encompass the fluctuations of waves.     

Extended Riccati Equation Rational Expansion Method and Its Application to Nonlinear Stochastic Evolution Equations

In this work, by means of a new more general ansatz and the symbolic computation system Maple, we extend the Riccati equation rational expansion method [Chaos, Solitons & Fractals 25 (2005) 1019] to uniformly construct a series of stochastic nontravelling wave solutions for nonlinear stochastic evolution equation. To illustrate the effectiveness of our method, we take the stochastic mKdV equation as an example, and successfully construct some new and more general solutions including a series of rational formal nontraveling wave and coefficient functions' soliton-like solutions and trigonometric-like function solutions. The method can also be applied to solve other nonlinear stochastic evolution equation or equations.     

Convergence to SPDEs in Stratonovich Form

We consider the perturbation of parabolic operators of the form ∂ _t  + P(x, D) by large-amplitude highly oscillatory spatially dependent potentials modeled as Gaussian random fields. The amplitude of the potential is chosen so that the solution to the random equation is affected by the randomness at the leading order. We show that, when the dimension is smaller than the order of the elliptic pseudo-differential operator P(x, D), the perturbed parabolic equation admits a solution given by a Duhamel expansion. Moreover, as the correlation length of the potential vanishes, we show that the latter solution converges in distribution to the solution of a stochastic parabolic equation with multiplicative noise that should be interpreted in the Stratonovich sense. The theory of mild solutions for such stochastic partial differential equations is developed. The behavior described above should be contrasted to the case of dimensions larger than or equal to the order of the elliptic pseudo-differential operator P(x, D). In the latter case, the solution to the random equation converges strongly to the solution of a homogenized (deterministic) parabolic equation as is shown in [2]. A stochastic limit is obtained only for sufficiently small space dimensions in this class of parabolic problems.     

Uniformly Accelerated Charge in a Quantum Field: From Radiation Reaction to Unruh Effect

We present a stochastic theory for the nonequilibriurn dynamics of charges moving in a quantum scalar field based on the worldline influence functional and the close-time-path (CTP or in-in) coarse-grained effective action method. We summarize (1) the steps leading to a derivation of a modified Abraham-Lorentz-Dirac equation whose solutions describe a causal semiclassical theory free of runaway solutions and without pre-acceleration patholigies, and (2) the transformation to a stochastic effective action, which generates Abraham-Lorentz-Dirac-Langevin equations depicting the fluctuations of a particle’s worldline around its semiclassical trajectory. We point out the misconceptions in trying to directly relate radiation reaction to vacuum fluctuations, and discuss how, in the framework that we have developed, an array of phenomena, from classical radiation and radiation reaction to the Unruh effect, are interrelated to each other as manifestations at the classical, stochastic and quantum levels. Using this method we give a derivation of the Unruh effect for the spacetime worldline coordinates of an accelerating charge. Our stochastic particle-field model, which was inspired by earlier work in cosmological backreaction, can be used as an analog to the black hole backreaction problem describing the stochastic dynamics of a black hole event horizon.     

Linear response to perturbation of nonexponential renewal processes

We study the linear response of a two-state stochastic process, obeying the renewal condition, by means of a stochastic rate equation equivalent to a master equation with infinite memory. We show that the condition of perennial aging makes the response to coherent perturbation vanish in the long-time limit.     

Stability of Semiclassical Gravity Solutions with Respect to Quantum Metric Fluctuations

We discuss the stability of semiclassical gravity solutions with respect to small quantum corrections by considering the quantum fluctuations of the metric perturbations around the semiclassical solution. We call the attention to the role played by the symmetrized 2-point quantum correlation function for the metric perturbations, which can be naturally decomposed into two separate contributions: intrinsic and induced fluctuations. We show that traditional criteria on the stability of semiclassical gravity are incomplete because these criteria based on the linearized semiclassical Einstein equation can only provide information on the expectation value and the intrinsic fluctuations of the metric perturbations. By contrast, the framework of stochastic semiclassical gravity provides a more complete and accurate criterion because it contains information on the induced fluctuations as well. The Einstein–Langevin equation therein contains a stochastic source characterized by the noise kernel (the symmetrized 2-point quantum correlation function of the stress tensor operator) and yields stochastic correlation functions for the metric perturbations which agree, to leading order in the large N limit, with the quantum correlation functions of the theory of gravity interacting with N matter fields. These points are illustrated with the example of Minkowski space-time as a solution to the semiclassical Einstein equation, which is found to be stable under both intrinsic and induced fluctuations.     

Adiabatic Elimination in Quantum Stochastic Models

We consider a physical system with a coupling to bosonic reservoirs via a quantum stochastic differential equation. We study the limit of this model as the coupling strength tends to infinity. We show that in this limit the solution to the quantum stochastic differential equation converges strongly to the solution of a limit quantum stochastic differential equation. In the limiting dynamics the excited states are removed and the ground states couple directly to the reservoirs.     

A proposed solution to the tail problem of dynamical reduction models

We show that it is possible to avoid the tail problem in dynamical reduction models by considering, in addition to the stochastic equation for the wavefunction, an equation for position in a manner similar to Bohm, but with a velocity subjected to a stochastic process.     

A Variational Formulation for the Navier-Stokes Equation

In this paper we prove a new variational principle for the Navier-Stokes equation which asserts that its solutions are critical points of a stochastic control problem in the group of area-preserving diffeomorphisms. This principle is a natural extension of the results by Arnold, Ebin, and Marsden concerning the Euler equation.Supported in part by FCT/POCTI/FEDER     

Poisson-Vlasov: stochastic representation and numerical codes

A stochastic representation for the solutions of the Poisson-Vlasov equation, with several charged species, is obtained. The representation involves both an exponential and a branching process and it provides an intuitive characterization of the nature of the solutions and its fluctuations. Here, the stochastic representation is also proposed as a tool for the numerical evaluation of the solutions. An erratum to this article is available at .     

A Stochastic Mechanics Based on Bohm‧s Theory and its Connection with Quantum Mechanics

We construct a stochastic mechanics by replacing Bohm‧s first-order ordinary differential equation of motion with a stochastic differential equation where the stochastic process is defined by the set of Bohmian momentum time histories from an ensemble of particles. We show that, if the stochastic process is a purely random process with n-th order joint probability density in the form of products of delta functions, then the stochastic mechanics is equivalent to quantum mechanics in the sense that the former yields the same position probability density as the latter. However, for a particular non-purely random process, we show that the stochastic mechanics is not equivalent to quantum mechanics. Whether the equivalence between the stochastic mechanics and quantum mechanics holds for all purely random processes but breaks down for all non-purely random processes remains an open question.     

Convergence from Boltzmann to Landau Processes with Soft Potential and Particle Approximations

Our aim in this paper is to show how a probabilistic interpretation of the Boltzmann and Landau equations gives a microscopic understanding of these equations. We firstly associate stochastic jump processes with the Boltzmann equations we consider. Then we renormalize these equations following asymptotics which make prevail the grazing collisions, and prove the convergence of the associated Boltzmann jump processes to a diffusion process related to the Landau equation. The convergence is pathwise and also implies a convergence at the level of the partial differential equations. The best feature of this approach is the microscopic understanding of the transition between the Boltzmann and the Landau equations, by an accumulation of very small jumps. We deduce from this interpretation an approximation result for a solution of the Landau equation via colliding stochastic particle systems. This result leads to a Monte-Carlo algorithm for the simulation of solutions by a conservative particle method which enables to observe the transition from Boltzmann to Landau equations. Numerical results are given.     

Exact solution of a Langevin equation with nonlinear periodic noise

An exact stochastic average of a Langevin equation with a multiplicative nonlinear periodic noise is performed. The noise is described by an arbitrary periodic function of the diffusion Wiener-Lévy stochastic process. The solution of this stochastic equation is given by periodic solutions of the Hill equation.