Thermo-Quantum Diffusion

A new approach to the thermo-quantum diffusion is proposed and a nonlinear quantum Smoluchowski equation is derived, which describes classical diffusion in the field of the Bohm quantum potential. A nonlinear thermo-quantum expression for the diffusion front is obtained, being a quantum generalization of the classical Einstein law. The quantum diffusion at zero temperature is also described and a new dependence of the position dispersion on time is derived. A stochastic Bohm-Langevin equation is also proposed.     

Nonlinear Theory of Quantum Brownian Motion

A nonlinear theory of quantum Brownian motion in classical environment is developed based on a thermodynamically enhanced nonlinear Schrödinger equation. The latter is transformed via the Madelung transformation into a nonlinear quantum Smoluchowski-like equation, which is proven to reproduce key results from the quantum and classical physics. The application of the theory to a free quantum Brownian particle results in a nonlinear dependence of the position dispersion on time, being quantum generalization of the Einstein law of Brownian motion. It is shown that the time of decoherence from quantum to classical diffusion is proportional to the square of the thermal de Broglie wavelength divided by the classical Einstein diffusion constant.     

Nonlinear dynamics of filamentation instability and current filament merging in a high density current-driven plasma

The magnetic field generation due to the filamentation instability (FI) of a high density current-driven plasma is studied through a new nonlinear diffusion equation. This equation is obtained on the basis of quantum hydrodynamic model and numerically solved by applying the Crank–Nicolson method. The spatiotemporal evolution of the magnetic field and the electron density distribution exhibits the current filament merging as a nonlinear phase of the FI which is responsible for the strong magnetic fields in the current-driven plasmas. It is found that the general behaviour of the FI is the same as that of the classical case but the instability growth rate, its magnitude, and the saturation time are affected by the quantum effects. It is eventually concluded that the quantum effects can play a stabilizing role in such situation.     

Friction and diffusion coefficients in coordinate in nonequilibrium nuclear processes

The influence of different sets of friction and diffusion coefficients on the dynamics of a nuclear system is investigated. Taking as an example a dinuclear system we show in a “classic” investigation that with zero diffusion in the coordinate, the uncertainty relation can be violated during short initial times. Sets of diffusion coefficients are found for which the “classic” and quantum diffusion equations give close physical results. The tunneling through an energy barrier is sensitively influenced by the friction and diffusion coefficients in coordinate in the diffusion equation.     

耗散子运动方程与福克-普朗克量子主方程

发展非线性耦合环境下精确、非微扰的量子耗散方法仍然是一个巨大的挑战.本文针对线性和二次耦合热浴模型,介绍了两种刻画系统与环境耦合动力学的有效方法.一个是耗散子运动方程（DEOM）理论,最近已被扩展到处理非线性耦合环境.另一个是推广的福克-普朗克量子主方程（FP-QME）方法,将在这项工作中基于DEOM推导给出.本文对这两种方法进行了详细的比较,并重点介绍了所涉及的准粒子图像、物理含义以及实现方案.     

Application of the theory of open quantum systems to nuclear physics problems

Quantum diffusion equations with transport coefficients explicitly depending on time are derived from the generalized non-Markovian Langevin equations. The asymptotic behavior of the friction and diffusion coefficients is investigated in the case of the FC and RWA couplings between the collective and internal subsystems. An asymptotic expression is obtained for the propagator of the density matrix of the open quantum system with the general quadratic Hamiltonian, linearly coupled (in coordinate and momentum) to internal degrees of freedom. The effect of different sets of transport coefficients on the decoherence and decay rate of the metastable state is investigated using the master equation for the reduced density matrix of open quantum systems. The developed approach is used to study the capture of the projectile nucleus by the target nucleus at energies near the Coulomb barrier. Capture cross sections in asymmetric reactions are well described with allowance for the calculated capture probabilities. Particular cases where dissipation favors penetration through the potential barrier are found. The generalized Kramers formula for the quasi-stationary decay rate of the quantum metastable systems is analytically derived.     

A fundamental constraint on quantum mechanical diffusion coefficients

A general quantum mechanical master equation for the damped oscillator, which can be represented as a phase space Fokker-Planck equation for the Wigner function, will be investigated with respect to the uncertainty principle. This leads to a condition to be imposed on the diffusion coefficients, which (i) is not fulfilled by an often quoted quasi-classical choice, and (ii) provides a uniqueness criterium in Dekker's theory.     

Lagrangian form of Schrödinger equation

Lagrangian formulation of quantum mechanical Schrödinger equation is developed in general and illustrated in the eigenbasis of the Hamiltonian and in the coordinate representation. The Lagrangian formulation of physically plausible quantum system results in a well defined second order equation on a real vector space. The Klein–Gordon equation for a real field is shown to be the Lagrangian form of the corresponding Schrödinger equation.     

The foundations of relativity

In a previous paper a stochastic foundation was proposed for microphysics: the nonrelativistic and relativistic domains were shown to be connected with two different approximations of diffusion theory; the relativistic features (Lorentz contraction for the coordinate standard deviation, covariant diffusion equation) were not derived from the relativistic formalism introduced at the start, but emerged from diffusion theory itself. In the present paper these results are given a new presentation, which aims at elucidating not the foundations of quantum mechanics, but those of relativity. This leads to a discussion of points still controversial in the interpretation of relativity. In particular two problems appear in a new light: the character of time and length alterations, and the privileged role of the velocityc. Besides, the question of a possible limitation of relativity (and more generally of the laws of mechanics) in the domain of particle substructure is raised and supported by exemples drawn from the hydrodynamical model of a spinned particle. Suggestions are presented for the possibility of a deeper conceptual unification of special and general relativity.     

A Novel Method to Solve Nonlinear Klein-Gordon Equation Arising in Quantum Field Theory Based on Bessel Functions and Jacobian Free Newton-Krylov Sub-Space Methods

The Klein-Gordon equation arises in many scientific areas of quantum mechanics and quantum field theory.In this paper a novel method based on spectral method and Jacobian free Newton method composed by generalized minimum residual(JFNGMRes) method with adaptive preconditioner will be introduced to solve nonlinear Klein-Gordon equation. In this work the nonlinear Klein-Gordon equation has been converted to a nonlinear system of algebraic equations using collocation method based on Bessel functions without any linearization, discretization and getting help of any other methods. Finally, by using JFNGMRes, solution of the nonlinear algebraic system will be achieved. To illustrate the reliability and efficiency of the proposed method, we solve some examples of the Klein-Gordon equation and compare our results with other methods.     

Localized solution of a simple nonlinear quantum Langevin equation

A simple nonlinear quantum Langevin equation is introduced as phenomenological equation for quantum brownian motion. Easy calculations yield a unique localized wave function in the stationary regime. The given example may encourage more general use of nonlinear quantum Langevin equations for damped quantum systems, e.g. in measurement theory, in heavy ion physics, etc.     

Multi-wave trains and Sasa - Satsuma freak events generation in an optical metamaterial

In this work, we present a new Ansatz function called ”type I Ansatz function” with seven collective coordinates compared to the conventional Gaussian Ansatz function with six collective coordinates. We show that the additional coordinate called ”nonlinear phase” will allow us to improve the technique of measurement of internal excitation leading to the generation of rogue events. Moreover, the nonlinear phase coordinate will give supplemental details on the phase distortions during the generation of specific rogue events such as wall of waves, tree structures, double Kuznetsov - Ma breathers and Sasa - Satsuma rogue waves. The propagation will be modeled by a higher - order nonlinear Schrödinger equation when second - and third - order dispersions, self - and quintic - phase modulations and self - steepening come into play for negative index and absorption regimes. The conditions of stability of the soliton light pulse will also been investigated. The validity of our results is confirmed numerically by using a systematic computational approach.     

On the quantum mechanical description of scattering by a dissipative and diffusive scattering centre

A partial integro-differential equation is formulated for the Wigner transform of the quantum mechanical reduced density operator describing the time evolution of a “macroscopic” coordinate under the influence of coupling to a large number of “intrinsic” degrees of freedom. The equation contains integral operators which lead to energy dissipation and diffusion and reduces to a transport equation of the Fokker-Planck type if the form factors in the integrands are treated in appropriate (harmonic) approximations. The stationary solution of the partial integro-differential equation is obtained numerically for scattering by a conservative potential and by a dissipative and diffusive scattering centre in one spatial dimension.     

A general nonlinear Fokker-Planck equation and its associated entropy

A recently introduced nonlinear Fokker-Planck equation, derived directly from a master equation, comes out as a very general tool to describe phenomenologically systems presenting complex behavior, like anomalous diffusion, in the presence of external forces. Such an equation is characterized by a nonlinear diffusion term that may present, in general, two distinct powers of the probability distribution. Herein, we calculate the stationary-state distributions of this equation in some special cases, and introduce associated classes of generalized entropies in order to satisfy the H-theorem. Within this approach, the parameters associated with the transition rates of the original master-equation are related to such generalized entropies, and are shown to obey some restrictions. Some particular cases are discussed.     

A Finite Volume Method for the Approximation of Diffusion Operators on Distorted Meshes

A new finite volume method is presented for discretizing general linear or nonlinear elliptic second-order partial-differential equations with mixed boundary conditions. The advantage of this method is that arbitrary distorted meshes can be used without the numerical results being altered. The resulting algorithm has more unknowns than standard methods like finite difference or finite element methods. However, the matrices that need to be inverted are positive definite, so the most powerful linear solvers can be applied. The method has been tested on a few elliptic and parabolic equations, either linear, as in the case of the standard heat diffusion equation, or nonlinear, as in the case of the radiation diffusion equation and the resistive diffusion equation with Hall term.     

Quantum fluctuations of spatial dissipative solitons in a nonlinear interferometer

A quantum Langevin equation is derived that makes it possible to study the radiation field in a large-aperture nonlinear interferometer excited by external classical radiation. This equation is linearized in the vicinity of the solution for a stationary soliton. A mathematical formalism for obtaining a spectral representation of the solution to the linearized problem is constructed. It is shown that, in general, the excitation spectrum of a soliton consists of three branches, two of which belong to a continuous spectrum, while the third branch is discrete. The spectral representation obtained makes it possible to rigorously define the operator of soliton coordinate fluctuations, since, as is shown in the study, the traditional definition of this operator leads to a divergence in the vicinity of the solution. A new type of dissipative soliton is found, which is a natural generalization of a stationary soliton and takes into account its motion. A relation is found between this soliton and the contribution to the solution for field fluctuations from the discrete spectrum expansion. The mean squares of fluctuations of the soliton coordinate and momentum are calculated. A range of parameters is determined where the momentum of the soliton can always be measured with a spread smaller than the standard quantum limit. This possibility is related to the occurrence of states squeezed with respect to the soliton momentum. A scheme is proposed for the experimental observation of these states.     

Master equation with two times: Diffusion in external field

The master equation for diffusion involving two times applies to the problem of diffusion in a time-dependent (in general inhomogeneous) external field. We consider the case of the quasi Fokker-Planck approximation, when the probability transition function for diffusion (PTD-function) does not possess a long tail in coordinate space and can be expanded as the function of instantaneous displacements. For relatively weak external field the linear expansion of the PTD function leads to a simple generalization of diffusion equation, containing the retardation factors.     

Quantum fluctuations,master equation and Fokker-Planck equation

In order to describe quantum fluctuations a general method is developed, which also may be applied to nonstationary systems as well as to states far from thermodynamic equilibrium. After a concise derivation of the master equation quantum mechanically determined dissipation and fluctuation coefficients are introduced, for which several theorems and relations are given. By using these coefficients there is set up a general Fokker-Planck equation for the diffusion of the statistical operator due to quantum fluctuations.