scholarly journals Evolution of Correlations of Many-Particle Quantum Systems in Condensed States

2018 ◽  
Author(s):  
Viktor Gerasimenko
Keyword(s):  
Evolution Equations ◽  
Bbgky Hierarchy ◽  
Quantum Systems ◽  
Nonlinear Evolution Equations ◽  
Von Neumann ◽  
Correlation Operator ◽  
Nonperturbative Solution ◽  
Rigorous Description ◽  
The Cauchy Problem ◽  
Dynamics Of Correlations

We review some new approaches to the description of the evolution of states of many-particle quantum systems by means of the correlation operators. Using the denition of marginal correlation operators within the framework of dynamics of correlations governed by the von Neumann hierarchy, we establish that a sequence of such operators is governed by the nonlinear quantum BBGKY hierarchy. The constructed nonperturbative solution of the Cauchy problem to this hierarchy of nonlinear evolution equations describes the processes of the creation and the propagation of correlations in many-particle quantum systems. Moreover, we consider the problem of the rigorous description of collective behavior of many-particle quantum systems by means of a one-particle (marginal) correlation operator that is a solution of the generalized quantum kinetic equation with initial correlations, in particular, correlations characterizing the condensed states of systems.

scholarly journals Low-Density Asymptotic Behavior of Observables of Hard Sphere Fluids

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Viktor Gerasimenko ◽  
Igor Gapyak
Keyword(s):  
Asymptotic Behavior ◽  
Hard Sphere ◽  
Particle Distribution ◽  
Scaling Limit ◽  
Bbgky Hierarchy ◽  
Boltzmann Kinetic Equation ◽  
Low Density ◽  
Nonperturbative Solution ◽  
Rigorous Description ◽  
The Cauchy Problem

The paper deals with a rigorous description of the kinetic evolution of a hard sphere system in the low-density (Boltzmann–Grad) scaling limit within the framework of marginal observables governed by the dual BBGKY (Bogolyubov–Born–Green–Kirkwood–Yvon) hierarchy. For initial states specified by means of a one-particle distribution function, the link between the Boltzmann–Grad asymptotic behavior of a nonperturbative solution of the Cauchy problem of the dual BBGKY hierarchy for marginal observables and a solution of the Boltzmann kinetic equation for hard sphere fluids is established. One of the advantages of such an approach to the derivation of the Boltzmann equation is an opportunity to describe the process of the propagation of initial correlations in scaling limits.

scholarly journals Frequency-Uniform Decomposition, Function Spaces , and Applications to Nonlinear Evolution Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Shaolei Ru ◽  
Jiecheng Chen
Keyword(s):  
Cauchy Problem ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Function Spaces ◽  
Nonlinear Evolution Equations ◽  
Ginzburg Landau ◽  
New Class ◽  
The Cauchy Problem ◽  
Ginzburg Landau Equations

By combining frequency-uniform decomposition with (), we introduce a new class of function spaces (denoted by ). Moreover, we study the Cauchy problem for the generalized NLS equations and Ginzburg-Landau equations in .

scholarly journals Generalized quasilinear hyperbolic equations and Yosida approximations

2003 ◽  
Vol 74 (1) ◽  
pp. 69-86 ◽  
Author(s):  
Jong Yeoul Park ◽  
Il Hyo Jung ◽  
Yong Han Kang
Keyword(s):  
Cauchy Problem ◽  
Boundary Conditions ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Hyperbolic Equations ◽  
Global Solutions ◽  
Global Solvability ◽  
Nonlinear Evolution Equations ◽  
Damping Term ◽  
The Cauchy Problem

AbstractWe will show the existence, uniqueness and regularity of global solutions for the Cauchy problem for nonlinear evolution equations with the damping term .As an application of our results, we give the global solvability and regularity of the mixed problem with Dirichiet boundary conditions:

Traveling wave solutions for the Couple Boiti-Leon-Pempinelli System by using extended Jacobian elliptic function expansion method

2015 ◽  
Vol 11 (3) ◽  
pp. 3134-3138 ◽  
Author(s):  
Mostafa Khater ◽  
Mahmoud A.E. Abdelrahman
Keyword(s):  
Elliptic Function ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Jacobian Elliptic Function ◽  
Function Expansion

In this work, an extended Jacobian elliptic function expansion method is pro-posed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics.

Hamilton-Jacobi Theory

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Phase Space ◽  
Bbgky Hierarchy ◽  
Distribution Functions ◽  
Symplectic Integrators ◽  
Partition Functions ◽  
Von Neumann ◽  
Equipartition Theorem ◽  
Recurrence Theorem ◽  
Phase Amplitude ◽  
Canonical Ensembles

This chapter focuses on Liouville’s theorem and classical statistical mechanics, deriving the classical propagator. The terms ‘phase space volume element’ and ‘Liouville operator’ are defined and an n-particle phase space probability density function is constructed to derive the Liouville equation. This is deconstructed into the BBGKY hierarchy, and radial distribution functions are used to develop n-body correlation functions. Koopman–von Neumann theory is investigated as a classical wavefunction approach. The chapter develops an operatorial mechanics based on classical Hilbert space, and discusses the de Broglie–Bohm formulation of quantum mechanics. Partition functions, ensemble averages and the virial theorem of Clausius are defined and Poincaré’s recurrence theorem, the Gibbs H-theorem and the Gibbs paradox are discussed. The chapter also discusses commuting observables, phase–amplitude decoupling, microcanonical ensembles, canonical ensembles, grand canonical ensembles, the Boltzmann factor, Mayer–Montroll cluster expansion and the equipartition theorem and investigates symplectic integrators, focusing on molecular dynamics.

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