scholarly journals A nonperturbative solution of the nonlinear BBGKY hierarchy for marginal correlation operators

2013 ◽  
Vol 36 (17) ◽  
pp. 2311-2328 ◽  
Author(s):  
V.I. Gerasimenko ◽  
D.O. Polishchuk
Keyword(s):  
Bbgky Hierarchy ◽  
Nonperturbative Solution

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.

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.

scholarly journals Transport of pseudothermal photons through an anharmonic cavity

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Dmitriy S. Shapiro
Keyword(s):  
Pauli Principle ◽  
Optical Systems ◽  
Frequency Noise ◽  
Continuous Transition ◽  
Noise Current ◽  
Nonperturbative Solution ◽  
Occupation Numbers ◽  
Fundamental Reason ◽  
The Rich ◽  
Quantum Optical

AbstractUnder nonequilibrium conditions, quantum optical systems reveal unusual properties that might be distinct from those in condensed matter. The fundamental reason is that photonic eigenstates can have arbitrary occupation numbers, whereas in electronic systems these are limited by the Pauli principle. Here, we address the steady-state transport of pseudothermal photons between two waveguides connected through a cavity with Bose–Hubbard interaction between photons. One of the waveguides is subjected to a broadband incoherent pumping. We predict a continuous transition between the regimes of Lorentzian and Gaussian chaotic light emitted by the cavity. The rich variety of nonequilibrium transport regimes is revealed by the zero-frequency noise. There are three limiting cases, in which the noise-current relation is characterized by a power-law, $$S\propto J^\gamma$$ S ∝ J γ . The Lorentzian light corresponds to Breit-Wigner-like transmission and $$\gamma =2$$ γ = 2 . The Gaussian regime corresponds to many-body transport with the shot noise ($$\gamma =1$$ γ = 1 ) at large currents; at low currents, however, we find an unconventional exponent $$\gamma =3/2$$ γ = 3 / 2 indicating a nontrivial interplay between multi-photon transitions and incoherent pumping. The nonperturbative solution for photon dephasing is obtained in the framework of the Keldysh field theory and Caldeira-Leggett effective action. These findings might be relevant for experiments on photon blockade in superconducting qubits, thermal states transfer, and photon statistics probing.

The Cauchy problem for BBGKY hierarchy of quantum kinetic equations with coulomb potential

2009 ◽  
Author(s):  
M. Brokate ◽  
M. Yu. Rasulova ◽  
A. H. Siddiqi ◽  
M. Brokate ◽  
A. K. Gupta
Keyword(s):  
Cauchy Problem ◽  
Coulomb Potential ◽  
Kinetic Equations ◽  
Bbgky Hierarchy ◽  
The Cauchy Problem

Statistical Theory of the SASE Fel Based on the Two-Particle Correlation Function Equation

2013 ◽  
Vol 8 (4) ◽  
pp. 25-34
Author(s):  
Oleg Shevchenko ◽  
Nikolay Vinokurov
Keyword(s):  
Correlation Function ◽  
Initial Conditions ◽  
Initial Density ◽  
Bbgky Hierarchy ◽  
Density Fluctuations ◽  
Distribution Functions ◽  
Time Coordinate ◽  
Particle Correlation ◽  
Sase Fel ◽  
Special Time

The startup from noise problem in SASE FELs is usually treated in linear approximation. In this case amplification of initial density fluctuations may be calculated, and averaging over initial conditions may be fulfilled explicitly. In general nonlinear case the direct averaging is not applicable. During last years we developed the approach based on the BBGKY hierarchy for the n-particle distribution functions. The interaction of particles in FEL is retarded. Nevertheless, using special time-coordinate transformation, it is possible to eliminate the interaction lag and then to write down the BBGKY equations. Similar to plasma physics, the equations may be truncated after the second one (for the two-particle correlation function). Using this approach we consider several particular cases which illustrate some peculiar features of the SASE FEL operation

Correlation Funktions of a System with Different Temperatures of the Particle Components

1964 ◽  
Vol 19 (13) ◽  
pp. 1447-1451 ◽  
Author(s):  
G. Ecker ◽  
W. Kröll
Keyword(s):  
Correlation Function ◽  
Electron Correlation ◽  
Pair Correlation ◽  
Bbgky Hierarchy ◽  
Ion Pair ◽  
Pair Correlations ◽  
The Poisson Equation ◽  
Average Potential ◽  
Different Temperatures ◽  
Dressed Particles

We consider a plasma consisting of particle components with different temperatures. The components are uniformly distributed in the configuration space and MAXWELLIAN in the velocity space. Pair correlations are assumed to be small and higher order correlations negligible. It is shown from the BBGKY-hierarchy that the influence of the electrons on the ion kinetics can be taken into account by treating the ions as dressed particles. The hierarchy for these dressed particles provides the ion-ion correlation function. The electron-ion pair correlation is calculated from the POISSON equation using the ion-ion correlation and relating the electron-ion pair distribution to the average potential. By the same procedure we derive the electron-electron correlation making use of the electron-ion correlation. The results are compared with those of other authors.

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