Derivation Of Quantum Kinetic Equation Research Articles

New approach to derivation of quantum kinetic equations with initial correlations

We propose a new approach to the derivation of kinetic equations from dynamics of large particle quantum systems, involving correlations of particle states at initial time. The developed approach is based on the description of the evolution within the framework of marginal observables in scaling limits. As a result the quantum Vlasov-type kinetic equation with initial correlations is constructed and the statement relating to the property of a propagation of initial correlations is proved in a mean field limit.

A Derivation of Quantum Kinetic Equation from Bohm Potential

In Bohm's interpretation of Quantum Mechanics, quantum effects are governed by a “quantum potential” (known as Bohm potential), and particles follow definite trajectories. In the present work, Liouville's theorem is invoked, an appropriate Liouville equation is derived, and following the BBGKY method a quantum kinetic equation (QKE) is derived. To demonstrate the working of the QKE, two examples of application are presented: the thermal equilibrium of a quantum gas and the propagation of disturbances in a force free gas of non-interacting bosons. In contrast to the classical collisionless Boltzmann equation, waves are found to be possible in the absence of interaction or external forces, due only to Bohm potential (zero sound propagation).

Derivation of quantum kinetic equation from the Liouville-von Neumann equation

A quantum Markov kinetic equation is derived from the Liouville—von Neumann equation in the framework of second-order kinetic perturbation theory. The proposed method can be used to obtain an equation that describes the quantum diffusion of a particle in a crystal.

On the statistical derivation of the Schr�dinger equation

The transition from reversible microscopic operator equations to irreversible equations for a deterministic density matrix is considered for examples of “simple systems”—the hydrogen atom or a free electron in an electromagnetic field. As a result of the transition, a system of a particle and field oscillators is replaced by a continuous medium. The Schrodinger equation for the deterministic wave function also describes the evolution of a continuum but without allowance for dissipative terms. In this sense, there is an analogy between the Schrodinger equation in quantum mechanics and Euler's equation in hydrodynamics. The smallest size of a “point” of a continuous medium is described by the classical electron radiusr e . It also determines the effective Thomson cross section for scattering of photons by free electrons. The lengthr e and the corresponding time interval τ e =r e /c play the role of “hidden parameters” in quantum mechanics. Two methods of calculating the effective Thomson cross section in terms of the extinction coefficient are considered. The first of them is based on the equation of motion of a free electron in a field with allowance for radiative friction. This equation leads to well-known difficulties. Moreover, the velocity fluctuations calculated on its basis lead to a contradiction with the second law of thermodynamics. The second method is based on the introduction of a constant friction coefficient Ń = τ e −1 , the presence of which reflects loss of information on smoothing over the volume of a “point” of the continuous medium. Such a method of calculation leads to the same expression for the effective cross section but makes it possible to avoid the difficulties with the second law of thermodynamics. In the derivation of quantum kinetic equations, the physically infinitesimally small scales are determined by the Compton length λC and the corresponding time interval. The introduction of these scales makes it possible to separate and eliminate small-scale fluctuations, the “collision integrals” being expressed in terms of the correlation functions of these fluctuations.

A derivation of quantum kinetic equations for a gas of composite particles I

The formalism of non-equilibrium quantum statistical mechanics developed by R. Balescu provides a convenient framework for deriving kinetic equations. It requires the definition of concepts such as vacuum of correlation and free motion operator. We propose a realization of such concepts for bound states in the concrete case of quantum gases starting from a description in terms of electrons and nuclei. Polarized hydrogen and unpolarized helium gases are considered.

The derivation of the quantum kinetic equation and the two-time resolvent method

Van Hove's partial density matrix,ρ E (t), in his generalized master equation is interpreted as a Wigner representation of “two-time dyad” for “energy E” and “time t”. This interpretation enables us to integrate the “energy”E in Van Hove's master equation. The resultant equation is of non-Markov type on two time parameters. Starting with this master equation, the derivation of quantum kinetic equations, including the second-order approximation in the density expansion, is discussed. The scaling of the quantum kinetic equation is examined in detail for a system in which particles interact through the delta shell potential. It is shown that the quantum kinetic equation, including three-particle scattering, may exist for the physical situations of low-energy scattering,high-energy scattering, and for resonance scattering for time scales of the system sufficiently separated. In deriving the quantum kinetic equation, a factorization theorem form-particle distribution functions is proved to arbitrary order in perturbation expansion.