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arXiv:1106.5616

Difference Boltzmann Equation

A. A. Klyukanov

Published Jun 28, 2011• 0 citations
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Abstract

Difference Boltzmann Equation is derived in a plane wavelets representation with account of two-particle correlations. It is shown that the set of plane wavelet orthonormal functions is complete. The set of ket vectors is used as the second quantization basis allowing introducing the positively definite distribution function. It is obtained that inhomogeneous system is described by numbers of particles at quantized positions with quantized momenta. Difference Boltzmann Equation transforms into the classical Boltzmann Equation in the limit, where distribution function varies little in and

DOI
10.48550/arxiv.1106.5616

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Difference Boltzmann Equation is derived in a plane wavelets representation with account of two-particle correlations. It is shown that the set of plane wavelet orthonormal functions is complete. The set of ket vectors is used as the second quantization basis allowing introducing the positively definite distribution function. It is obtained that inhomogeneous system is described by numbers of particles at quantized positions with quantized momenta. Difference Boltzmann Equation transforms into the classical Boltzmann Equation in the limit, where distribution function varies little in and
Difference Kinetic Equations are derived quantum mechanically in a plane wavelets representation with account of two-particle correlations. It is shown that the set of plane wavelet orthonormal functions is complete. The set of ket vectors is used as the second quantization basis allowing introducing the positively definite distribution function. It is obtained that inhomogeneous system is described by numbers of particles at quantized positions with quantized momenta. Difference Kinetic Equation for distribution function transforms into the classical Boltzmann Equation in the limit, where expectation value of particles number varies little.
Here we continue our investigation aimed at the derivation of the Boltzmann equation from the first principles of quantum mechanics. The key difficulty that we overcome in this manuscript is the incompatibility between traditional methods employed in the derivation of classical mechanics from quantum mechanics and methods used to describe transitions between quantum states due to interaction. Here we use a novel technique that is similar to the wavelet transform method that originally appeared in the context of signal processing. With this method, we can simultaneously work with slowly varying potentials that give rise to classical-like wave-function behaviour, motion along a prescribed trajectory, and sharp potentials that lead to behaviour that is best understood in terms of quantum transitions. Working with smooth potentials in this framework, we derive the Liouville equation from the Schrödinger equation. When using this method for sharp potentials, we naturally obtain the description of system dynamics in terms of a transition-rate matrix. The transition rates between system states, which characterize the process, are determined by Fermi's golden rule. We observe that, through the Liouville equation, we can deduce the non-collision part of the Boltzmann equation, and that, through the transition-rate matrix, we can deduce the collision integral. We combine the results to derive the Boltzmann equation, the main formula of physical kinetics.
Equations for correlation functions in nonequilibrium systems are derived using a generalization of the diagrammatic Keldysh method. The four field operators of the correlation function are ordered on a time contour that consists of four axes extending from -\ensuremath{\infty} to +\ensuremath{\infty} and from +\ensuremath{\infty} to -\ensuremath{\infty} (doubled Keldysh contour). For those correlation functions that may be considered as correlation functions of one-particle Green-functions' fluctuations, a closed system of equations is obtained in the case of weak interaction of the particles with the scattering system. Its structure may be related to the Langevin approach: on the left-hand side a linear operator acts on each of two pairs of the correlation-function arguments, each of these two operators being just the operator of the linearized quantum kinetic equation. The expression for the right-hand side of this equation is derived. It has the meaning of the correlation function of extraneous Langevin sources, and the equation may be called the quantum Boltzmann-Langevin equation. In the quasi-classical limit it reduces to the usual Boltzmann-Langevin equation for the particles' distribution-function fluctuations. Possible applications of the method are discussed.
A derivation of the quantum-mechanical analog of the Maxwell-Boltzmann equation of transport for gases at low density is presented. The analysis requires: the definition of a time differential which is necessary for the selection of only secular variations in the time dependence of the distribution functions (coarse-graining in time); a binary collision approximation; the connection between the phase-space transformation function, relating the Wigner distribution function at different times, and the transition matrix in the theory of scattering of two particles; and a coarse graining in configuration space leading to a kinetic equation for spatially nonuniform systems. A consequence of the binary collision approximation is the exclusion of the effects of symmetrization (Bose-Einstein or Fermi-Dirac statistics) resulting in density-dependent terms.
The inversion method to derive the quantum kinetic equation is extended to the case of an inhomogeneous system. Within the closed-time-path formalism, the source is introduced to probe the (Fourier transform of the) Wigner distribution function and the equation of motion for the Wigner function is derived perturbatively by the inversion formulae. The equation of motion has the form of the generalized Boltzmann equation, which is non-local in space and time. For the obtained kinetic equation, the local conservation laws of number, energy and momentum are confirmed.
Nothing can be made out of nothing. William Shakespeare , King Lear Historically, quantization of the motion of particles was developed first. The state was described by a wave function and observables by operators. When dealing with interactions between particles and fields, such as the electromagnetic field, the fields were treated classically. Classical field equations look like the quantum mechanical equations for the wave function of the field quanta. For example, the Klein–Gordon classical field equation is similar to the quantum mechanical wave equation for a relativistic spinless particle. Quantizing the fields, leading to quantum field theory, appears to be quantizing a theory that has already been quantized; hence the name “second quantization.” In reality, there is only one quantization and one quantum theory. The method of second quantization is important in the study of many-particle systems. It enables us to express many-body operators in terms of creation and annihilation operators, thus rendering calculations less cumbersome. Moreover, the method makes it possible to treat systems with a variable number of particles; that is why the method initially emerged in the context of quantum field theory. In Chapter 1 we indicated that any one-particle wave function may be expanded in a complete set of states. In this chapter, we show that products of single-particle states, when properly symmetrized, form an orthonormal basis for the expansion of the wave function of an N -particle system.