A quantum kinetic equation is obtained which includes scattering events beyond the binary collision process of the Boltzmann equation. The time rate of change of the Wigner function due to collisions is given as the difference of two scattering rates analogous to the collisional side of the Boltzmann equation. These total rates are given in terms of partial rates which are identified with probability amplitudes for scattering events in the medium. Thus, the method is inherently free of the divergence difficulties of theories which calculate the total rates in terms of scattering events in vacuum. Special attention is devoted to the question in what sense does some form of Fermi's golden rule apply to the calculation of the partial rates for a given event. It is shown that certain modifications to this picture are necessary, but the changes may be understood from a simple physical point of view.
A quantum kinetic equation is obtained which includes scattering events beyond the binary collision process of the Boltzmann equation. The time rate of change of the Wigner function due to collisions is given as the difference of two scattering rates analogous to the collisional side of the Boltzmann equation. These total rates are given in terms of partial rates which are identified with probability amplitudes for scattering events in the medium. Thus, the method is inherently free of the divergence difficulties of theories which calculate the total rates in terms of scattering events in vacuum. Special attention is devoted to the question in what sense does some form of Fermi's golden rule apply to the calculation of the partial rates for a given event. It is shown that certain modifications to this picture are necessary, but the changes may be understood from a simple physical point of view.
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 relationship between the collision kernel of the three-body quantum mechanical kinetic equation (quantum version of the Choh-Uhlenbeck equation) and modern scattering theory is examined utilizing the work of Faddeev and Weinberg. Recognizing that the collision kernel is related to the M⊘ller scattering operator it is shown that the collision kernel is “well-defined” for suitable interaction potentials and contains only connected three particle collisions in the sense of Weinberg. The collision kernel is interpreted physically and is shown to contain a generalization of the ordinary Boltzmann collision term to three particle processes plus a distorted-wave-impulse approximation correction to the two particle Boltzmann collision term.
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.
A bstract We study the kinetic theory of a weakly interacting quantum field. Assuming a state that is close to homogeneous and stationary, we derive a closed kinetic equation for the rate of change of the occupation numbers, perturbatively in the coupling. For a dilute gas, this reproduces the quantum Boltzmann equation, which only accounts for two-to-two scattering processes. Our expression goes beyond this, with terms accounting for multi-particle scattering processes, which are higher order in the density.
We calculate the scattering rate of a quantum particle, based on Fermi’s golden rule for a quantum particle as a distribution of matter, where the density of states is a function of the particle Lagrangian, not of the Hamiltonian, as in the conventional theory based on the Schrödinger equation. Thus, the density of states does not refer to the energies as in the conventional form, but to the Lagrangian functions, which, besides the energies, include the momentum-velocity products, that are also constants of motion for the initial and the final states. We obtain the relativistic dependence on the momentum, and the two scattering rates for the two possible cases, with the spin conservation, or the spin inversion.
This article is the second of a series where we propose a theory of the dynamics of quantum gases with a precise treatment of the short range two body correlations. For this purpose, we use a variant of the Wigner transform, the "free transform" , which has a useful property in this context: if, before collision, two particles are uncorrelated, the incoming part of the free Wigner transform remains always exactly factorized. Therefore, instead of asssuming a factorization of the two particle distribution, it is less restrictive to make a similar assumption on the free transform. Time symmetry is broken when the correlations of the outgoing part of the free transform are ignored; this approximation, which is well in the spirit of Boltzmann, amounts to considering successive collisions as independent processes (the correlations present in the outgoing part could play a role if the same particles rapidly collided again, an unlikely event in dilute gases). Therefore, the succession of collisions is not treated exactly, but each collision is, even during interaction. One then obtains a closed set of equations which leads directly to a kinetic equation for the free distribution $ f $. The equation includes retardation and quantum refraction effects, and provides a generalization of the Boltzmann equation. The reconstruction of the two particle distribution function $ f_{II} $ from $ f $ reintroduces the effects of short range correlations and, more generally, gives the values of all one particle and two particle physical observables. The price to pay for not using the usual one particle distribution is that the relation between the local physical quantities (density of particles, energy density, etc.) becomes more complicated than in the usual theory, and includes terms which are non linear in $ f $. We nevertheless show how this formalism automatically satisfies the local conservation of all hydrodynamic quantities: particle, momentum and energy densities. The pressure tensor includes quadratic terms which correspond to second virial corrections. The temperature of the gas is defined from the kinetic energy of the particles when they are far apart from each other. We finally briefly discuss the connections of this formalism with the classical Enskog theory, or mean field type theories such as the Landau theory (in particular, we note the presence of recoil effects in the "molecular field")