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Journal de physique

La transformée de Wigner libre, évolution dans une collision

Franck Laloë

Published Jan 1, 1989• 0 citations
Publisher

Abstract

This is the first of a series of articles where we develop a theory for the description of the dynamics of dilute quantum gases, at or out of equilibrium, with an accurate treatment of the correlation effects introduced by binary collisions (these correlations are not taken into account when the Boltzmann equation is used). For this purpose, we propose the introduction of a variant of the Wigner transform, the « free Wigner transform », which wipes out the effects of the potential at short distances. If initially factorized, the entering part of the free transform remains exactly so during all collision, so that it is better adapted to a molecular chaos assumption than the ordinary transform. Once the free transform is known, the effects of the short distance correlations can be reconstructed from it. Therefore, the problem is to find an appropriate equation of evolution which gives its time variations. With this aim in mind, we study in the present article the evolution of the free Wigner transform in a binary collision. We show that its evolution equation includes coefficients which can be exactly obtained from the « on shell » terms of the T matrix (S matrix, phase shifts), even while the particles are interacting. In addition to the usual terms which involve the scattering cross sections, as usual with the Boltzmann equation, the calculation provides other terms which correspond to collision retardation effects and « refraction » effects (refraction of the wave function of the test atom by density gradients of the collision partners). These terms will play an important role in the following article, and introduce the second virial corrections to the equilibrium and transport properties of the gas.

DOI
10.1051/jphys:0198900500140185100/pdf

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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")

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La transformée de Wigner libre, évolution dans une collision

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Le but général de la série d'articles dont celui-ci est le premier est de décrire la dynamique hors d'équilibre des gaz quantiques dilués en tenant compte de façon précise des corrélations binaires entre particules ; ces corrélations sont généralement ignorées dans les approches théoriques du type Boltzmann. Nous proposons pour cela l'utilisation d'une variante de la transformée de Wigner, la transformée de Wigner libre, qui « efface » les corrélations à courte distance créées par le potentiel d'interaction entre particules. Ainsi, chaque fois qu'avant collision la transformée libre était factorisée, une factorisation subsiste rigoureusement pendant toute la collision ; elle se prête donc mieux que la transformée habituelle à une hypothèse de « chaos moléculaire » à la Boltzmann. Une fois connue la transformée libre, les corrélations entre particules peuvent être reconstruites et leurs effets physiques calculés. L'étude de l'évolution de la transformée de Wigner libre dans une collision binaire conduit à des résultats qui s'expriment complètement en fonction des caractéristiques de la matrice T « sur la couche de masse » (matrice S, déphasages), même lorsque les particules sont en train d'interagir. En sus des termes habituels de collision où apparaissent comme dans l'équation de Boltzmann les sections efficaces de collision, le calcul donne des termes dus aux effets de retard, et aux effets « de réfraction » dans la collision. Ces termes joueront un rôle essentiel dans l'équation cinétique proposée dans l'article suivant, et dans les corrections du viriel aux propriétés d'équilibre et de transport du gaz.

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Kinetic equations for a nonideal quantum system

Th. Bornath, D. Kremp, W. D. Kraeft +1 more

In the framework of real-time Green's functions, the general kinetic equations are investigated in a first-order gradient expansion. Within this approximation, the problem of the reconstruction of the two-time correlation functions from the one-time Wigner function was solved. For the Wigner function, a cluster expansion is found in terms of a quasiparticle distribution function. In equilibrium, this expansion leads to the well-known generalized Beth-Uhlenbeck expression of the second virial coefficient. As a special case, the T-matrix approximation for the self-energy is investigated. The quantum kinetic equation derived thus has, besides the (Markovian) Boltzmann collision integral, additional terms due to the retardation expansion which reflect memory effects. Special interest is paid to the case that bound states exist in the system. It is shown that the bound state contribution, which can be introduced via a bilinear expansion of the two-particle T matrix, follows from the first-order retardation term in the general kinetic equation. The full Wigner function is now a sum of one function describing the unbound particles and another one for the bound state contribution. The latter two functions have to be determined from a coupled set of kinetic equations. In contrast to the quantum Boltzmann equation, energy and density of a nonideal system are conserved. \textcopyright{} 1996 The American Physical Society.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Oct 1, 1996•57 citations
This is the first of a series of articles where we develop a theory for the description of the dynamics of dilute quantum gases, at or out of equilibrium, with an accurate treatment of the correlation effects introduced by binary collisions (these correlations are not taken into account when the Boltzmann equation is used). For this purpose, we propose the introduction of a variant of the Wigner transform, the « free Wigner transform », which wipes out the effects of the potential at short distances. If initially factorized, the entering part of the free transform remains exactly so during all collision, so that it is better adapted to a molecular chaos assumption than the ordinary transform. Once the free transform is known, the effects of the short distance correlations can be reconstructed from it. Therefore, the problem is to find an appropriate equation of evolution which gives its time variations. With this aim in mind, we study in the present article the evolution of the free Wigner transform in a binary collision. We show that its evolution equation includes coefficients which can be exactly obtained from the « on shell » terms of the T matrix (S matrix, phase shifts), even while the particles are interacting. In addition to the usual terms which involve the scattering cross sections, as usual with the Boltzmann equation, the calculation provides other terms which correspond to collision retardation effects and « refraction » effects (refraction of the wave function of the test atom by density gradients of the collision partners). These terms will play an important role in the following article, and introduce the second virial corrections to the equilibrium and transport properties of the gas.
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")
Le but général de la série d'articles dont celui-ci est le premier est de décrire la dynamique hors d'équilibre des gaz quantiques dilués en tenant compte de façon précise des corrélations binaires entre particules ; ces corrélations sont généralement ignorées dans les approches théoriques du type Boltzmann. Nous proposons pour cela l'utilisation d'une variante de la transformée de Wigner, la transformée de Wigner libre, qui « efface » les corrélations à courte distance créées par le potentiel d'interaction entre particules. Ainsi, chaque fois qu'avant collision la transformée libre était factorisée, une factorisation subsiste rigoureusement pendant toute la collision ; elle se prête donc mieux que la transformée habituelle à une hypothèse de « chaos moléculaire » à la Boltzmann. Une fois connue la transformée libre, les corrélations entre particules peuvent être reconstruites et leurs effets physiques calculés. L'étude de l'évolution de la transformée de Wigner libre dans une collision binaire conduit à des résultats qui s'expriment complètement en fonction des caractéristiques de la matrice T « sur la couche de masse » (matrice S, déphasages), même lorsque les particules sont en train d'interagir. En sus des termes habituels de collision où apparaissent comme dans l'équation de Boltzmann les sections efficaces de collision, le calcul donne des termes dus aux effets de retard, et aux effets « de réfraction » dans la collision. Ces termes joueront un rôle essentiel dans l'équation cinétique proposée dans l'article suivant, et dans les corrections du viriel aux propriétés d'équilibre et de transport du gaz.
Abstract We apply the formalism of the free Wigner transform to the derivation of a kinetic equation valid for dilute quantum gases; this formalism is compatible with a precise treatment of two‐body correlations and allows the inclusion of second virial corrections. We generalize the results obtained in previous articles where the calculation of the collision integral was limited to zero (“local”) and first order gradient terms; here we use a more general (“non‐local”) form for the interaction term which is an integral of higher dimension (18). The conservation laws are studied in this formalism, which implies non‐linear expressions for the local hydrodynamic quantities in terms of the function appearing in the kinetic equation. The pressure at thermal equilibrium is then evaluated and found to be equal to the value predicted by the Beth‐Uhlenbeck formula, without adjusting any free parameter of the theory.
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.
We generalize our previous work on the compatibilty of kinetic equations with second virial corrections to the inclusion of spin and particle indistinguishability; the system is supposed to be sufficiently dilute for higher order virial density corrections (interactions and statistics) to be negligible. We show that the general idea of the “free Winger transform” can be extended to this situation; the function which appears in the kinetic equation becomes here a matrix which acts in the space of spin states of the particles. Assuming that the collisions are described by a hamiltonian which does not act on the spins (a very good approximation for nuclear spins), we write explicitly a kinetic equation which is valid for this case. The right hand side of the equation is an 18 dimension integral, as for spinless distinguishable particles, but here it contains an additional term due to statistics, which introduces commutators and anticommutators. We discuss the local conservation laws in this formalism and find, as expected, a total number of 8 conserved quantities for spin 1/2 particles (including three components of the magnetization). When the gas is at equilibrium, we obtain a pressure dependence which is in agreement with known calculations on spin polarized gases. We finally study the gradient expansion of the collision integral, and show that the zero-order (local) part is identical with the 4 terms (including identical spin rotation terms) obtained previously by Lhuillier et al. The first order (non-local) part contains many terms, wich we compare with those obtained by Silin in a context more closely related to the Landau theory.
In the framework of real-time Green's functions, the general kinetic equations are investigated in a first-order gradient expansion. Within this approximation, the problem of the reconstruction of the two-time correlation functions from the one-time Wigner function was solved. For the Wigner function, a cluster expansion is found in terms of a quasiparticle distribution function. In equilibrium, this expansion leads to the well-known generalized Beth-Uhlenbeck expression of the second virial coefficient. As a special case, the T-matrix approximation for the self-energy is investigated. The quantum kinetic equation derived thus has, besides the (Markovian) Boltzmann collision integral, additional terms due to the retardation expansion which reflect memory effects. Special interest is paid to the case that bound states exist in the system. It is shown that the bound state contribution, which can be introduced via a bilinear expansion of the two-particle T matrix, follows from the first-order retardation term in the general kinetic equation. The full Wigner function is now a sum of one function describing the unbound particles and another one for the bound state contribution. The latter two functions have to be determined from a coupled set of kinetic equations. In contrast to the quantum Boltzmann equation, energy and density of a nonideal system are conserved. \textcopyright{} 1996 The American Physical Society.