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.
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.
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.
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")
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.
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.
Close to equilibrium, a normal Bose or Fermi fluid can be
described by an exact kinetic equation whose kernel is nonlocal in space and time. The general expression derived for the kernel is evaluated to second order in the interparticle potential. The result is a wavevector- and frequency-dependent generalization of the linear
Uehling-Uhlenbeck kernel with the Born approximation cross section. The theory is formulated in terms of second-quantized phase space operators whose equilibrium averages are the n-particle Wigner distribution functions. Convenient expressions for the commutators and anticommutators of the phase space operators are obtained. The two-particle equilibrium distribution function is analyzed in terms of momentum-dependent quantum generalizations of the
classical pair distribution function h(k) and direct correlation function c(k). The kinetic equation is presented as the equation of motion of a two -particle correlation function, the phase space density-density anticommutator, and is derived by a formal closure of the quantum BBGKY hierarchy. An alternative derivation using a projection operator is also given. It is shown that the method used for approximating the kernel by a second order expansion preserves all the sum rules to the same order, and that the second-order kernel satisfies the appropriate positivity and symmetry conditions.
A dense interacting Bose or Fermi gas is considered. Within the Kadanoff-Baym real-time Green-function technique a generalized kinetic equation is derived in the nonequilibrium ladder approximation. As a Markovian limit the Boltzmann-Uehling-Uhlenbeck equation is recovered. The relevance of retardation effects in the kinetic equation of dense systems is shown. It turns out that the stationary solution of the generalized kinetic equation reproduces the second virial coefficient of the equation of state, including the Beth-Uhlenbeck quantum corrections. We establish a generalized kinetic equation describing correlation effects in a consistent approach to nonequilibrium for Bose and Fermi systems. In this way we give a generalization of the Beth-Uhlenbeck formula that applies to nonequilibrium and to finite systems. The derivation makes use of the time dependent T-matrix equation, which establishes a generalization of the Bethe-Goldstone equation. The correlated parts of the density and energy, which can be interpreted as virial corrections, are thermodynamically consistent. In this way global energy conservation is fulfilled.