J. Phys. I France

**1**(1991) 181-209

## A kinetic equation for quantum gases (spin and statistics)

**P.J. Nacher, G. Tastevin and F. Laloë**

Département de Physique de l'ENS, 24 rue Lhomond, F 75005 Paris, France

(Received 2 October 1990, accepted 31 October 1990)

** Abstract **

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.

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*Les Editions de Physique 1991*