Physica I08B (1981) 1205.1206 SE 3 North-Holland Publishing Company DYNAMIC FLUCTUATIONS IN NORMAL LIQUID 3He Oriol T. Valls School of Physics and Astronomy University of Minnesota Minneapolis, Minnesota 55455 Harvey Gould Department of Physics Clark University Worcester, Massachusetts 01610 Gene F. Mazenko James Franck Institute and the Department of Physics University of Chicago Chicago, Illinois 60637 A kinetic theory analysis of normal liquid 3He is presented. Its relationship to other theories is discussed, and the importance of nonlocal effects is emphasized. 1. INTRODUCTION Recent measurements 1,2 of inelastic neutron scattering from liquid 3He have stimulated re- newed interest in the theoretical description of the elementary excitations in normal Fermi liquids. Landau theory 3 has provided a power- ful description of the elementary excitations in the limit of small wavevector k and low fre- quency m. However, Landau theory is not ap- plicable for the values of k and ~ associated with the collective zero sound mode excited by coherent neutron scattering. We discuss here a quantum kinetic theory4, ~ approach to the elementary excitations at intermediate k and w, the relation of this approach to the polariza- tional potential approach of Pines and collab- orators% 7, and the results of our kinetic theory for the low temperature shear viscosity of low temperature liquid '3He. 2. FORMAL BACKGROUND As we have shown elsewhere 4,5, the Kubo func- tion ~[(k,pp',z) is the quantity which satisfies a well-behaved kinetic equation over a wide range of k,~, and temperature T. The dynamic structure function S(k,~) observed in neutron scattering is related to ~ by nS(k,m) =fd3pd3p '8~cothSm/2 Im~(k,pp' ,~+i0 +) , (i) l+e-8~ where 8 = (kDT)-l and n is the density. The kinetic equation for ~ has the general form (z-~-~) 27(k ,pp', z) - fd 3p0 (k, pp, z)a~(k,pp', z) = ~(k,pp'), (2) where ~ (k,pp ')=X(k,pp' ,t=O)=-8-1x(k,pp ' , z=O. The generalized susceptibility ×(k,pp',z) is the usual commutator function between tVigner operators. The non-Iocal memory function 0(k,pp',z) separates into a sum of two terms, 0(k,pp' ,z)=0(s) (k,pp')+0 (c) (k,pp' ,z). The static term 0 (s) represents instantaneous mean field effects and is related to ×(k,pp',0) by k~P x (k, pp', 0) +ld 3p0 (s) (k, pp) X(k, pp', 0) = -6(p-p')[f(p-k/2)-f(p+k/2)], (3) where f(p) is the momentum distribution func- tion. The frequency-dependent term 0(c) des- cribes the effects of collisions. Microscopic expressions for 0 (s) and 0(c) are given in Ref. 4. 3. PHENOMENOLOGICAL THEORIES Because of the difficulties associated with a microscopic analysis of @ for quantum liquids, we develop a phenomenological model of @ which retains non-local effects. The model is based on the correspondence of the formal theory to the analogous small k and ~ results of the Landau Theory. We know that in the limit k,~+0, low temperature normal liquid 3He can be represented as a dilute gas of quasiparticles with effective mass m* different from the mass m of an individual 3He atom. Hence we assume that f(p) is the usual Fermi-Dirac distribution function for particles of mass m*. In order to make contact with Landau Theory, we write ¢(s) as ¢(s) ck,pp,)=~, (p-p')+~, *(k,pp'), (4) where 1/m' = 1/m*-l/m. If we substitute (4) into (21, ignore 0 (c), and let k~0, we obtain the usual Landau kinetic equation with 0*(k,pp9 identified with the Landau quasiparticle inter- action energy 3 fpp,. Based in part on this identification, we expand 0" for k#0 in terms of a complete set of func- tions {~o } and retain only the first two terms. We write for the spin-symmetric part of 0": 0378-4363/81/0000-0000/$0250 © North-HollandPublishingCompany 1205