16.3 Viscosity and Non-equilibrium Alignment Phenomena 319
∂
∂tpμν+ 2 p 0 ∇μvν+νppμν +νpa√
2 p 0 aμν= 0 , (16.59)
∂
∂taμν−ωBHμν,μ′ν′aμ′ν′+(√
2 p 0)− 1
νappμν +νaaμν= 0.The fourth rank tensorHμν,μ′ν′, defined by (14.26), emerges from the computation of
the commutatorhλ[Jλ,Jμ′Jν′]−by analogy to (13.17), see also (13.19). The colli-
sion frequenciesν..can be expressed in terms of collision integrals which involve the
scattering amplitude in a binary way. The non-diagonal coefficients obey the Onsager
symmetry relationνap=νpaand the inequalitiesνa>0,νp>0,νaνp>ν^2 ap.
For a stationary situation and in the absence of the magnetic field, the equations
(16.59)imply
pμν =− 2 η∇μvν,η=ηiso( 1 −Apa)−^1 ,ηiso=p 0
νp, Apa=νpaνap
νpνa.
(16.60)
The viscosityηis larger thanηisowhich would be the value of the viscosity for an
absolutely isotropic state whereaμν=0.
For a stationary situation, with a magnetic field present, the solution of the coupled
equations (16.59) with the methods discussed in Sect.14.4, yields a viscosity tensor
of the form (16.47) with the coefficients given as functions ofφa=ωB/νaby
η(^0 )=η, η(m+)−η=−ηApa(mφa)^2
1 +(mφa)^2,η(m−)=−ηApamφa
1 +(mφa)^2,
(16.61)
form= 1 ,2. Clearly, here the coefficientη(^0 )is not affected by the magnetic field.
The even coefficientsη(^1 +)andη(^2 +)decrease with increasing field strength from
the zero field valueηto the valueηiso. The ratioApaof the relaxation frequencies
determines the magnitude of the relative change of the viscosity. The odd coefficients
η(^1 −)andη(^2 −)vanish both for weak and for very strong magnetic fields and they
have an extremum atmφa=1, i.e., where the precession frequencymωBis equal
to the collision frequencyνa=τa−^1. The relaxation timeτais of the order of the
average time between two collisions, which is the longer, the lower the pressure
p 0 is. Due toνa∼p 0 , one hasφa=ωB/νa∼B/p 0. Thus the smaller magnetic
moment of diamagnetic gases, compared with paramagnetic ones, is compensated
by smaller pressuresp 0 , while the coefficientsηandηisoare independent ofp 0 ,in
gases at moderate pressures, say between 10−^3 and 10 times the ambient pressure at
room temperature.
Some historical remarks with a personal touch: The occurrence of Hall-effect like
transverse termsη(^1 −)andη(^2 −)is surprising for a fluid without free electric charges.
In fact, after the publication of the first measurements with paramagnetic gases, Max
von Laue discussed the tensorial behavior of the viscosity, following the symmetry
argumentsofW.VoigtfortheshearmodulusandheclaimedthatHall-effectliketerms
should not exist in this case. The transverse effects for transport in molecular gases
were first treated theoretically in 1964 in the diploma thesis of the present author, in