16.1 General Principles 303
where the tensorsJ(..)andF(..)are referred to asthermodynamic fluxesandther-
modynamic forces, respectively. Examples for such fluxes are the heat flux and the
friction pressure tensor, the pertaining “forces” are the temperature gradient and
velocity gradient tensor. Also a non-equilibrium alignment and its time derivative
is such a force-flux pair, cf. Sect.16.4.5. Typically, the pertaining forces and fluxes
have opposite time reversal behavior, i.e.TJ=−TF. Here two force-flux pairs are
considered. The reduction to just one such pair is obvious. The generalization to
more than two pairs can be formulated along the lines presented here.
Inirreversible thermodynamics, the linear ansatz
−Jμ(^11 )...μ
1
=Cμ(^111 ...μ)
1 ν^1 ...ν 1
Fν( 11 ...ν)
1
+C(μ^121 ...μ)
1 ν^1 ...ν 2
Fν(^21 ...ν)
2
−Jμ(^21 )...μ
2
=Cμ(^211 ...μ)
2 ν^1 ...ν 1
Fν( 11 ...ν)
1
+C(μ^221 ...μ)
2 ν^1 ...ν 2
Fν(^21 ...ν)
2, (16.6)
is made for the constitutive relations. Positive entropy production requires
(
δs
δt)
irrev> 0 , (16.7)
and consequently, in symbolic notation,
F(^1 )·C(^11 )·F(^1 )+F(^2 )·C(^22 )·F(^2 )+F(^1 )·C(^12 )·F(^2 )+F(^2 )·C(^21 )·F(^1 )> 0 .(16.8)This imposes conditions on the coefficient tensors, in particular, the parts of the
diagonal tensorsC(^11 )andC(^22 ), which are symmetric under the exchange of the
front and back indices, have to be positive definite. Furthermore, the magnitude of
the non-diagonal tensorsC(^12 )andC(^21 )is bounded by the diagonal ones.
Now it is assumed that 1 = 2 =and that both forcesF(^1 )andF(^2 )have the
same behavior under time reversal, in obvious notationTF1=TF2, and both fluxes
J(^1 )andJ(^2 )have the opposite behavior, vizTJ1=−TF1,TJ2=−TF2. Then the
Onsager symmetry relation, also calledreciprocal relations[109]
C(^12 )=C(^21 ) (16.9)holds true. This symmetry relation for coefficients governing irreversible macro-
scopic behavior is based on the time reversal invariance of the underlying microscopic
dynamics, For 1 = 2 still a symmetry relation like (16.9) applies, but the back and
front indices have to be transposed on one side. Examples for the applications of
Onsager symmetry relation are e.g. given in Sects.16.3.6and16.4.5.
When the two forces have opposite time reversal behavior, i.e. when one has
TF1=−TF2,theOnsager-Casimir symmetry relation
C(^12 )=−C(^21 ) (16.10)applies, instead of (16.9).