Tensors for Physics

302 16 Constitutive Relations

16.1.2 Energy Principle

Consider a constitutive relation (16.1) where=kapplies and where the scalar
product

bμ 1 ...μaμ 1 ...μ

is proportional to a contribution to the energy, which has to be positive. From

bμ 1 ...μaμ 1 ...μ=aμ 1 ...μCμ 1 ...μν 1 ...νaν 1 ...ν> 0 , (16.4)

follows: the part of the tensorCwhich is symmetric under the interchange of the
front and back set ofindices, viz.

Csymμ 1 ...μν 1 ...ν=

1

2

(Cμ 1 ...μν 1 ...ν+Cν 1 ...νμ 1 ...μ)

is positive definite. Notice, the part of the tensorCwhich is antisymmetric under the
interchange is not necessarily zero.
As an example the dielectric tensorεμνof a linear medium is considered, where
the relation

Dμ=ε 0 εμνEν

applies. In this case, the energy density isuel=^12 DμEμ,cf.(8.118), and conse-
quently

uel=

1

2

EμεμνEν=

1

4

(εμν+ενμ)EμEν≥ 0.

In general, in particular in the presence of an external or an internal magnetic field,
the dielectric tensor possess an antisymmetric partεμνasym =^12 (εμν−ενμ).The
conditionuel>0 poses a condition on the symmetric part of the dielectric tensor,
viz.εμνsym=^12 (εμν+ενμ)has to be positive definite.

16.1.3 Irreversible Thermodynamics, Onsager Symmetry

Principle

The density of the entropy production caused by irreversible processes is given by
expressions of the type [108]

(

δs

δt

)

irrev

=−

(

Jμ(^11 )...μ

1

Fμ(^11 )...μ

1

+Jμ(^21 )...μ

2

Fμ(^21 )...μ

2

)

, (16.5)