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latter case. The applications of the linear thermodynamics of irreversible processes
now follow from consideration of the quantities in the terms of Tr and of the
inequality expressed in (
2.7
).
The physics of the processes is expressed in the relationships between the forces
and fluxes, X
a
;
J
a
;
known as the phenomenological, constitutive, or kinetic rela-
tions or as the thermodynamic equations of motion; their role is in many respects
analogous to that of the equations of state relating extensive and intensive quan-
tities in equilibrium thermodynamics. One of the main activities of thermodynamic
theory has been to place constraints on the relations between the thermodynamic
variables and to discuss criteria for stability and stationary states. However,
whereas the constraints governing the quantities entering the equilibrium equations
of state can generally be stated independently of the particular nature of these
equations, it is very difficult to establish laws of general validity governing the
quantities entering the phenomenological equations for an arbitrary non-equilib-
rium situation. Substantial progress has only been made for certain classes of
situations, chiefly for those close to equilibrium. We therefore restrict consider-
ations to the latter and in particular to the situations in which the relationships
between forces and fluxes can be written in the linear form
J
a
¼
X
b
L
ab
X
b
b
¼
1
;
2
;
3...
ð
2
:
9
Þ
where L
ab
are constants, often called the phenomenological or kinetic coefficients. In
writing the phenomenological relations in the form (
2.9
), the quantities X
b
;
J
a
are
treated as scalars but in practice they can represent scalar quantities or the Cartesian
components of vector or tensor quantities. The relations (
2.9
) take into account the
possibility of coupling effects between non-conjugate forces and fluxes, for example,
coupling between heat flow and diffusion or between the diffusion of different spe-
cies. It is now possible, using (
2.9
), to write (
2.7
) in the form
Tr
¼
X
a
;
b
L
ab
J
a
J
b
0
ð
2
:
10
Þ
which has important consequences.
The principal initial success of the linear thermodynamics of irreversible pro-
cesses lies in the enunciation of the Onsager (
1931a
) reciprocal relations
L
ab
¼
L
ba
ð
2
:
11
Þ
which express a symmetry between coupling effects. Note that some elaboration of
(
2.11
) is needed when magnetic fields or rotational effects are present or the forces or
fluxes are mixed in respect of being odd or even in tensorial character; see Casimir
(
1945
) and Meixner and Reik (
1959
). Much has been written concerning the statistical
mechanical derivation of (
2.11
) using a fundamental principle of microscopic
reversibility; for critical discussion and references, see Lavenda (
1978
,
Chap. 2
): also
Callen (
1960
). However, (
2.11
) can also be treated as an empirical axiom of
