Geology Reference
In-Depth Information
The alternative and more conservative approach is to retain the concept of local
state in giving the physical specification of a system at any instant. The postulate
that a local state exists is often taken as being equivalent to assuming some form of
local equilibrium (Glansdorff and Prigogine
1971
, p. 14) but it can have a wider
meaning (Lavenda
1978
, p. 77). Attempts to justify this postulate usually point to
the relaxation time for fluctuations at the atomic scale being short compared with
the timescale of the macroscopic processes to which the theory is applied. Where
the local state cannot be fully specified in terms of measurable macroscopic
variables it is assumed that there exist additional internal or hidden variables (for
example, dislocation density) which complete the description of the local state.
The importance of the postulate of a local state is that it enables many concepts to
be carried over from classical equilibrium thermodynamics, such as the concepts
of entropy and energy as scalar potentials and the Gibbs-Duhem and the Gibbs
relations (
2.3a
) and (
2.7
). The production of entropy envisaged by the Second Law
for irreversible processes is then discussed with a view to placing constraints on
the laws governing these processes, especially in relation to their stability. The
applications have commonly been confined to processes in systems not very far
from equilibrium, the theory for which is termed the linear thermodynamics of
irreversible processes. The theoretical situation for more general applications is
less well developed, and still to a considerable extent the subject of research. Thus
there have been attempts to develop the theory of the nonlinear thermodynamics of
irreversible processes applicable to systems far from equilibrium; see for example,
Glansdorff and Prigogine (
1971
), who introduce the concept of dissipative struc-
tures; also Lavenda (
1978
). Other applications have been to processes such as
friction or ideal plasticity where dissipation is equally important no matter how
slowly the process proceeds, for example, Kestin (
1966
) and Nemat-Nasser
(
1974
). The remainder of these notes will concern linear thermodynamic theory
under the assumption of the existence of a local state, as expounded by Denbigh
(
1951
), Meixner and Reik (
1959
), de Groot and Mazur (
1962
), Katchalsky and
Curran (
1965
), Prigogine (
1967
), Fisher and Lasaga (
1981
), Kuiken (
1994
),
Martyushev and Seleznev (
2006
), Holyst (
2009
), Kleidon (
2009
), and others.
The starting point for the linear thermodynamics of irreversible processes is the
Second Law and the concept of entropy production in an irreversible process. In
any irreversible change in a system, the rate of change in entropy is made up of a
part due to entropy flow from the surroundings and a part due to changes within the
system. The latter part is known as the rate of entropy production, or simply the
entropy production, and designated r per unit volume; according to the Second
Law it must be positive. In the energy representation of the evolution of the
system, the corresponding quantity is Tr
;
which is sometimes called a dissipation
function or potential since it represents the rate at which irrecoverable energy or
work must be supplied or done to maintain the process. The dissipation function
can be written in the form
Tr
X
a
J
a
X
a
[ 0
a
¼
1
;
2
;
3...
ð
2
:
7
Þ
