On the mathematical structure of thermodynamics
T. Matolcsi
Citation: J. Math. Phys. 41, 2021 (2000); doi: 10.1063/1.533224
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JOURNAL OF MATHEMATICAL PHYSICS
VOLUME 41, NUMBER 4
APRIL 2000
On the mathematical structure of thermodynamics
T. Matolcsia)
Department of Applied Analysis, Eötvös Loránd University,
1088 Múzeum krt 6-8, Hungary
共Received 1 July 1997; accepted for publication 19 November 1999兲
A mathematically exact dynamical theory of classical thermodynamics of homogeneous bodies is presented in which processes are time-dependent functions, governed by an ordinary differential equation. The fundamental objects of the mathematical structure of a thermodynamical system are the dynamical law, the
thermodynamical force, and the constraints; all the other usual notions, too, such as
substances, bodies, linear approximation by Onsager, etc. have got a mathematical
definition. Equilibria are the constant processes; their stability is investigated by
Lyapunov’s method. © 2000 American Institute of Physics.
关S0022-2488共00兲01304-9兴
I. INTRODUCTION
Classical mechanics is based on the Newtonian equation and constraints that define the processes unambiguously; then classical mechanics becomes an elegant mathematical theory by the
use of contact or symplectic manifolds. Quantum mechanics is based on the Schrödinger equation
that defines the processes unambigously; Hilbert spaces or C * algebras offer a complete mathematical formulation of quantum mechanics. Classical electrodynamics is based on the Maxwell
equations that define processes unambiguously; differential forms on manifolds admit a nice
mathematical formulation of classical electrodynamics. Continuum 共irreversible兲 thermodynamics
is based on the balance equations, partial differential equations that define processes unambiguously by boundary conditions and initial values.1–5
All these theories are mathematically well defined and have a clear mathematical structure; the
physical notions have an exact mathematical definition.
Classical 共equilibrium兲 thermodynamics is a theory to which—at present—no clear mathematical structure is assigned and many physical notions are intuitive ones without a mathematical
definition, e.g., equilibrium, processes, reversible, irreversible, quasistatic, trend to equilibrium. It
is well known how intuitive notions can mislead us 共Richard paradox6兲, which can be demonstrated by an excellent example taken from thermodynamics, too. In usual treatments of thermodynamics one ‘‘proves’’ that the Kelvin–Planck formulation and the Clausius formulation of the
second law are equivalent;7 a rigorous mathematical examination shows, however, that the
Kelvin–Planck formulation follows at once if the heating has an integrating factor and thus does
not necessarily presume or imply thermodynamic axioms of any kind.8
Several attempts have been made for a mathematically correct theory of thermodynamics,
starting from different points of view.9–13 Though some relations have been clarified and obtained
an elegant form, the whole theory cannot be treated in those ways satisfactorily. The reason is the
following. The physical theories enumerated above—in particular, continuum thermodynamics—
are dynamical theories: they describe what will happen under given circumstances. On the contrary, in spite of its name, classical thermodynamics—either in its usual treatments or in the
mentioned mathematical approaches—does not involve dynamics.
In order to have a satisfactory formulation and to find a convenient mathematical structure of
thermodynamics, we must establish a dynamical theory in which processes are described by a
differential equation. We know that the Onsager formalism describes nonequilibrium processes
a兲
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© 2000 American Institute of Physics
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J. Math. Phys., Vol. 41, No. 4, April 2000
T. Matolcsi
near equilibria and the relation between forces and fluxes outlines a strong mathematical structure.
However, the Onsager formalism is only a linear approximation, and the approximation procedure
is not well clarified from a mathematical point of view.
Recently a nonlinear theory has been proposed,14–19 called ordinary thermodynamics, in
which the fundamental notion is the process governed by an ordinary differential equation called
the dynamical law; thus processes, solutions of the dynamical equation, are functions in time.
Equilibria are constant processes. The well-defined linear approximation of the dynamical law
gives the usual Onsager formalism in special cases.
To clarify the physical meaning of ordinary thermodynamics, we make the following comments; further details can be found in Ref. 14.
Let us consider a continuous medium consisting of identical, spinless, chargeless particles. A
process of such a medium is the field (u,e, v ), the velocity, the specific internal energy, and the
specific volume as functions defined in space–time. If the body force and the body heating are
taken to be zero, then the balances of momentum, energy, and mass yield the partial differential
equations,
D u u⫽⫺ v “"P,
D u e⫽⫺ v共 “"k⫹P:“u兲 ,
D u v ⫽ v “"u,
where D u denotes the the ‘‘substantial time derivative’’ with respect to the velocity field u, k and
P are the heating flux and the pressure tensor, respectively, given by constitutive relations as
functionals of (u,e, v ).
The process (u,e, v ) can be determined, at least in theory, from initial and boundary values by
these balance equations that form a complete dynamical law.
In ordinary thermodynamics we consider the bodies as homogeneous i.e., all quantities depend only on time, not on space. Let us insert the conditions “u⫽0,“e⫽0,“ v ⫽0,“k⫽0,“P
⫽0 into the equations of continuum thermodynamics; we find that the quantities do not depend on
time either, that is, nothing happens. There is no nonconstant homogeneous process. Ordinary
thermodynamics cannot be obtained from continuum thermodynamics as a special case. Perhaps
one could even say then that the theory of homogeneous bodies is meaningless, because it is an
experimental fact as well that bodies out of equilibrium are never homogeneous; for example, the
temperature of a cooling body is always lower on the surface than in the interior of the body.
However, we know as well that a rigid body does not exist: all bodies are deformed under forces;
still, certain bodies in certain circumstances can be considered as rigid. The rigid body model is
simple, much simpler than the model of deformable bodies, and it is suitable for many purposes.
Similarly, ordinary thermodynamics offers simpler models than continuum thermodynamics and
they are applicable for a large class of phenomena. The theory of ordinary thermodynamics gives
a good approximation when the inner motion of the bodies is insignificant, and it has the advantage that we can use ordinary differential equations that are much simpler than partial differential
equations. Of course, the homogeneous model is rougher than the continuum model; however, it
derives results in several questions where the continuum theory seems useless because of its
complexity.
A similar point of view is accepted in the theory of chemical reactions that are mostly
described by ordinary differential equations20,21 as if the materials in chemical reactions were
homogeneous though, evidently, they are far from being homogeneous. Nevertheless, a lot of
basic features of chemical reactions are well reflected in such a description. Some other properties,
of course, can be deduced only from a continuum theory.22 Then comparing the results we can see
clearly where the inhomogeneity plays a fundamental role. In reaction kinetics the differential
equations concern only the concentrations, that is, only the concentrations are considered as
dynamic variables; thermodynamical properties of reactions are taken into account in another way.
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J. Math. Phys., Vol. 41, No. 4, April 2000
On the mathematical structure of thermodynamics
2023
The theory of chemical reactions in ordinary thermodynamics involves the dynamical description
of all thermodynamical quantities, so it is an extension of usual reaction kinetics.
Ordinary thermodynamics was formulated and applied in Refs. 14–18 to particular systems of
homogeneous bodies 共one body in an environment, interacting bodies, phase transitions兲. Now a
general mathematical framework will be given and investigated for systems consisting of singlecomponent bodies. Sections II and III are devoted to the basic notions; Definition 5 includes all the
special systems considered in the earlier papers and can be applied for diffusion processes as well
that have not yet been treated. The thorough examination of thermodynamic forces leads us to the
conclusion that we have to make a clear distinction between nominal forces and effective forces,
and suggests how we have to formalize constraints, which is one of the main results of the present
paper. In Sec. IV, we present the abstract mathematical structure of ordinary thermodynamics,
which can be summarized briefly as follows. There are given an open subset of a vector space 共the
set of states兲; a covector field on the set of states 共the nominal thermodynamical force兲; a vector
field on the set of states 共the dynamical quantities兲, satisfying some conditions, the most important
of which is the dissipative property; and a subspace field on the set of states 共the constraint兲.
The effective thermodynamical force is the restriction of the nominal force onto the constraint
subspaces; the dynamical quantities determine the dynamical law, a differential equation whose
solutions are the processes. The fundamental properties of this structure are demonstrated in Sec.
IV. In Sec. V general theorems on the stability of equilibria are proved.
II. SUBSTANCES, PHASES, BODIES
Let us recapitulate the most important notions and results indispensable for the mathematical
treatment.
To have a mathematically exact and unambiguous formulation, we shall take into account the
‘‘physical dimension’’ of the quantities that will be measured in SI units. For instance, the values
of energy are real multiples of J⫽Joule, i.e., they are elements of
共 J 兲 ª 兵 ␣ J 兩 ␣ 苸R其 .
Similarly, we shall use the notations (m 3 ) ⫹ ,(K) ⫹ for the 共positive兲 values of volume, temperature,
etc. An exact mathematical meaning can be given23 to the product and quotient of units of
measurements 共e.g., to J/K).
For the sake of perspicuity, here we shall consider the mathematical description of singlecomponent materials. A generalization to multicomponent materials is straightforward from a
conceptual point of view 共but its composition is more complicated兲.
The attribute ‘‘specific’’ will mean ‘‘per particle’’ 共molecule兲.
Definition 1: A single-component substance is a quintet 共D,T,P,␮,R兲, where
共i兲 D, called the constitutive domain, is a nonvoid subset of (J) ⫹ ⫻(m 3 ) ⫹ ; the first and second
variables in D 共usually denoted by e and v , respectively兲, are the specific internal energy and the
specific volume, respectively; the elements of D are called states of the substance;
共ii兲 T:D→共K兲⫹ , the temperature, P:D→共Pa兲, the pressure, ␮:D→共J兲, the chemical potential, the constitutive functions, are continuous.
共iii兲 R, the regular constitutive domain, the subset of D on which the constitutive functions are
continuously differentiable and
⳵T
⬎0,
⳵e
⳵P ⳵T ⳵P ⳵T
⬍0,
⫺
⳵v ⳵e ⳵e ⳵v
共1兲
holds is an open set dense in D.
Definition 2: Let 共D,T,P,␮,R兲 be a single-component substance. Then
s:D→ 共 J/K 兲 ,
共 e, v 兲 哫
e⫹P共 e, v 兲v ⫺ ␮共 e, v 兲
T共 e, v 兲
共2兲
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J. Math. Phys., Vol. 41, No. 4, April 2000
T. Matolcsi
is the specific entropy. The substance is called entropic if Ds⫽(1/T,P/T) on the regular domain.
Here and in the following D denotes the derivative of functions.
Note that if the substance is entropic then the specific entropy is twice continuously differentiable on the regular domain and its second derivative is negative-definite.
Definition 3: A phase of a single-component substance (D,T,P, ␮,R) is a connected open
subset Z of R such that 共i兲 共T,P兲 is injective on Z.
共ii兲 Z is maximal with this property 共i.e., if N is a connected open subset of R containing Z and
(T,P) is injective on N, then N⫽Z).
Proposition 1: Every point of the regular domain is in a phase.24
The injectivity of 共T,P兲 in a phase implies that there the specific internal energy and the
specific volume can be given as functions of temperature and pressure. In particular, to every
phase Z we can define the chemical potential of the phase by
␮ Z ª ␮ⴰ„共 T,P兲 兩 Z …⫺1 .
共3兲
The phase connections 共‘‘transitions’’兲 have been examined and classified in Ref. 17.
A body means a certain amount of a substance: the triplet of variables (e, v ,N) describes a
body where (e, v ) is an element of the constitutive domain and N is the particle number, an
arbitrary positive number.
Definition 4: A body consisting of a single-component substance (D,T,P, ␮,R) is (D
⫻R⫹ ,T,P, ␮,R); the elements of D⫻R⫹ are called the states of the body.
It turns out that the description of processes of bodies in which mass varies will be simpler if
instead of the variables e and v we use the total internal energy E and the total volume, V,
respectively. More precisely, we establish the smooth bijection,
共 J 兲 ⫹ ⫻ 共 m 3 兲 ⫻R⫹ → 共 J 兲 ⫹ ⫻ 共 m 3 兲 ⫹ ⫻R⫹ ,
共 e, v ,N 兲 哫 共 Ne,N v ,N 兲 ⫽: 共 E,V,N 兲 ,
whose inverse,
共 E,V,N 兲 哫 共 E/N,V/N,N 兲 ,
is smooth as well.
We find convenient to introduce the notation
R⫹ * Hª 兵 共 Ne,N v ,N 兲 兩 共 e, v 兲 苸H,N苸R⫹ 其 ,
共4兲
for an arbitrary subset H of D.
Using the variables (E,V,N), we define
T̂共 E,V,N 兲 ªT共 E/N,V/N 兲 ,
共5兲
ˆ as well. For the sake of brevity and perspicuity, an abuse of
and similar expressions for P̂ and ␮
notations will be applied further on: the simple symbol T, etc. will be written instead of T̂, etc.,
i.e., two different functions will be denoted by the same letter. Then we easily derive that
⳵T 1 ⳵T
⫽
,
⳵E N ⳵e
⳵T 1 ⳵T
⫽
,
⳵V N ⳵v
共6兲
holds on R⫹ * R, where, according to the previously accepted abuse of notations, it is understood
that the variables on the left-hand side and on the right-hand side are (E,V,N) and (e, v )
⫽(E/N,V/N), respectively. Moreover, we have
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J. Math. Phys., Vol. 41, No. 4, April 2000
On the mathematical structure of thermodynamics
⳵T
E ⳵T V ⳵T
⫽⫺
⫺
,
⳵N
N ⳵E N ⳵V
2025
共7兲
and similar formulas for P and ␮ as well.
For the total entropy,
S共 E,V,N 兲 ªNs共 E/N,V/N 兲 ,
共8兲
we get the usual equalities if the substance is entropic:
⳵S 1
⫽ ,
⳵E T
冉
⳵S
␮
⫽⫺ .
⳵N
T
⳵S P
⫽ ,
⳵V T
共9兲
Then the second derivative of the total entropy is
D2 S⫽⫺
1
2
T
⳵T
⳵E
P
⳵T
⳵V
⳵T
⳵P
⫺T
⳵E
⳵E
⫺␮
P
⳵T
⳵P
⫺T
⳵V
⳵V
⳵T
⳵N
P
⳵T
⳵P
⫺T
⳵N
⳵N
⳵T
⳵␮
⳵T
⳵␮
⳵T
⳵␮
⫹T
⫺␮
⫹T
⫺␮
⫹T
⳵E
⳵E
⳵V
⳵V
⳵N
⳵N
冊
.
共10兲
Proposition 2: D2 S(E,V,N) is negative semidefinite for all (E,V,N)苸R⫹ * R, having a onedimensional kernel spanned by (E,V,N).
As usual, we call energy, volume, and mass the extensive variables, temperature, pressure, and
chemical potential the intensive variables.
III. AN OUTLINE OF ORDINARY THERMODYNAMICS
A. Heuristic considerations
In this paragraph we use rather loose notations.
The state of a body is the triplet (E,V,N); a process of a body is a function that assigns states
to instants: t哫„E(t),V(t),N(t)…. We assume that the domain of a process is a time interval.
The first law of thermodynamics is expressed in the form
Ė⫽Q⫹W⫹L,
where Q is the heating, W is the working, and L is the transferring; this last quantity expresses the
energy change of the body due to the particle change. The quantities on the right-hand side of the
equation are time rates, thus, e.g., the heating is the heat per unit time; working is the work per
unit time.
We shall deal with the ideal case only, i.e., when
W⫽⫺ PV̇,
L⫽ ␮ Ṅ.
The first law is conceived as a differential equation. Of course, this single equation is not
sufficient to determine processes consisting of three functions. Therefore we suppose that we have
equations for the time change of V and N as well:
V̇⫽F,
Ṅ⫽G,
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J. Math. Phys., Vol. 41, No. 4, April 2000
T. Matolcsi
where F and G are called the springing and the converting, respectively. The quantities Q, W, L,
F, and G are supposed to be given as functions of the state (E,V,N); so we have a complete set
of differential equations.
If n⭓2 bodies interact, then a process of the system of interacting bodies is the joint of the
processes of the bodies: t哫„(E i (t),V i (t),N i (t) 兩 i⫽1,...,n….
The processes are supposed to be governed by a system of differential equations,
Ė i ⫽Q i ⫹W i ⫹L i ,
共 W i ⫽⫺ P i F i ,
V̇ i ⫽F i
L i⫽ ␮ iG i ,
Ṅ i ⫽G i ,
i⫽1,...,n 兲 ,
called the dynamical law, where the ith heating Q i , etc. are given as functions of the states. Later
we examine the properties of these functions.
The bodies can be in contact with an environment, which may be thought as an ‘‘infinitely
large’’ body whose process is prefixed 共e.g., its temperature and pressure is constant兲, whose state
does not change in the interaction 共i.e., the environment acts on the bodies, the bodies do not act
on the environment兲. This means that the environment is always characterized by its given temperature T a , pressure P a , and chemical potential ␮ a , which can vary with time.
The heating of a body consists of the heatings from the other bodies and from the environment, so
n
Q i⫽
兺
k⫽0
Q ik ,
where the subscript 0 refers to the environment. Similarly, we have
n
F i⫽
兺
k⫽0
n
F ik ,
G i⫽
n
Wi
兺
k⫽0
兺
G ik ,
k⫽0
n
W ik ,
L i⫽
兺
k⫽0
L ik .
It is convenient to introduce the notation
n
A ik ªQ ik ⫹W ik ⫹L ik ,
A iª
兺
k⫽0
A ik .
Q ik , etc. are called the system constitutive functions or the dynamical quantities.
Evidently, the particle number passed from the ith body to the kth body is the opposite to the
particle number passed from the kth body to the ith body, and a similar statement is true for the
energy change and volume change of the ith body due to the kth body. Thus we accept—
roughly—that A ik ⫽⫺A ki , F ik ⫽⫺F ki , G ik ⫽⫺G ki . The exact formulation of these requirements
will be given later.
We underline that the heatings need not have the above property, which is a well-known fact
in classical thermodynamics: the ‘‘noncompensated heating’’ Q ik ⫹Q ki is not necessarily zero. A
similar remark is valid for workings and transferrings.
It is reasonable to suppose 共as it is done in mechanics, too兲 that the interaction of two bodies
can be characterized by the properties of the two bodies only, which means that the dynamical
quantities between two bodies depend only on the states 共i.e., on the extensive variables兲 of the
two bodies. Moreover, according to our experience, we accept that the dynamical quantities
depend on the extensive variables through the intensive variables and the particle numbers. Since
the intensive variables do not characterize the state 共the same intensive values can belong to
different states in different phases兲, the dynamical quantities depend on phases, too, which corre-
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J. Math. Phys., Vol. 41, No. 4, April 2000
On the mathematical structure of thermodynamics
2027
sponds to a trivial experimental fact: the heat conduction at a given temperature and pressure
between two ice bodies is different from that between two water bodies.
The second law must be reflected in the properties of the dynamical quantities. The dissipative
inequality 共Clausius–Duhem inequality兲 in nonequilibrium thermodynamics expresses the second
law 共positive entropy production兲; an analogon of this exact relation can be well defined in
ordinary thermodynamics, too.25
B. Thermodynamical systems
On the base of the heuristic considerations of the previous paragraph we can formulate an
exact definition.
Definition 5: Let n be a given positive integer. A thermodynamical system of n bodies in a
given environment consists of the following.
共1兲 A family of simple substances, (D i ,Ti ,Pi , ␮i ,R i ) (i⫽0,...,n); the zeroth substance is
called the environment; the body corresponding to the ith substance is called the ith body of the
system.
共2兲 A given phase Z 0 of the environment, and the given temperature and pressure of the
environment, as a continuous function defined on a time interval: t哫„T a (t), P a (t)…苸(T0 ,P0 )
关 Z0兴.
共3兲 The dynamical quantities, given for all phases Z i and Z k of the ith and the kth body
(i,k⫽0,1,...,n), respectively,
QZ i Z k :⌽ i ⫻⌽ k → 共 J/s 兲 ,
FZ i Z k :⌽ i ⫻⌽ k → 共 m3/s兲 ,
GZ i Z k :⌽ i ⫻⌽ k → 共 1/s兲 ,
共11兲
where
⌽ i ª 共 Ti ,Pi 兲关 Z̄ i 艚D i 兴 ⫻R⫹ ;
共12兲
these functions are continuous; moreover, they are continuously differentiable on the interior of
their domain.
The dynamical quantities satisfy for all i,k⫽0,1,...,n.
共i兲 The compatibility property: QZ i ,Z k and QZ ⬘ Z ⬘ , etc. are equal on the intersection of their
i k
domain for all phases Z i , Z ⬘i and Z k , Z k⬘ ; furthermore, with the notations
WZ i Z k 共 T i , P i ,N i ,T k , P k ,N k 兲 ª⫺ P i FZ i Z k 共 T i , P i ,N i ,T k , P k ,N k 兲 ,
LZ i Z k ª 共 T i , P i ,N i ,T k , P k ,N k 兲 ª ␮ Z i 共 T i , P i 兲 G ik 共 T i , P i ,N i ,T k , P k ,N k 兲 ,
AZ i Z k ªQZ i Z k ⫹WZ i Z k ⫹LZ i Z k ,
and then 共for the sake of brevity兲 with the subscripts ik instead of Z i Z k and
关 i,k 兴 ª 共 T i , P i ,N i ,T k , P k ,N k 兲 苸⌽ i ⫻⌽ k ,
the dynamical quantities satisfy the following.
共ii兲 The mutuality property,
Aik 共关 i,k 兴 兲 ⫽⫺Aki 共关 k,i 兴 兲 ,
Fik 共关 i,k 兴 兲 ⫽⫺Fki 共关 k,i 兴 兲 ,
Gik 共关 i,k 兴 兲 ⫽⫺Gki 共关 k,i 兴 兲 .
共13兲
共iii兲 The dissipative property:
⫺
Wik 共关 i,k 兴 兲
Lik 共关 i,k 兴 兲
Qik 共关 i,k 兴 兲
„␮ 共 T , P 兲 ⫺ ␮ k 共 T k , P k 兲 …⭓0,
共 T i ⫺T k 兲 ⫺
共 P i⫺ P k 兲⫺
Ti
Pi
␮ i共 T i , P i 兲 i i i
共14兲
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J. Math. Phys., Vol. 41, No. 4, April 2000
T. Matolcsi
where equality holds if and only if Qik ( 关 i,k 兴 )⫽0, Wik ( 关 i,k 兴 )⫽0, Lik ( 关 i,k 兴 )⫽0; this inequality
can be rewritten in the form
1
Pi Pk
⫺ ␮ i共 T i , P i 兲 ␮ k共 T k , P k 兲
1
⫺
⫹Fik 共关 i,k 兴 兲
⫺
⫹Gik 共关 i,k 兴 兲
⫹
⭓0,
Aik 共关 i,k 兴 兲
Ti Tk
Ti Tk
Ti
Tk
共15兲
冉
冊
冉
冊
冉
冊
where equality holds if and only if Aik ( 关 i,k 兴 )⫽0, Fik ( 关 i,k 兴 )⫽0, Gik ( 关 i,k 兴 )⫽0.
共4兲 The dynamical law,
Ė i ⫽Q i ⫹W i ⫹L i
V̇ i ⫽F i ,
Ṅ i ⫽G i
共 i⫽1,...,n 兲 ,
共16兲
where
n
Q i⫽
兺
k⫽0
n
W i⫽
Q ik ,
兺
k⫽0
n
W ik ,
n
F i⫽
兺
k⫽0
L i⫽
兺
k⫽0
L ik ,
n
F ik ,
G i⫽
兺
k⫽0
G ik ,
and
Q ik ªQZ i Z k „T i 共 E i ,V i ,N i 兲 ,Pi 共 E i ,V i ,N i 兲 ,N i ,Tk 共 E k ,V k ,N k 兲 ,Pk 共 E k V k ,N k 兲 ,N k …,
if k⫽0 and
Q i0 ªQZ i Z 0 „Ti 共 E i ,V i ,N i 兲 ,Pi , 共 E i ,V i ,N i 兲 ,N i ,T a , P a ,N 0 …,
etc., where Z i and Z k are the phases whose closure contains (E i /N i ,V i /N i ) and (E k /N k ,V k /N k ),
respectively.
Remarks: The dynamical quantities with subscripts 0i do not appear in the dynamical law,
thus they are superfluous; we involved them only for an economic formulation. If the the dynamical quantities with subscripts i0 are given, putting QZ 0 Z i ( 关 0 兴 , 关 i 兴 )ª⫺QZ i Z 0 ( 关 i 兴 , 关 0 兴 ), etc. we
make all the requirements satisfied.
The particle number N 0 of the environment is irrelevant to the interaction 共the environment is
‘‘infinitely large’’兲, the dynamical quantities do not depend on N 0 ; it is involved as a dummy
variable only for an economic formulation.
The dynamical quantities with subscripts ii are zero by the mutuality property.
The condition imposed on the equality in the dissipative property is a strong requirement for
the dynamical quantities, because equality holds evidently if T i ⫽T k , P i ⫽ P k , and ␮ i (T i , P i )
⫽ ␮ k (T k , P k ), so these relations must imply that the dynamical quantities take the zero value.
Since all the bodies are supposed to be single component, it is understood that if GZ i Z k ⫽0
then the substance of the ith body coincides with that of the kth body.
Definition 6: A constant solution of the dynamical law—i.e., a state at which the right-hand
side of the dynamical law takes a zero value—is called a standstill. A standstill is an equilibrium
if all the dynamical quantities take a zero value at the corresponding state.
Proposition 3: If at least one of the dynamical quantities does depend on the process of the
environment, then standstill can exist only if the process of the environment is constant.
C. Thermodynamical forces and the conductivity matrix
For the sake of brevity, in the following the subscripts Z i Z k will be substituted by ik.
The coefficients of the dynamical quantities in the dissipative inequality 共15兲 are known as the
thermodynamical forces; more precisely, we accept the following definition.
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J. Math. Phys., Vol. 41, No. 4, April 2000
On the mathematical structure of thermodynamics
2029
Definition 7: The function
冉
冊
1
␮i ␮k
1 Pi Pk
⫺ , ⫺ ,⫺ ⫹
:D i ⫻D k → 共 1/K 兲 ⫹ ⫻ 共 Pa/K 兲 ⫻ 共 J/K 兲 ,
Ti Tk Ti Tk
Ti Tk
is called the thermodynamical force between the ith and kth body.
Definition 8: The dynamical quantities Qik ,Fik ,Gik are called quasilinear if
冉 冊冉
␭ ik
Qik
Fik ⫽ ␣ ik
Gik
␳ ik
␬ ik
␽ ik
␤ ik
␥ ik
␴ ik
␸ ik
冉
冊冉
1
1
⫺
Ti Tk
冊
Pi Pk
⫺
Ti Tk
⫺
␮ i共 T i , P i 兲 ␮ k共 T k , P k 兲
⫹
Ti
Tk
where ␭ ik , etc. are continuous function defined on ⌽ i ⫻⌽ k . Equivalently,
冉 冊冉
Aik
Fik ⫽
Gik
˜␭ ik
˜␬ ik
˜ ik
␽
␣ ik
␤ ik
␥ ik
␳ ik
␴ ik
␸ ik
1
1
⫺
Ti Tk
Pi Pk
⫺
Ti Tk
⫺
␮ i共 T i , P i 兲 ␮ k共 T k , P k 兲
⫹
Ti
Tk
冊
冊
,
共17兲
,
共18兲
where
˜␭ ik ª␭ ik ⫺ P i ␣ ik ⫹ ␮ i ␳ ik ,
˜␬ ik ª ␬ ik ⫺ P i ␤ ik ⫹ ␮ i ␴ ik ,
˜ ik ª ␽ ik ⫺ P i ␥ ik ⫹ ␮ i ␸ ik .
␽
The matrix vector on the right-hand side of equality 共18兲 is called the conductivity matrix
between the ith and the kth body.
Note that the conductivity matrix is, in fact, a matrix-valued function. It is a simple fact that
the conductivity matrix is not uniquely defined. To see this, it suffices to show that
冉 冊
˜␭
˜␬
˜
␽
␣
␤
␥
␳
␴
␸
冉
1
1
⫺
Ti Tk
Pi Pk
⫺
Ti Tk
⫺
␮ i共 T i , P i 兲 ␮ k共 T k , P k 兲
⫹
Ti
Tk
冊
⫽0,
共19兲
can hold if the matrix above—which represents the difference of two conductivity matrices—is
not zero. This is the case, for example, if ˜␭ ª P i /T i ⫺ P k /T k , ˜␬ ª⫺1/T i ⫹1/T k , and the other
entries are zero.
Of course, if the conductivity matrix is constant then it is uniquely determined.
Proposition 4: Suppose (19) holds and the bodies are entropic; if ␮ i (T, P)⫽ ␮ k (T, P), then
we have
˜␭ 共 T, P,N i ,T, P,N k 兲 ⫹ ˜␬ 共 T, P,N i ,T, P,N k 兲 P⫹ ␽
˜ 共 T, P,N i ,T, P,N k 兲 „␮ i 共 T, P 兲 ⫹Tsi 共 T, P 兲 …⫽0,
␣ 共 T, P,N i ,T, P,N k 兲 ⫹ ␤ 共 T, P,N i ,T, P,N k 兲 P⫹ ␥ 共 T, P,N i ,T, P,N k 兲 „␮ i 共 T, P 兲 ⫹Tsi 共 T, P 兲 …⫽0,
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2030
J. Math. Phys., Vol. 41, No. 4, April 2000
T. Matolcsi
␳ 共 T, P,N i ,T, P,N k 兲 ⫹ ␴ 共 T, P,N i ,T, P,N k 兲 ⫹ ␸ 共 T, P,N i ,T, P,N k 兲共 ␮ i 共 T, P 兲 ⫹si 共 T, P 兲 T 兲 ⫽0,
˜ 共 T, P,N i ,T, P,N k 兲 vi 共 T, P 兲 ⫽0,
˜␬ 共 T, P,N i ,T, P,N k 兲 ⫺ ␽
␤ 共 T, P,N i ,T, P,N k 兲 ⫺ ␥ 共 T, P,N i ,T, P,N k 兲 vi 共 T, P 兲 ⫽0,
␴ 共 T, P,N i ,T, P,N k 兲 ⫺ ␸ 共 T, P,N i ,T, P,N k 兲 vi 共 T, P 兲 ⫽0,
where si and vi are the specific entropy and the specific volume of the ith body as a function of
temperature and pressure in the corresponding phase.
Proof: Let Pª P i ª P k ,TªT k ⫽T i , divide 共19兲 by T i ⫺T, and take the limit T i →T. According to our hypothesis ␮ i (T, P)⫽ ␮ k (T, P) and to the Gibbs–Duhem relations, we have
␮ i 共 T i , P 兲 ⫺ ␮ k 共 T, P 兲
⫽⫺si 共 T, P 兲 ,
T i ⫺T
T i →T
lim
from which we infer the first three equalities. The further ones follow from TªT i ªT k , Pª P k
⫽ P i , and
␮ i 共 T, P i 兲 ⫺ ␮ k 共 T, P 兲
⫽vi 共 T, P 兲 .
P i⫺ P
Pi→ P
lim
䊐
The elements (T, P,N i ,T, P,N k )苸⌽ i ⫻⌽ k for which ␮ i (T, P)⫽ ␮ k (T, P) holds will be called
central.
˜ ⫽0,␥ ⫽0,␳ ⫽0,␴ ⫽0, ␸ ⫽0—then
We see that if the bodies cannot change particles—i.e., ␽
the values of the conductivity matrix are uniquely defined at the central values; otherwise they are
not.
Definition 9: The conductivity matrix between the ith and kth body is 共1兲 strictly Onsagerian
if it is constant and symmetric, 共2兲 Onsagerian if its every value is symmetric, 共3兲 weakly Onsagerian if its values at central elements are symmetric.
Usually one considers strictly Onsagerian conductivity matrices; however, the Onsagerian
formalism is said to be a linear approximation around equilibrium, so the usual Onsager matrix
corresponds to a value of our conductivity matrix at a central element. Thus, the usual formalism
corresponds, in fact, to the weakly Onsagerian case.
It is a remarkable result26 that if a twice continuously differentiable conductivity matrix is
weakly Onsagerian then—using its nonuniqueness—we can take it to be symmetric in a neighborhood of the central values.
IV. GENERAL MATHEMATICAL FORMULAS
A. The dynamical law and the nominal thermodynamical force
A state of a system consisting of n bodies,
xª 共 x i ª 共 E i ,V i ,N i 兲 兩 i⫽1,...,n 兲 ,
共20兲
is in
X Dª
Ã共 R⫹ * D i 兲 ,
n
共21兲
i⫽1
which is a subset of the vector space,
Xª„共 J 兲 ⫻ 共 m 3 兲 ⫻R…n .
共22兲
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J. Math. Phys., Vol. 41, No. 4, April 2000
On the mathematical structure of thermodynamics
2031
Now we find it convenient to consider kT instead of the temperature T, where k is the
Boltzmann constant. The physical dimension of kT is that of energy, i.e., kT苸(J).
The dual of a one-dimensional vector space is its ‘‘reciprocal’’ 27 and the duality map is the
共tensorial兲 multiplication that is well reflected in our present notation: an element from (1/J)
multiplied by an element from 共J兲 results in a real number. The dual of a Cartesian product is
identified with the Cartesian product of the duals by the usual rule: take the sum of the products
of the components.
Thus we have that the intensive quantities are elements in the dual space of the extensive
quantities, namely,
冉 冉
冊
冊冏
1 P i ␮ i共 T i , P i 兲
,
,
i⫽1,...,n 苸X * ⫽
yª y i ª
kT i kT i
kT i
冉冉 冊 冉 冊 冊
1
1
⫻
⫻R
J
m3
n
.
共23兲
Consequently, the constitutive functions map from the state space into its dual:
yª
Ãyi ª Ã
n
n
i⫽1
i⫽1
冉
冊
1 Pi
␮i
,
,⫺
:X→X * ,
kTi kTi
kTi
共24兲
whose domain is X D .
Using the notation
y 0 ª 共 1/kT 0 , P 0 /kT 0 ,⫺ ␮ 0 共 T 0 , P 0 兲 /kT 0 兲 苸Z 0 ,
let us introduce
R ik 共 x i ,x k 兲 ª 共 Aik ,Fik ,Gik 兲 „yi 共 x i 兲 ,yk 共 x k 兲 …,
共25兲
R i0 共 x i ,y 0 兲 ª 共 Ai0 ,Fi0 ,Gi0 兲 „yi 共 x i 兲 ,y 0 …,
for x苸X D ,i,k⫽1,...,n.
Note that in these notations the mutuality property of the dynamical quantities becomes
R ik 共 x i ,x k 兲 ⫽⫺R ki 共 x k ,x i 兲
共26兲
共 x苸X D ,i,k⫽1,...,n 兲 .
Putting
冉
n
R 共 x,y 0 兲 ª R i 共 x 兲 ªR i0 共 x i ,y 0 兲 ⫹
兺
k⫽1
冏
冊
R ik 共 x i ,x k 兲 i⫽1,...,n ,
共27兲
for x苸X D and y 0 苸Z 0 , we can rewrite the dynamical law 共16兲 in the form
ẋ⫽R 共 x,y a 兲 .
共28兲
Note that if t哫x(t) is a function defined in time and having values in X, then ẋ has values in
X/s, where s denotes ‘‘second.’’ Thus, R(x,y 0 )苸X/s for all x and y 0 .
The collection of the thermodynamical forces between the bodies and the environment will
play and important role that is why we introduce the following notion.
Definition 10: The function
F:X D ⫻Z 0 →X * ,
F 共 x,y 0 兲 ª„yi 共 x i 兲 ⫺y 0 兩 i⫽1,...,n…,
共29兲
is called the nominal thermodynamical force in the system.
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2032
J. Math. Phys., Vol. 41, No. 4, April 2000
T. Matolcsi
B. The dissipative property and its consequences
The dissipative property of the dynamical quantities in the previously introduced concise
notations reads as follows:
„y共 x i 兲 ⫺yk 共 x k 兲 …•R ik 共 x i ,x k 兲 ⭓0
„yi 共 x i 兲 ⫺y 0 …•R i0 共 x i ,y 0 兲
共 x苸X D ,i,k⫽1,...,n 兲 ,
共 x苸X D ,y 0 苸Z 0 ,i⫽1,...,n 兲 ,
where equality holds if and only if R ik (x i ,x k )⫽0 and R i0 (x i ,y 0 )⫽0; the dot denotes the sum of
the product of the components.
Proposition 5: The dissipative property of the dynamical quantities imply
F 共 x,y 0 兲 •R 共 x,y 0 兲 ⭓0
共 x苸X D ,y 0 苸Z 0 兲 ,
共30兲
where equality holds if and only R ik (x i ,x k )⫽0 and R i0 (x i ,y 0 )⫽0 for all i,k⫽1,...,n.
Proof: Let us introduce the formal quantities R i0 (x i ,x 0 )ª⫺R 0i (x 0 ,x i )ªR i0 (x i ,y 0 ), and let
us put y k ªyk (x k ) for the sake of brevity. Then we have
共 y i ⫺y k 兲 •R ik 共 x i ,x k 兲 ⭓0
共 i,k⫽0,1...,n 兲 ,
from which we infer by the mutuality property of the dynamical quantities that
n
1
0⭐
共 y ⫺y k 兲 •R ik 共 x i ,x k 兲
2 i,k⫽0 i
兺
n
n
1
1
⫽
共 y ⫺y 0 兲 •R ik 共 x i ,x k 兲 ⫺
共 y ⫺y 0 兲 •R ik 共 x i ,x k 兲
2 i,k⫽0 i
2 i,k⫽0 k
兺
兺
n
⫽
兺
i,k⫽0
n
⫽
共 y i ⫺y 0 兲 •R ik 共 x i ,x k 兲
n
兺 共 y i ⫺y 0 兲 • k⫽0
兺 R ik共 x i ,x k 兲 ,
i⫽1
where equality holds if and only R ik (x i ,x k )⫽0 for all i,k⫽0,1,...,n. The last formula coincides
with the left-hand side of 共30兲.
Of course, equality holds if R(x,y 0 )⫽0; thus we have the following.
Proposition 6: R(x,y 0 )⫽0 if and only if R ik (x i ,x k )⫽0 and R i0 (x i ,y 0 )⫽0 for all i,k
⫽1,...,n.
This has an interesting and important consequence.
Proposition 7: Every standstill of the dynamical law (28) is an equilibrium.
Proof: There can be a standstill if and only if either all the dynamical quantities are independent of the environment or y a 共the process of the environment兲 is constant. In both cases the state
x is a standstill if and only if R(x,y a )⫽0 that is equivalent by the previous result to the fact that
all the dynamical quantities take a zero value at x, i.e., x is an equilibrium.
C. The quasilinear case
Let C ik denote the conductivity matrix between the ith and kth body as a function of the
extensive variables 关i.e., put T i ªTi (E i ,V i ,N i ), etc. in them兴. Then we have for x苸X D and y 0
苸Z 0 ,
R ik 共 x i ,x k 兲 ⫽Cik 共 x i ,x k 兲 •„yi 共 x i 兲 ⫺yk 共 x k 兲 …,
R i0 共 x i ,y 0 兲 ⫽C i0 共 x i ,y 0 兲 •„yi 共 x i 兲 ⫺y 0 …,
for i,k⫽1,...,n. It follows from the mutuality property that
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J. Math. Phys., Vol. 41, No. 4, April 2000
On the mathematical structure of thermodynamics
2033
Cik 共 x i ,x k 兲 •„yi 共 x i 兲 ⫺y共 x k 兲 …⫽⫺Cki 共 x k ,x i 兲 •„yk 共 x k 兲 ⫺yi 共 x i 兲 ⫽Cki 共 x k ,x i 兲 •„yi 共 x i 兲 ⫺yk 共 x k 兲 …,
which does not imply
Cik 共 x i ,x k 兲 ⫽Cki 共 x k ,x i 兲
共 x苸X D ,i,k⫽1,...,n 兲 ;
共31兲
since these matrices play a role only when multiplied by yi (x i )⫺yk (x k ), we do not restrict the
generality by requiring equality 共31兲.
Moreover, we have that
n
Ci0 „yi 共 x i 兲 ,y 0 …•„yi 共 x i 兲 ⫺y 0 …⫹
n
兺 Cik共 x i ,x k 兲 •„yi共 x i 兲 ⫺yk共 x k 兲 …⫽ i⫽1
兺 B ik共 x,y 0 兲 •„yk共 x k 兲 ⫺y 0 …,
i⫽1
where
B ik 共 x,y 0 兲 ª
再
Cik 共 x i ,x k 兲 ,
if i⫽k,
n
Ci0 共 x i ,y 0 兲 ⫹
兺 Cik共 x i ,x k 兲 ,
i⫽1
if i⫽k.
共32兲
Thus, introducing
B 共 x,y 0 兲 ª„B ik 共 x,y 0 兲 兩 i,k⫽1,...,n…,
共33兲
and using the nominal thermodynamical force defined in 共29兲, we get
R 共 x,y 0 兲 ⫽B 共 x,y 0 兲 •F 共 x,y 0 兲 ,
共34兲
and the dynamical law has the form
ẋ⫽B 共 x,y a 兲 •F 共 x,y a 兲 .
共35兲
Definition 11: The function B:X D ⫻Z 0 →Lin(X * ,X/s) defined in 共33兲 is called the nominal
conductivity matrix in the system.
Proposition 8: If C ik (x i ,x k ) and C i0 (x i ,y 0 ) are symmetric for all i,k⫽1,...,n for a given x and
y 0 , and (31) holds, then B(x,y 0 ) is symmetric as well.
D. Constraints
1. Heuristic considerations
A system of interacting bodies, in general, is subjected to some constraints. As examples let
us consider the following systems.
共1兲 A body with a constant particle number in a given constant environment, the pressure of
the body is held constant, equaling the pressure P a of the environment; then
Ṅ⫽0,
⳵ P共 E,V,N 兲
⳵ P共 E,V,N 兲
Ė⫹
V̇⫽0.
⳵E
⳵V
共2兲 A heat insulated body with a constant particle number in a given environment,
Ṅ⫽0,
E⫽⫺P共 E,V,N 兲 V̇.
共36兲
共3兲 A body with constant volume in a given environment,
V̇⫽0.
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2034
J. Math. Phys., Vol. 41, No. 4, April 2000
T. Matolcsi
共4兲 Two bodies that are totally insulated from the environment,
Ė 1 ⫹Ė 2 ⫽0,
V̇ 1 ⫹V̇ 2 ⫽0,
Ṅ 1 ⫹Ṅ 2 ⫽0.
共37兲
The constraints are not characterized completely by the previous formulas that concern the
‘‘fluxes,’’ i.e., the time derivative of the extensive variables. Constraints are intimately related to
forces, too. Namely, the real driving forces in the examples are 共1兲 the temperature difference
between the body and the environment; 共2兲 the pressure difference between the body and the
environment; 共3兲 the temperature difference between the body and the environment; 共4兲 the temperature difference, the pressure difference, and the chemical potential difference between the two
bodies.
Now we shall show how we get these real forces from the nominal one.
The nominal thermodynamical force for the examples 共1兲–共3兲, taken at the process of the
environment is—in a loose notation—
冉
冊
1
␮
␮a
1 P
Pa
⫺
⫹
, ⫺
,⫺
.
kT kT a kT kT a
kT kT a
共1兲 Equality 共36兲 and the first law Ė⫽Q⫺ P a V̇ result in that the heating is proportional to the
springing, Q⫽ ␣ F, thus the dynamical equation becomes
Ė⫽共␣⫺Pa兲F, V⫽F,
N⫽0.
The right-hand side of the equation is a multiple of the vector ( ␣ ⫺ P a ,1,0).
Let us apply the nominal thermodynamical force 共as an element of the dual space兲 to this
vector:
1
Pa Pa
1
1
1
⫽␣ ⫺
.
⫺
共␣⫺Pa兲⫹ ⫺
kT kTa
kT kTa
kT kTa
冉
冊
冉
冊 冉
冊
We have got 共a multiple of兲 the really acting thermodynamical force whose zero value determines the equilibrium if the environment is constant.
共2兲 In the case of a heat insulated body with a constant particle number, the right-hand side of the
dynamical law,
Ė⫽⫺PF,
V̇⫽F, Ṅ⫽0,
is a multiple of the vector (⫺ P,1,0).
Applying the nominal thermodynamical force to this vector, we get
1
P Pa
1
1
⫽ 共 P⫺Pa兲,
⫺
共⫺Pa兲⫹ ⫺
kT kTa
kT kTa kt
冉
冊
冉
冊
which is 共a multiple of兲 the really acting thermodynamical force whose zero value determines
the equilibrium if the environment is constant.
共3兲 In this example the dyamical law has the form
Ė⫽Q⫹ ␮ G,
V̇⫽0,
Ṅ⫽G,
whose right-hand side is spanned by the multiple of the vectors 共1, 0, 0兲 and 共0, 0, 1兲.
Applying to these vectors the nominal thermodynamical force, we get
冉
冊
1
␮
␮a
1
,⫺
,
⫺
⫹
kT kT a
kT kT a
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J. Math. Phys., Vol. 41, No. 4, April 2000
On the mathematical structure of thermodynamics
2035
which is the really acting thermodynamical force whose zero value determines the equilibrium
if the environment is constant.
共4兲 In this example the dynamical law has the form
Ė 1 ⫽Q 1 ⫺ P 1 F 1 ⫹ ␮ 1 G 1 ,
V̇ 1 ⫽F 1 ,
Ṅ 1 ⫽G 1 ,
Ė 2 ⫽Q 2 ⫺ P 2 F 2 ⫹ ␮ 2 G 2 ,
V̇ 2 ⫽F 2 ,
Ṅ 2 ⫽G 2 ,
moreover, relation 共37兲 holds; thus the right-hand side of the dynamical equation is spanned by the
multiple of the vectors 共1, 0, 0, ⫺1, 0, 0兲, 共0, 1, 0, 0, ⫺1, 0兲, and 共0, 0, 1, 0, 0, ⫺1兲. Applying to
these vectors the nominal thermodynamical force,
冉
冊
1
␮1
␮a 1
␮2
␮a
1 P1
Pa
1 P2
Pa
⫺
,
⫺
,⫺
⫹
,
⫺
,
⫺
,⫺
⫹
,
kT 1 kT a kT 1 kT a
kT 1 kT a kT 2 kT a kT 2 kT a
kT 2 kT a
we get
冉
冊
1
␮1
␮2
1 P1
P2
⫺
,
⫺
,⫺
⫹
,
kT 1 kT 2 kT 1 kT 2
kT 1 kT 2
which is the really acting thermodynamical force whose zero value determines the equilibrium.
2. Mathematical formulation of constraints
Definition 12: We say that the dynamical law 共28兲 is subjected to a constraint if there is a set
⌫ of continuous maps X D →X * , such that 兵 p(x) 兩 p苸⌫ 其 is linearly independent for all x苸X D ;
p(x)ẋ⫽0 holds for all p苸⌫ and for all processes x 关solutions of the dynamical law 共28兲兴, which
is equivalent to
R 共 x,y a 兲 苸K 共 x 兲 /s
共 x苸X D 兲 ,
共38兲
where
K 共 x 兲 ª 艚 Ker p 共 x 兲 .
共39兲
p苸⌫
is the constraint subspace at x; R(x,y a )⫽0 if and only if F(x,y a ) 兩 K(x) ⫽0.
The function
X D →X * ,
x哫F ⌫ 共 x,y a 兲 ªF 共 x,y a 兲 兩 K 共 x 兲 ,
共40兲
is called the effective thermodynamical force corresponding to the constraint.
Now we apply well-known notions of the theory of manifolds. If U is a u-dimensional C r
submanifold in X D , then a parametrization 共the inverse of a local coordinatization兲 of U around
x 0 苸U is a map p:Ru →X, such that x 0 苸Ran p傺U; p is injective, p ⫺1 is continuous; p is r times
continuously differentiable; Dp(x) is injective for all x苸Dom p.
One of the most important relations is that Ran Dp(x)⫽T x (U), where the last symbol denotes
the tangent space of U at x.
A submanifold will mean a C 1 submanifold.
A submanifold U in X D is called a constraint manifold if T x (U)⫽K(x) for all x苸U. A
constraint manifold U is invariant for the dynamical law, i.e., every process starting from U
remains in U.
If the constraint is a foliation i.e., for every x in the interior of X D there is a 共maximal兲
constraint manifold containing x then the interior of X D is the disjoint union of constraint manifolds.
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2036
J. Math. Phys., Vol. 41, No. 4, April 2000
T. Matolcsi
The constraint is called holonom, if for all p苸⌫ there is a ⌽ p :X D →R, continuously differentiable on the interior of X D , such that p傻D⌽ p . Then the constraint is a foliation and the
constraint manifolds are the subsets that ⌽ p is constant on for all p苸⌫.
If ⌫⫽⭋ then there is no constraint, i.e., K(x)⫽X for all x苸X D , and there is a single
constraint manifold; the interior of X D .
In the quasilinear case R(x,y a )⫽B(x,y a )F(x,y a )苸K(x)/s does not imply, in general, that
Ran B(x,y a )傺K(x)/s even if F(x,y a )⫽0. Moreover, in general, R(x,y a ) cannot be given as a
quasilinear function of the effective thermodynamical force. That is why we introduce the following notion.
Definition 13: The nominal conductivity matrix fits the constraint if for all x苸X D there is a
B ⌫ (x,y a )苸Lin„K(x) * ,K(x)/s…, such that
B 共 x,y a 兲 F 共 x,y a 兲 ⫽B ⌫ 共 x,y a 兲 F ⌫ 共 x,y a 兲 .
共41兲
In this case the map x哫B ⌫ (x,y a ) is called the effective conductivity matrix.
The relation between the nominal conductivity matrix and the effective conductivity matrix
can be expressed by the canonical embedding i(x):K(x)→X and its transpose i(x) * :X *
→K(x) * as follows:
i 共 x 兲 B ⌫ 共 x,y a 兲 i 共 x 兲 * ⫽B 共 x,y a 兲
共 x苸X D 兲 .
共42兲
Then we have the following results.
Proposition 9: (i) The nominal conductivity matrix fits the constraint if and only if for all x
苸X D we have Ran B(x,y a )傺K(x)/s and Ker B(x,y a )傻(K(x))°, which is equivalent to the fact
that p(x)B(x,y a )⫽0 and B(x,y a )p(x)⫽0 for all p苸⌫, where „K(x)…°ª 兵 y苸X * 兩 i(x) * y⫽0 其 is
the annullator of K(x).
(ii) The nominal conductivity matrix fits the constraint if Ran B(x,y a )傺K(x) and B(x,y a ) is
symmetric for all x苸X D .
Proof: 共i兲 If the kernel of B(x,y a ) contains the annullator of K(x), then B ⌫ (x,y a ) is well
defined by B ⌫ (x,y a )i(x) * yªB(x,y a )y (y苸X *兲; so the stated relations are sufficient. The statement concerning the necesssity is trivial.
It is evident that the annullator of K(x) is spanned by 兵 p(x) 兩 p苸⌫ 其 , thus the equivalent
statement holds true as well.
共ii兲 If B(x,y a ) is symmetric, then its kernel contains the annullator of its range.
Simple arguments prove the following statements, too.
Proposition 10: Let the nominal conductivity matrix fit the constraint and use the previous
notations. Then for all x苸X D : 共i兲 Ker B ⌫ (x,y a )⫽ 兵 0 其 if and only if Ker B(x,y a )⫽(K(x)) 0 ;
B ⌫ (x,y a ) is symmetric if and only if B(x,y a ) is symmetric; 共iii兲 B ⌫ (x,y a ) is positive semidefinite
if and only if B(x,y a ) is positive semidefinite.
Since R(x,y a )苸K(x)/s, inequality 共30兲 can be rewritten in the form
F ⌫ 共 x,y a 兲 R 共 x,y a 兲 ⭓0
共 x苸X D 兲 ,
共43兲
where equality holds if and only if R(x,y a )⫽0, which is equivalent to F ⌫ (x,y a )⫽0 by the
definition of constraints.
In the quasilinear case if the effective conductivity matrix exists then the above inequality
becomes
F ⌫ 共 x,y a 兲 B ⌫ 共 x,y a 兲 F ⌫ 共 x,y a 兲 ⭓0
共 x苸X D 兲 ,
共44兲
where equality holds if and only if F ⌫ (x,y a )⫽0.
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J. Math. Phys., Vol. 41, No. 4, April 2000
On the mathematical structure of thermodynamics
2037
E. Onasager formalism
We call the constraint affine if every constraint subspace is the same: there is a linear subspace
K of X such that K(x)⫽K for all x苸X D . The constraint is affine in the previous examples 3 and
4 and is not affine in the examples 1 and 2.
Note the important fact that the effective conductivity matrix cannot be constant if the constraint is not affine; indeed, B ⌫ (x,y a ) is a linear map from K(x) * into K(x)/s; thus, if K(x)
⫽K(x ⬘ ) then B ⌫ (x,y a )⫽B ⌫ (x ⬘ ,y a ).
If the constraint is affine then the effective conductivity matrix can but evidently need not be
constant.
Let us consider an affine constraint with constraint subspace K. Let x 0 be an equilibrium. Put
y 0 ªy(x 0 ) and y na ª(y a ,...,y a ). Then
0⫽F 共 x 0 ,y a 兲 兩 K ⫽ 共 y 0 ⫺y na 兲 兩 K ,
consequently,
F ⌫ 共 x,y a 兲 ⫽F ⌫ 共 x,y 0 兲 ª„y共 x 兲 ⫺y 0 …兩 K ,
i.e., in this case the effective thermodynamical force can be expressed by the deviation of the
intensive quantities from their equilibrium values. In the quasilinear case the effective conductivity matrix B ⌫ (x,y a ) maps K into K for all x; near the equilibrium x 0 it can be approximated by its
equilibrium value, which is a linear map K→K, too; thus the dynamical law can be approximated
by
ẋ⫽B ⌫ 共 x 0 ,y a 兲 F ⌫ 共 x,y 0 兲 .
This is the usual form of the Onsager formalism: the conductivity matrix is constant, the thermodynamical force is expressed by the difference between the intensive variables and their equilibrium values 共and the fluxes correspond to the time derivatives of the extensive quantities兲.
On the other hand, if the constraint is not affine then F ⌫ (x,y a )⫽„y(x)⫺y 0 …兩 K(x) , i.e., the
effective thermodynamical force cannot be expressed by the the deviation of the intensive quantities from their equilibrium values. Moreover, any approximation of B ⌫ (x,y a ):K(x)→K(x)/s
must be a map K(x)→K(x)/s; thus the effective conductivity matrix cannot be approximated by
its equilibrium value B ⌫ (x 0 ,y a ):K(x 0 )→K(x 0 )/s. All these mean that the Onsger formalism does
not work for nonaffine constraints.
V. STABILITY
A. Some comments
One of the main problems of thermodynamical systems is the stability of equilibria. The
stability investigations of classical thermodynamics are not satisfactory because stability is not
defined in a mathematically exact way and, indeed, the results concern only some constitutive
properties of the substances 共intrinsic stability兲 and do not take into account dynamical properties
共which are formulated here by the dissipative property兲. In continuum 共irreversible兲 thermodynamics the notion of stability is defined in a mathematically exact way referring to the balance
equations; however, the investigations are very hard from a mathematical point of view because
the equations describing the processes are partial differential equations.28–33
In ordinary thermodynamics stability investigations are based on mathematically correct notions, and are much more easier than in continuum thermodynamics because here ordinary differential equations govern the processes. Trend to equilibrium—a fundamental concept—corresponds
to the asymptotic stability of equilibria.
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2038
J. Math. Phys., Vol. 41, No. 4, April 2000
T. Matolcsi
B. Notions and results from the theory of stability
For the sake of simplicity, from now on we omit y a from the notations of the functions
introduced up to now, i.e., we write R(x), B(x), F(x), etc. instead of R(x,y a ), B(x,y a ),
F(x,y a ), etc.; then, in particular, the dynamical law has the form
ẋ⫽R 共 x 兲 .
Recall that x 0 is an equilibrium if and only if R(x 0 )⫽0.
Definition 14: Let U be a subset invariant for the dynamical equation 共in particular, a constraint manifold兲.
共1兲 An equilibrium x 0 苸U is stable with condition U if for each neighborhood N of x 0 there is a
neighborhood G of x 0 such that for every process starting from G艚U proceeds in N艚U.
共2兲 An equilibrium x 0 苸U is asymptotically stable with condition U if it is stable with condition
U and there is a neighborhood V of x 0 such that for every process r starting from V艚U we
have limt→⬁ r(t)⫽x 0 .
共3兲 A set E傺U of equilibria is strictly asymptotically stable34 with condition U if every equilibrium in E is stable with condition U; every equilibrium in E has a neighborhood V such that
if r is a process starting from V艚U then limt→⬁ r(t) is in the closure of E.
Besides the well-known and fundamental results of stability theory;35 we shall apply a less
common one,36 which reads in our formulation as follows.
Consider the differential equation
␰˙ ⫽⌽ 共 ␰ 兲 ,
in Ru , where ⌽ is continuously differentiable. Let ⌬ be the set of its equilibria and suppose 共i兲
there is a nontrivial linear subspace Z傺Ru , an element a of Ru such that ⌬⫽(a⫹Z)艚Dom ⌽; 共ii兲
for all ␰ 苸⌬:—Ker D⌽( ␰ )⫽Z,—the algebraic multiplicity and the geometric multiplicity of the
zero eigenvalue of D⌽( ␰ ) are equal—the real part of the nonzero eigenvalues of D⌽( ␰ ) is
negative.
Then ⌬ is strictly asymptotically stable.
•
If L is a differentiable scalar-valued function defined in X D , then L (x)ªDL(x)R(x) is called
the derivative of L along the dynamical equation.
Proposition 11: Let U be a submanifold in X D , invariant for the dynamical law (in particular,
a constraint manifold). If x 0 苸U is an equilibrium and there is a continuously differentiable
real-valued function L, defined in a neighborhood of x 0 , such that (i) L has a strict local maxi•
mum at x 0 with condition U, i.e., L(x)⬍L(x 0 ) for all x苸U in a neighborhood of x 0 ; (ii) L has a
(strict) local minimum at x 0 with condition U, then the equilibrium x 0 is (asymptotically) stable
with condition U.
Proof: Let u be the dimension of U and take a local parametrization p:Ru →U around x 0 .
Then the restriction of the dynamical law onto the invariant submanifold U is reduced to the
differential equation
␰˙ ⫽D p 共 ␰ 兲 ⫺1 R„p 共 ␰ 兲 …,
共45兲
for the function ␰ ª p ⫺1 ⴰx.
Then ␰ 0 ª p ⫺1 (x 0 ) is an equilibrium of the reduced dynamical equation. It is trivial that ⌳
ªLⴰp is a continuously differentiable function that has a strict maximum at ␰ 0 .
Since D⌳⫽(DLⴰp)Dp, the derivative of ⌳ along the reduced equation,
•
•
⌳ ⫽ 共 DLⴰ p 兲共 Rⴰ p 兲 ⫽L ⴰ p,
共46兲
has a 共strict兲 minimum at ␰ 0 .
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J. Math. Phys., Vol. 41, No. 4, April 2000
On the mathematical structure of thermodynamics
2039
These imply by Lyapunov’s theory that ␰ 0 is an 共asymptotically兲 stable equilibrium of the
reduced equation from which it follows that x 0 is an 共asymptotically兲 stable equilibrium with
condition U.
Proposition 12: Let U be a submanifold, invariant for the dynamical equation (in particular,
a constraint manifold) and let E be the set of equilibria in U. Suppose that 共1兲 E is a submanifold;
共2兲 for all x苸E, the linear map DR(x) 兩 T x (U) :T x (U)→T x (U) has the following properties: (i) its
kernel is T x (E); (ii) the algebraic multiplicity and the geometric multiplicity of its zero eigenvalue
coincide; (iii) its nonzero eigenvalues have negative real part.
Then E is strictly asymptotically stable with condition U.
Proof: Let e and u be the dimensions of E and U, respectively. There is a local parametrization p:Re ⫻Ru⫺e →U such that p(•,0):Re →E is a parametrization of E. Then
Ran„Dp 共 ␩ ,0兲 兩 Re ⫻ 兵 0 其 …⫽T p 共 ␩ ,0兲 共 E 兲 .
The set of equilibria of the reduced dynamical equation 共45兲 is the manifold
p ⫺1 共 E 兲 ⫽Re ⫻ 兵 0 其 艚Dom p,
whose tangent space at 共␩, 0兲 equals
Re ⫻ 兵 0 其 ⫽Dp 共 ␩ ,0兲 ⫺1 关 T p 共 ␩ ,0兲 共 E 兲兴 .
The derivative of the right-hand side of the reduced dynamical equation at an equilibrium 共␩,
0兲, is
D 共 ␩ ,0兲 ªDp 共 ␩ ,0兲 ⫺1 DR 共 p 共 ␩ ,0兲兲 Dp 共 ␩ ,0兲 .
As a consequence, the spectral properties of D( ␩ ,0) coincide with those of DR„p( ␩ ,0)… 共i.e.,
they have the same eigenvalues and multiplicities兲; moreover,
Ker D 共 ␩ ,0兲 ⫽„Dp 共 ␩ ,0兲 …⫺1 关 Ker DR 共 p 共 ␩ ,0兲 …兴 ⫽Re ⫻ 兵 0 其 .
Thus, according to the theorem cited above, the set of equilibria of the reduced dynamical equation is strictly asymptotically stable, which implies that our assertion is true.
C. Stability in ordinary thermodynamics
There are nice stability results for several phenomena, including phase transitions 共see Refs.
14–19兲. It is remarkable that some of them is obtained without the use of entropy. The entropic
property, however, admits general results on stability.
The entropy of the environment—in a loose notation—is
S 0⫽
E 0⫹ P 0V 0⫺ ␮ 0N 0
.
T0
The total energy, the total volume and the total particle number of the bodies, and the environment together are constant,
n
兺
i⫽0
n
E i ⫽const,
兺
i⫽0
n
V i ⫽const,
兺 N i ⫽const.
i⫽0
Let us suppose that the temperature T a , the pressure P a 共thus the chemical potential ␮ a , too兲,
in the given process of the environment are constant 共which is a necessary condition of the
existence of equilibrium if at least one of the dynamical quantities is not independent of the
environment兲. Then
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2040
J. Math. Phys., Vol. 41, No. 4, April 2000
冉
n
Lª
兺
i⫽1
S i⫺
T. Matolcsi
E i⫹ P aV i⫺ ␮ aN i
Ta
冊
is the total entropy of the bodies and the environment together, up to an additive constant.
Using the notations introduced previously, we can write the above function in the form
n
L共 x 兲⫽
兺 „Si共 x i 兲 ⫺y a •x i …
i⫽1
共47兲
共 x苸X D 兲 .
Further on we refer frequently to the set of states corresponding to the regular domains:
n
X R ª X 共 R⫹ * R i 兲 .
共48兲
i⫽1
Suppose the bodies of the thermodynamical system are entropic. Then L is twice differentiable
on X R ; the derivative of L equals the nominal thermodynamical force,
DL 共 x 兲 ⫽F 共 x 兲 ,
共49兲
for x苸X R . Moreover,
n
D L共 x 兲⫽
2
兺 D2 Si共 x i 兲
共50兲
i⫽1
is negative semidefinite; its kernel is spanned by the vectors (x 1 ,0,0,...,0),
(0,x 2 ,0,...,0)¯(0,0,0,...,x n ).
Proposition 13: Let U be a constraint manifold and let us apply the previously introduced
notations. Let x 0 be an equilibrium in U艚X R . If (i) the bodies are entropic; (ii) L has a strict
local maximum at x 0 with condition U, then x 0 is asymptotically stable with condition U.
Proof: Condition 共ii兲 implies that x 0 has a neighborhood in which F ⌫ (x)⫽⫽DL(x) 兩 T x(U)
•
⫽0 for x 0 ⫽x苸U. As a consequence, the derivative of L along the dynamical law, L ⫽(DL)R
⫽FR⫽F ⌫ R has a strict local minimum at x 0 by 共43兲; hence we infer the desired result from
Proposition 11.
䊐
Next, we give easily verifiable relations, which imply condition 共ii兲 and will be useful in
applications.
Proposition 14: Let the constraint manifold U be a C 2 submanifold. Suppose x 0 is an equilibrium in U艚X R and 共i兲 Ker„D2 L(x 0 )…艚T x 0 共U兲⫽0; (ii) there is a parametrization of U around
x 0 such that DL(x 0 )D2 p„p ⫺1 (x 0 )… is negative semidefinite; then L has a strict local maximum at
x 0 with condition U.
Proof: The function ⌳ªLⴰp is twice differentiable, and
D⌳ 共 ␰ 兲 ⫽DL„p 共 ␰ 兲 …Dp 共 ␰ 兲 ,
共51兲
D2 ⌳ 共 ␰ 兲 ⫽D2 L„p 共 ␰ 兲 …ⴰ„Dp 共 ␰ 兲 ⫻Dp 共 ␰ 兲 …⫹DL„p 共 ␰ 兲 …D2 p 共 ␰ 兲 ,
共52兲
for all ␰ 苸Dom p.
Put ␰ 0 ª p ⫺1 (x 0 ). Since x 0 is an equilibrium, we have DL(x 0 ) 兩 T x
0
(U) ⫽0,
consequently
D⌳( ␰ 0 )⫽0. Moreover, at ␰ ⫽ ␰ 0 , the first term on the right-hand side of 共52兲 is negative definite
by 共i兲; the second one is negative semidefinite by 共ii兲, so the sum is negative definite.
As a consequence, ⌳ has a strict maximum at ␰ 0 that is equivalent to that L has a strict
maximum at x 0 with condition U.
䊐
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J. Math. Phys., Vol. 41, No. 4, April 2000
On the mathematical structure of thermodynamics
2041
Property 共ii兲 in the previous proposition holds if U is a subset of an affine subspace; then p can
be taken to be affine and its second derivative is zero.
Proposition 13 can be applied when every body has a constant particle number. Indeed, then
the third, sixth, ninth, etc. components of the elements in K(x) are zero, thus only the zero is both
in K(x) and in the kernel of D2 L(x). A similar assertion holds when everybody has constant
volume.
Asymptotic stability of an equilibrium implies that the equilibrium is locally unique. In general, e.g., in the first-order phase transitions and in diffusions when the particle number and the
volume of the bodies changes, equilibria are not locally unique; they constitute a nonzero dimensional submanifold.37 Then instead of the previous proposition we can apply the following one.
Proposition 15: Let the constraint manifold U be a C 2 submanifold and let E be the set of
equilibria in U艚X R . If (i) the bodies are entropic, (ii) the dynamical quantities are quasilinear
and the nominal conductivity matrix fits the constraint, (iii) E is a submanifold; and for all x 0
苸E, (iv) B ⌫ (x 0 ) is symmetric and positive definite, (v) Ker„D2 L(x 0 )…艚T x 0 (U)⫽T x 0 (E), (vi)
there is a parametrization p of U around x 0 such that DL(x 0 )D2 p„p ⫺1 (x 0 )… is negative semidefinite; then E is strictly asymptotically stable with condition U.
Proof: The dynamical equation is of the form
ẋ⫽B ⌫ 共 x 兲共 DL 兲 ⌫ 共 x 兲
and
E⫽ 兵 x苸U 兩 共 DL 兲 ⌫ 共 x 兲 ⫽0 其 .
Let x 0 be an arbitrary element of E and let p be a parametrization of U around x 0 . Then for
the function ⌳ªLⴰp and ␰ 0 ª p ⫺1 (x 0 ) we have the equalities 共51兲 and 共52兲.
The dynamical equation reduced by the parametrization becomes
␰˙ ⫽⌿ 共 ␰ 兲 D⌳ 共 ␰ 兲 ,
共53兲
where
⌿ 共 ␰ 兲 ªDp 共 ␰ 兲 ⫺1 B ⌫ „p 共 ␰ 兲 …„Dp 共 ␰ 兲 * …⫺1 .
The set of equilibria of the reduced dynamical equation is
␾ ª 兵 ␰ 苸Dom p 兩 D⌳ 共 ␰ 兲 ⫽0 其 .
共54兲
Now we show that
T ␰ 0 共 ␾ 兲 ⫽Ker D2 ⌳ 共 ␰ 0 兲 .
The relation 傺 follows from 共54兲 trivially. The relation 傻 can be verified as follows: if p has
property 共vi兲 and v is in the kernel of D2 ⌳( ␰ 0 ), then 共52兲 implies that Dp( ␰ 0 ) v 苸T p( ␰ 0 ) (U) is in
the kernel of D2 L(x 0 ), thus it is an element of T x 0 (E) according to property 共v兲; this is equivalent
to that v 苸T ␰ 0 ( ␾ ).
The derivative of the right-hand side of the reduced dynamical equation 共53兲 at ␰ 0 equals
⌿ 共 ␰ 0 兲 D2 ⌳ 共 ␰ 0 兲 .
Property 共iv兲 implies that ⌿( ␰ 0 ) is symmetric and positive definite, and according to 共52兲 and
property 共vi兲, D2 ⌳( ␰ 0 ) 共which is necessarily symmetric兲 is negative semidefinite. Therefore the
kernel of ⌿( ␰ 0 )D2 ⌳( ␰ 0 ) equals the kernel of D2 ⌳( ␰ 0 ), which is the tangent space of ␾ at ␰ 0 ;
and38 the geometric multiplicity and the algebraic multiplicity of the zero eigenvalue of
⌿( ␰ 0 )D2 ⌳( ␰ 0 ) coincide, all the nonzero eigenvalues of ⌿( ␰ 0 )D2 ⌳( ␰ 0 ) are negative.
As a consequence of Proposition 12, the set of equilibria of the reduced dynamical equation is
strictly asymptotically stable 共without a condition兲, which implies that E is strictly asymptotically
stable with condition U.
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J. Math. Phys., Vol. 41, No. 4, April 2000
T. Matolcsi
Property 共iv兲 in the previous proposition is equivalent to 共see Proposition 10兲 共iv兲 B(x 0 ) is
symmetric, positive semidefinite and KerB(x 0 )⫽T x 0 (U) ⴰ , which is very useful in practice.
VI. DISCUSSION
A general mathematical theory of thermodynamical systems consisting of single-component
homogenous bodies has been expounded. Substances, phases, bodies, systems, thermodynamical
forces, constraints, etc., all the usual notions have got a mathematical definition. The basic object
is the dynamical law, a differential equation whose solutions are the processes of the system.
Equilibria are the constant processes. The first law is a part of the dynamical law 共as in continuum
thermodynamics, where the first law is one of the balance equations兲; the second law is expressed
by a condition imposed on the dynamical quantities, called the dissipative property 共an analogon
of the Gibbs–Duhem relation in continuum thermodynamics兲; the dissipative property is something like positive entropy production but is formulated without the notion of entropy and can be
applied for some systems of nonentropic bodies, too.39 Constraints can be treated by the notions of
manifolds. The Onsager formalism is a well-defined linear approximation of the dynamical law
around equilibrium if and only if the constraint is affine. General results on 共strict兲 asymptotic
stability 共trend to equilibrium兲 are obtained for a system of entropic bodies.
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See Ref. 12.
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See Ref. 18.
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See Refs. 14 and 15.
1
2
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