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Further remarks on convexity of thermodynamic functions
Physica
42 (1969)
242-244
0 North-Holland
LETTER
FURTHER
Publishing
TO THE EDITOR
REMARKS
ON CONVEXITY
THERMODYNAMIC L. Istituto
di Fisica
Euratom,
Ispra,
Italia,
GALGANI Milano,
Italia
SCOTT1
and Istituto
Received
OF
FUNCTIONS*
dell’Unzversit& A.
C.C.R.
Co., Amsterdam
di Fisica
2 September
Milano,
dell’ Universitd,
Italia
1968
Synopsis Convexity deduced
properties
from
Legendre
convexity
transforms.
of
thermodynamic
properties
potentials
of energy
Then corresponding
and
and entropy
extremum
Massieu
functions
are
by means of a lemma
properties
follow
on
via extensivity.
In a previous letter 1) we established formally the connection between the extremum and the convexity properties of entropy and energy; this geometrization of the second principle allowed us in addition to deduce the minimum property of energy from the maximum property of entropy. The task of obtaining analogous convexity and extremum properties for all other thermodynamic functions immediately presents itself: we are going to accomplish it in the present note making use of the following Lemma: the Legendre transformation on a completely convex or concave function leaves unchanged the convexity properties in the variables which are not transformed. To prove this lemma let us consider, for definiteness, a completely convex function of two variables, i.e. a function f(x, y) that satisfies the following inequality : f{B(Xl
+
X2)> &(Yl +
Y2,> I
6WJ
Yl)
+
/(x2>
Yz));
(1)
then the Legendre transform (for example with respect to x) g(~ y) is defined2) bY
g(w y) = inf{f(x, y) - ~4, r
* Work supported in part by grant from the C.N.R. (Consiglio Ricerche) to the “Gruppo di Meccanica Statistica e Teoria dei Molti 242
Nazionale Corpi”.
delle
CONVEXITY
OF THERMODYNAMIC
FUNCTIONS
243
where (~//EJx)+,_is the right and the left partial derivative, respectively. noticing
Now,
that for any given zt, yr, ys there exist XI, x2 such that
inf{f(x, yr) -
+
= /(xl, yr) -
uxr, inf{&
y2) -
4
=/(x2,
~2) -
f4~2
z
z
and that inf{@,
B(yr + ~2)) - ux} I f(x, 4(y1 + y2)) - ux,
2
for any x, whe have, choosing x = 4(x1 + 24 and using eq. (l), g(~ +(yl
+
~2))
=
inf{f(x,
Hyl
I
ricx1
+
I
t{f(xl,
=
&(%
+
y2))
-
+
x2),
i(Yl
+
Yl)
-
@Xl
+
Yl)
+
g(f4
Yz)}.
5
Y2))
-
4(x1
/(x2, y2) -
+
x2) I
24x2> =
Analogous conclusions can be reached for a completely concave function, for which, of course, the Legendre transformation is defined trough sup instead of inf. Recall now the following properties and definitions: i) the Legendre transformation inverts the convexity properties on the transformed variablesa) ; ii) energy (entropy) is a completely convex (concave) functionr); iii) its Legendre transforms are called thermodynamic potentials (Massieu functions) 4) ; iiii) the new variables are intensive. We may then state the following Theorem: the thermodynamic potentials (Massieu functions) are convex (concave) in the extensive variables and concave (convex) in the intensive ones. For example, the free energy F(T, V, IV) = inf{U(S, 9
TS)
V, N) -
satisfies the inequalities F{T, !#‘I
+ v2), WI
F{+(Tl
T2), V", N>
+
N2)) _( i{F(T,
'VI, NI)
+
F(T,
v2, Nz)}, (2)
Noticing
+
2
:{F(L
'v, N)
+
F(T2,
in addition that, from the extensivity
F(T, IV, IN) = inf{U(ilS, IV, IN) -
J"',N)}.
of U, one has
TIS} = ilF(T, V, N),
S
it follows also that F is subadditive
in the extensive variables:
F(T, VI + 'v2, N1 + N2) < F(T,
VLNI)
So in general we may state the following
+
F(T,
1"2, N2).
244
CONVEXITY
OF THERMODYNAMIC
FUNCTIONS
: the thermodynamic potentials (Massieu functions) are subadditive (superadditive) in the extensive variables; that is to say, they possess extremum properties in these variables. In conclusion we note that convexity relations such as relations more general than the usual thermodynamic stability conditions
(2) are
which indeed follow on the assumption that second derivatives exist; such more general convexity relations could be proven up to now, as far as we know, only on the basis of statistical mechanics considerations5).
REFERENCES
1) Galgani, L. and Scotti, A., Physica 40 (1968) 150. 4 Griffiths, R. B., J. math. Phys. 6 (1965) 1447.
3) Mandelbrojt, S., Compt. Rend. (Paris) 209 (1939) 977. 4) Callen, H. B., Thermodynamics (Wiley, New York, 1960) p 101. 5) see ch. 3.1, Stability and convexity, p. 644 of Fisher, M. E., Rep. Progr. Phys. 30 (1967) 615.
42 (1969)
242-244
0 North-Holland
LETTER
FURTHER
Publishing
TO THE EDITOR
REMARKS
ON CONVEXITY
THERMODYNAMIC L. Istituto
di Fisica
Euratom,
Ispra,
Italia,
GALGANI Milano,
Italia
SCOTT1
and Istituto
Received
OF
FUNCTIONS*
dell’Unzversit& A.
C.C.R.
Co., Amsterdam
di Fisica
2 September
Milano,
dell’ Universitd,
Italia
1968
Synopsis Convexity deduced
properties
from
Legendre
convexity
transforms.
of
thermodynamic
properties
potentials
of energy
Then corresponding
and
and entropy
extremum
Massieu
functions
are
by means of a lemma
properties
follow
on
via extensivity.
In a previous letter 1) we established formally the connection between the extremum and the convexity properties of entropy and energy; this geometrization of the second principle allowed us in addition to deduce the minimum property of energy from the maximum property of entropy. The task of obtaining analogous convexity and extremum properties for all other thermodynamic functions immediately presents itself: we are going to accomplish it in the present note making use of the following Lemma: the Legendre transformation on a completely convex or concave function leaves unchanged the convexity properties in the variables which are not transformed. To prove this lemma let us consider, for definiteness, a completely convex function of two variables, i.e. a function f(x, y) that satisfies the following inequality : f{B(Xl
+
X2)> &(Yl +
Y2,> I
6WJ
Yl)
+
/(x2>
Yz));
(1)
then the Legendre transform (for example with respect to x) g(~ y) is defined2) bY
g(w y) = inf{f(x, y) - ~4, r
* Work supported in part by grant from the C.N.R. (Consiglio Ricerche) to the “Gruppo di Meccanica Statistica e Teoria dei Molti 242
Nazionale Corpi”.
delle
CONVEXITY
OF THERMODYNAMIC
FUNCTIONS
243
where (~//EJx)+,_is the right and the left partial derivative, respectively. noticing
Now,
that for any given zt, yr, ys there exist XI, x2 such that
inf{f(x, yr) -
+
= /(xl, yr) -
uxr, inf{&
y2) -
4
=/(x2,
~2) -
f4~2
z
z
and that inf{@,
B(yr + ~2)) - ux} I f(x, 4(y1 + y2)) - ux,
2
for any x, whe have, choosing x = 4(x1 + 24 and using eq. (l), g(~ +(yl
+
~2))
=
inf{f(x,
Hyl
I
ricx1
+
I
t{f(xl,
=
&(%
+
y2))
-
+
x2),
i(Yl
+
Yl)
-
@Xl
+
Yl)
+
g(f4
Yz)}.
5
Y2))
-
4(x1
/(x2, y2) -
+
x2) I
24x2> =
Analogous conclusions can be reached for a completely concave function, for which, of course, the Legendre transformation is defined trough sup instead of inf. Recall now the following properties and definitions: i) the Legendre transformation inverts the convexity properties on the transformed variablesa) ; ii) energy (entropy) is a completely convex (concave) functionr); iii) its Legendre transforms are called thermodynamic potentials (Massieu functions) 4) ; iiii) the new variables are intensive. We may then state the following Theorem: the thermodynamic potentials (Massieu functions) are convex (concave) in the extensive variables and concave (convex) in the intensive ones. For example, the free energy F(T, V, IV) = inf{U(S, 9
TS)
V, N) -
satisfies the inequalities F{T, !#‘I
+ v2), WI
F{+(Tl
T2), V", N>
+
N2)) _( i{F(T,
'VI, NI)
+
F(T,
v2, Nz)}, (2)
Noticing
+
2
:{F(L
'v, N)
+
F(T2,
in addition that, from the extensivity
F(T, IV, IN) = inf{U(ilS, IV, IN) -
J"',N)}.
of U, one has
TIS} = ilF(T, V, N),
S
it follows also that F is subadditive
in the extensive variables:
F(T, VI + 'v2, N1 + N2) < F(T,
VLNI)
So in general we may state the following
+
F(T,
1"2, N2).
244
CONVEXITY
OF THERMODYNAMIC
FUNCTIONS
: the thermodynamic potentials (Massieu functions) are subadditive (superadditive) in the extensive variables; that is to say, they possess extremum properties in these variables. In conclusion we note that convexity relations such as relations more general than the usual thermodynamic stability conditions
(2) are
which indeed follow on the assumption that second derivatives exist; such more general convexity relations could be proven up to now, as far as we know, only on the basis of statistical mechanics considerations5).
REFERENCES
1) Galgani, L. and Scotti, A., Physica 40 (1968) 150. 4 Griffiths, R. B., J. math. Phys. 6 (1965) 1447.
3) Mandelbrojt, S., Compt. Rend. (Paris) 209 (1939) 977. 4) Callen, H. B., Thermodynamics (Wiley, New York, 1960) p 101. 5) see ch. 3.1, Stability and convexity, p. 644 of Fisher, M. E., Rep. Progr. Phys. 30 (1967) 615.