Thermodynamic Stability - A note on a footnote in Ruelle's book
UWThPh-2007-09
Thermodynamic Stability – A note on a
footnote in Ruelle’s book
Bernhard Baumgartner1
Institut für Theoretische Physik, Universität Wien
Boltzmanngasse 5, A-1090 Vienna, Austria
April 3, 2007
Abstract
Thermodynamic stable interaction pair potentials which are not of the
form “positive function + real continuous function of positive type” are
presented in dimension one. Construction of such a potential in dimension
two is sketched. These constructions use only elementary calculations. The
mathematical background is discussed separately.
PACS numbers: 05.20.-y, 02.20.-a, 02.40.Ft
Keywords: thermodynamic stability, convex cone
1 Introduction
In Ruelle’s book [R69] on statistical mechanics, in section 3.2 concerning one
species of classical particles in Rν , you can read:
1 PROPOSITION. If the pair potential Φ can be written in the form
Φ = Φ1 + Φ2 (1)
where Φ1 is positive, and Φ2 is a real continuous function of positive type, then
Φ is stable.
1Bernhard.Baumgartner@univie.ac.at
http://arxiv.org/abs/0704.0417v1
TDS BB April 3, 2007 2
“Positive” is meant here and throughout this paper as nowhere negative,
“stable” means
∃E0 ∈ R such that ∀N, ∀{x1...xN} ⊂ Rν : U(x1 · · ·xN) ≧ N · E0, (2)
where U(x1 · · ·xN ) =
i 6=j
Φ(xj − xi). (3)
This proposition is accompanied by the
2 FOOTNOTE. It seems to be an open problem to construct a stable potential
which is not of the form (1).
We solve this problem in dimension 1, considering particles either in Z or in
R, giving a detailed proof. In dimension 2 the problem can also be solved, but
we give only a sketch of the ideas. 1
To make it simple, we consider only pair potentials which are bounded con-
tinuous functions and state the stability property as
3 DEFINITION. A bounded continuous real valued function V on Rν is stable,
E(ρ) :=
ρ(x)V (x− y)ρ(y)dνx dνy ≥ 0 (4)
for every positive finite measure ρ(x)dνx on Rν. A bounded real valued function
V on Zν is stable, if
E(ρ) :=
ρ(~m)V (~m− ~n)ρ(~n) ≥ 0 (5)
for every positive bounded function ρ(~m) on Zν .
The stability property used in Ruelle’s Theorem is an immediate consequence.
With ρ =
i=1 δ(xi − xj) put into equation (4) one gets
U(x1 · · ·xN ) = E(ρ)−N · V (0) ≥ −N · V (0).
The main result of our considerations is stated as
4 THEOREM. Each of the following functions is a stable pair potential, but
not a sum of a positive and a real valued positive definite function.
1. The function V : Z → R, defined as
V (0) = V (2) = V (−2) = 1, V (1) = V (−1) = −1, (6)
V (n) = 0 ∀n with |n| ≥ 3,
2. The function W : R → R, defined as
W (x) =
V (n)f(n− x+ y)f(y)dy, (7)
with f a positive continuous function (−1
) → R and V as defined in (6).
1 Construction in higher dimensions is still an open problem.
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2 Properties of the interaction potentials
Proof. Of part (1) of 4 Theorem.
Denote the distribution of particles on the chain by the “density” ρ, a function
Z → Z+. The interaction energy U becomes smaller, when the system is cut into
non-interacting pieces: If ρ(n) ≥ ρ(n+1) divide the chain, cutting between n+1
and n+ 2. Moving the pieces apart, one looses the energy
2[ρ(n)− ρ(n + 1)]ρ(n+ 2) + 2ρ(n+ 1)ρ(n + 3) ≥ 0.
The symmetric procedure of cutting between n − 2 and n − 1 lowers the energy
if ρ(n− 1) ≤ ρ(n).
Now there remains a set of pieces of no more than three lattice points, with
densities like
0 ≤ ρ(n− 1) ≤ ρ(n) ≥ ρ(n + 1) ≥ 0.
Including the “self-energies” N · V (0) one gets for each piece, centered around n,
E = ρ(n−1)2+ρ(n)2+ρ(n+1)2+2[ρ(n−1)ρ(n+1)−ρ(n−1)ρ(n)−ρ(n)ρ(n+1)]
= [ρ(n− 1)− ρ(n) + ρ(n+ 1)]2 ≥ 0.
Proving the stability of V .
If V were the sum of a positive and a positive definite function, it would give
V (n)µ(n) ≥ 0, (8)
for each µ being both positive and positive definite. Now consider
µ(5ν) = 1, µ(5ν ± 1) =
, µ(5ν ± 2) = 0, (9)
which is obviously positive. Positive definiteness is seen by using Bochner’s the-
orem [RN55] and calculating the Fourier-Transform, with α ∈ (−π,+π]:
µ̂(α) =
µ(n)e−inα
5 δ(α) +
),+δ(α+
> 0. (10)
But it does not give a positive value in (8):
V (n)µ(n) = 2−
5 < 0.
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The appearance of the numbers 5 and
5 may seem mysterious. Demystifying
is the next section, where we present the “origin” of these V and µ.
In this section we develop further use of these functions in R and in R2.
Proof. Of part (2) of 4 Theorem.
For N particles at x1 . . . xN consider the measure
ρ(x) =
δ(x− xj). (11)
Adding the self-energies N ·W (0), we study
ρ(x)W (x− y)ρ(y) dxdy
V (n)
ρf (x+ n)ρf (x) dx, (12)
with ρf(x) :=
f(x − y)ρ(y) dy. Splitting the integral in (12) into pieces of
intervals with unit length and defining ρf,x(m) = ρf(x+m) gives
V (n)ρf (x+m+ n)ρf(x+m)
ρf,x(p)V (p−m)ρf,x(m) ≥ 0,
by part (1) of the theorem. So the potential W is stable.
Now consider the distribution
µD(x) =
µ(m)δ(x−m), (13)
using the sequence µ defined in (9). This distribution is positive and positive
definite, as can be seen at its Fourier transform, which is (up to a factor) the
same as in (10), now with µ̂D(α+2π) = µ̂D(α) periodically extended to all α ∈ R.
This µD is used to show that the potential is not a sum of positive and positive
definite functions:
W (x)µD(x)dx
V (n)
∫ + 1
dx δ(x−m)f(n− x+ y)f(y)
V (n)µ(n) ·
f 2(y)dy < 0. (14)
In the last step the final support of f is essential.
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Construction of a stable pair potential in R2 being a function of the particle
distances only may be done in the following way:
• Use W (x) defined in (7), now with an f supported on (−1
), convolute it
twice with the distribution
h(x) =
e−ǫ|n|δ(x− 5n) :
W1(x) =
h(x− y)W (y − z)h(z)dy dz.
• Take the mean value (times 2π) of all rotated versions: Wr(~x) = 1rW1(|~x|).
• Smoothen out Wr with a positive continuous function g(r) with support on
[0, 1
W2(~x) =
g(|~x− ~y|)
W1(|~y − ~z|)
|~y − ~z|
g(|~z|)d2y d2z.
That the stability is not destroyed by the double convolution with h follows
from a consideration as it is used in the equation (12). Written in a formal way:
〈ρ|W1 |ρ〉 = 〈ρ| h ∗W ∗ h− |ρ〉 = 〈ρ ∗ h|W |ρ ∗ h〉.
Considering only smooth densities ρ(~x) one may take W1(x1)δ(x2) as a stable
distribution in R2:
〈ρ|W1 · δ |ρ〉dim=2 =
〈ρy|W1 |ρy〉dim=1 d y ≥ 0.
Now rotating the axes and taking the mean value does not destroy the stability.
Once more a double convolution is done, now with g in order to get W2 as a
bounded continuous potential acting in R2.
〈ρ|W2 |ρ〉 = 〈ρ| g ∗W ∗ g− |ρ〉 = 〈ρ ∗ g|W |ρ ∗ g〉 ≥ 0.
Smoothing by convolution with g enables to consider again sets of particles rep-
resented by delta-functions in ρ.
To disprove the possibility of splitting W2 into a sum of a positive and a
positive definite function one may use the µD of equ. (13) embedded into R
µD(x, y) = µD(x)δ(y).
Due to the smoothing of Wr by g and due to its decrease given by the decrease
of h, the integral
W2µD is finite:
W2µD(x) dx = W2(0) + 2W2(1)µ(1) + 2 ·
W2(|5ν + n|)µ(n)
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The bounded support of f and g is needed here as it was in equ. (14). The
exponential decrease implies
W2(|5ν + n|) = const. · e−5ǫ ν
V (n) ·
1 +O(
The “const.” factor involves the integrals over f 2 and g2, the error term O( 1
gives the difference between e−5ǫ ν/5ν and e−ǫ (5ν+n)/(5ν + n). The summations
over ν and n give
≈ 2 · const. ·
V (n)µ(n) · log(1/ǫ) +O
e−5ǫ ν
The first part is negative and increases without limit when ǫ → 0, while the other
term remains finite. So W2 with small ǫ can not be a sum of positive and positive
definite functions.
3 Mathematical background
Only in applying Proposition 1 in statistical mechanics the Thermodynamic Limit
is considered, not yet in the investigations of “stability”. Moreover, in the re-
formulation in 3 Definition there is no mentioning of “particles”. What is used
of properties of space are: A distance relation between points and an invariant
measure. This allows for a more general version of the definition, concerning
functions on groups. We keep the notation we used above: x and y are elements
of the group, their “group product” is x+ y, the “inverse” of x is −x.
5 DEFINITION. Consider a bounded continuous real valued function V on a
locally compact abelian group G which has the Haar measure dx. V is stable, if
〈ρ|V |ρ〉 :=
ρ(x)V (x− y)ρ(y)dx dy ≥ 0 (15)
for every finite positive Borel measure ρ(x)dx.
Stable functions can be added, multiplied by positive numbers, and limits
may be formed. So they form a closed convex cone, which we call STB. This
cone STB contains POS, the cone of positive functions, also PDF, the cone of
positive definite functions and sums thereof.
STB ⊃ POS + PDF (16)
An investigation of the relations between these cones may proceed via investi-
gation of the dual cones (see [V64, R62, G03]). The dual cones are subsets of
V ′, the space of finite Borel measures µ(x)dx, which is the dual space to V, the
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Banach space of bounded continuous functions. The dual cone to POS is POS′,
the set of finite positive Borel measures, dual to PDF is PDF′, the set of finite
positive definite Borel measures. The cone STB′ is given as the closure of the
cone of convex combinations of “correlation measures”
µ(x) =
ρ(x)ρ(y + x)dy, (17)
i.e. convolutions of finite positive Borel measures ρ(x)dx with their reflected ver-
sion ρ(−x)dx. These correlation measures are both positive and positive definite:
STB′ ⊂ POS′ ∩ PDF′ (18)
Now the question of equality or inequality in this relation is related to the
central problem which is our concern in this investigation, the question of equality
or inequality in (16). If the closed cone POS′∩PDF′ contains an element µ which
is not in the closed cone STB′, then, by definition of “dual cone”, there exists
an element V ∈ STB such that
V µ < 0, incompatible with a decompostion
V = f + g, f ∈ POS, g ∈ PDF.
For the groups Z2, Z3, Z4 there is equality in the equations (16) and (18), but
not for Z5.
6 PROPOSITION. The intersection of POS′ ∩ PDF′ with the plane
{(µ(−2) . . . µ(2))|µ(0) = 1} is completely characterized by its extremal points
(0, 0, 1, 0, 0), (0, γ, 1, γ, 0), (γ, 0, 1, 0, γ), (1, 1, 1, 1, 1), with γ = (
5 − 1)/2 =
1/(2| cos 4π/5|).
Proof. By using Bochner’s theorem and analyzing the Fourier transform
µ̂(k) =
µ(n)e−2π k n/5. (19)
On the other hand there is a bound for STB′ which cuts off a triangular subset
of this convex quadrangle:
7 LEMMA. Each element of STB′ obeys the inequality
µ(1) ≤
µ(n)/4. (20)
Proof. STB′ is defined by its extremal rays, formed as correlation measures of
positive densities.
µ ∈ STB′, µ extremal ⇔ ∃ρ ≥ 0, µ(n) =
ρ(m)ρ(m + n).
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Assume, w.l.o.g., that ρ(−1) ≥ ρ(−2). then
µ(1) = [ρ(−1) + ρ(1)] · [ρ(−2) + ρ(0) + ρ(2)]− [ρ(−1)− ρ(−2)] ρ(2)− ρ(−2)ρ(1)
− x)(s
+ x) ≤ s
Here s =
m ρ(m), x = [ρ(−2) + ρ(0) + ρ(2)− ρ(−1)− ρ(1)] /2.
Observe
n µ(n) = s
Remark: Also µ(2) obeys this inequality and µ(−1) = µ(1), µ(−2) = µ(2).
Closer inspection reveals moreover two rounded edges of STB′.
Now the extremal point with µ(n) as in equation (9) with ν = 0 is outside
this boundary. And V (n) as in equation (6) is an element of STB, but outside of
POS+PDF.
4 Conclusion
For pair potentials which are bounded continuous functions the property of being
“stable” can be reformulated without mention of particles. In this way it can be
studied for abstract abelian groups. At the heart of the present investigation is
the observation of a function V in Z5 which is stable, but indecomposable into a
sum of positive and positive definite functions. This function V can also be used
on Z. With some smoothing it can be used on R, and in damped periodically
extended, rotationally symmetrized and again smoothed form on R2. Of course
it is possible find sets of other examples nearby. So V (−1) = V (1) in Theorem 4
could be a little bit higher than −1. Only at −(
5+1)/4 ≈ −0.8 does it become
decomposable.
The construction of a rotationally invariant example for dimension two is not
so simple. A nicer one, or one for higher dimension, is not yet known.
References
[R69] D. Ruelle: Statistical Mechanics: Rigorous Results (W. A. Benjamin, inc.,
New York) 1969.
[RN55] F. Riesz and B. Sz. Nagy: Functional Analysis (Ungar, New York) 1955
[V64] Frederick A. Valentine: Convex sets (McGraw-Hill, NY (McGraw-Hill se-
ries in higher mathematics)) 1964
[R62] W. Rudin: Fourier Analysis on Groups (Interscience, New York) 1962
[G03] “Convex cones and their faces” Chapter 3 in: H. Glöckner: Positive Def-
inite Functions on Infinite-Dimensional Convex Cones; Memoirs AMS,
166, Number 789, 2003
	Introduction
	Properties of the interaction potentials
	Mathematical background
	Conclusion
