This document discusses the continuity of pressure in quantum statistical mechanics. It proves that under certain conditions on the pairwise interaction potential, the pressure in the canonical ensemble is a continuous function that satisfies a Lipschitz condition. Specifically, it shows that if the interaction potential is twice continuously differentiable and satisfies an additional inequality, then: 1) The pressure exists for all specific volumes and temperatures. 2) In any closed region of the specific volume-temperature plane, the pressure satisfies a Lipschitz inequality, meaning the difference in pressure between any two points is bounded by a constant times the difference in specific volume. 3) Sufficient conditions are given for the interaction potential to satisfy the required inequality, such as

1. ON THE EXISTENCE AND CONTINUITY OF THE
PRESSURE IN QUANTUM STATISTICAL MECHANICS
L. A. Pastur
It is shown that in the case of all three statistics (Maxwell-Boltzmann; Bose-Einstein, and
Fermi-Dirac) the pressure in the canonical ensemble is a continuous function that satisfies
a Lipschitz condition provided the pair interaction potential O(r) for r -> a (a -> 0 is the hard-
core radius) is a twice continuously differentiable function. Apart from the usual conditions
needed to ensure the existence of the thermodynamic limit, this function satisfies for some
e > 0 the further inequality
u,c(xl, X B>~0,
where r = r + e(2r~'(r)Lr247 '' (r)). Some sufficient conditions to be imposed on r
for this inequality to hold are given.
We consider a system of N identical pairwise interacting particles in a cube V of three-dimensional
space. The statistical properties of such a system are described by the canonical distribution
exp{-~H(N, V) + pN-V(N, 7)}, p > 0,
where H(N, V) is the Hamiltonian of the system and f(N, V) is the free energy per particle. For a fairly
large class of interaction potentials in both the classical and the quantum-mechanical case there exists the
limit[l]. Pin what follows we shall omit the argument fi, since the dependence on fi in the matter under con-
sideration is of no importance. In this formula and all that follows the cube and its volume are denoted by
the same letter V.]
lira [(N, V)= [(v, fi),
WINdy
which is a nondeereasing and convex function of v. One of the important conditions that must be imposed on
the interaction potential UN(X1, x2..... xN) in this case is the condition that it be stable. This requires the
existence of a nonnegative constant B such that
Up (x,, x...... x~) =~ q~(Ix~- xjl)/> -BN (1)
i<j
all N >-- 1 and all N, x2 .... xN [here r is the energy of the pair interaction between particles]. However,
the existence and even continuity of f(v) are not, in general, sufficient to construct a thermodynamics of
the system, since the pressure p(v) plays an important role (equation of state); the pressure is defined by
p(v) = ol / ov
and is assumed to be a continuous fimction and even one that satisfies a Lipsehitz condition.
It can be shown that in the framework of classical statistical mechanics this is indeed so [2, 3], and
moreover under virtually the same conditions on the interaction potential. However, on the transition to
the quantum-meehanicaI case the situation becomes more complicated since the methods used to prove this
in classical statistical mechanics cannot be applied directly in the quantum-mechanical situation. The dif-
ficulties that are encountered when these methods are transferred have so far been only partly overcome
and then only for Maxwell-Boltzmannand Bose-Einstein statistics [4].
Physicotechnical Institute of Low Temperatures, Academy of Sciences of the Ukrainian SSR. Trans-
lated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 14, No. 2, ppo 211-219, February, 1973.
Original article submitted December 20, 1971.
9 Consultants Bureau, a division of Plenum Publishing Corporation, 227 g:est ]7th Street, New York, Y. Y. 10011.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by an)" means,
electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. :.1
copy of this article is available from the publisher [or $i5.00.
157

2. In the present paper, we use a method that differs from that of [2-4] to prove the continuity of ~he
pressure. The conditions that we find for the interaction potential to satisfy are rather unusual and this is
evidently a shortcoming of the method. Its advantages, in our opinion, are its "insensitivity" to the form
of the quantum-mechanical statistics and its conceptional simplicity.
Note that in the classical case a similar method was proposed for the first time in [5].
To be specific we shall consider the case when null boundary conditions (which are the natural condi-
tions in quantum mechanics) are imposed at the cube boundary. One could also consider the cases of
Neumann boundary conditions or periodic conditions.
THEOREM. Suppose that the pair interaction potential r r >--0, is a twice continuously differenti-
able funetioneverywhere except, perhaps, in the neighborhood r -< a, a >- 0, of the origin and such that in
addition to (i) the following inequality is satisfied for some e > 0:
U~(xl,...,x~)=Z ~P(Ix,-x~I)>~:-BN, B~O, (2)
where r = r + e(2rr Then the pressure p(v) exists for all v > aa 3, fi > 0, (The constant
(~ is the specific volume of dense packing of spheres of unit radius in three-dimensional space. ) In every
closed finite subregion S of this part of the (v, fl) plane the pressure satisfies the inequality
lp(~,)-p(~41 <cl~,-~I (3)
with a constant C that, in general, depends on S.
Before we turn to the proof of the theorem, let us discuss the condition (2), which is obviously that
not only r but also the "modified" potential r be stable. Since the stability is largely determined by
the sufficiently rapid decrease at infinity and sufficiently rapid increase as the hard-core is approached, i.e.,
as the region r <- a, a -> 0, is approached, fulfilment of the condition (2) means roughly that the addition
a(2rr162 '') decreases at infinity not slower than r and increases not slower than r as r ~a. The
condition (2) can clearly be verified directly by establishing whether r satisfies any of the well-known
stability criteria. One such criteria is [6]
3~/~
5
~P(r) r2dr + ~ (I)z,/, (r) r z dr > O, (4)
4~ 0 5
where
or the simpler
(P(r)= in[ (I)(y), cl)~(r)= in[ (I)(9),
Iq~(r) l~<C/r'% r>~R~, n,>3;
q3(r) >~C/r% r<~Rz, n..>3, R~<R~.
(5)
For example, using (4), it is easy to show that condition (2) is satisfied by the Lennard-Jones potential:
cl)(r) =C~/r',--C2/f% C,>0, C2~>0, n~>n2>3
and the screened Coulomb potential
@(r)=e-~/r, a>0.
Using (4) and (5) and making some simple estimates, we can obtain different sufficient conditions that must
be satisfied by r for the inequality (2) to be satisfied; for example:
Condition 1:
Condition 2:
~(r)={ul(r)/f~, r>~R,, nt>5, v~)l<~C,(~) k=0, t,2,
u~(r)ir'% r<~R2, n2>3, v2 <~Cr-~, k=0,1,2.
~(r) -Cr-% n2>3, (-t)k~Pr k=0, t, 2, 3, r<R2,
]qb(r)] <~Cr-"', nl>3, (--t)kO(k)~<O for (--l)aOck)~>0,
k=0, t, 2,3, R~<r.
158

3. Condition 3: ~, r20 '', and rr are bounded for r >- 0 and have an integrable monotonic majorant, i.e.,
c~
there exists a positive nondecreasing function (p(r) such that q0(0) < ~, .! ~r2dr < ~ and I 9 [, r I ~'1 and r 2
0
I ~" J do not exceed ~0.
This case is evidently fairly general, since in the second-quantization formalism that is usually used
in quantum statistical mechanics the potential has a Fourier transform that occurs explicitly and is, as a
rule, a rapidly decreasing function.
An example of a potential for which the condition (2) can be verified by means of the last criterion is
the Morse potential:
<D(r) = C[e -2~(~-R~- 2e-~(T-R):], aR > t6.
A simple example of a potential with a hard-core that satisfies the Condition 2 is a function #(r) that tends
sufficiently rapidly and smoothly to zero at infinity (for exampie, a function of compact support or one that
satisfies one of the Conditions 1-3) and remains bounded with its two derivatives as r ~ a.
We now prove the theorem. The main stage in the proof is as follows.
LEMMA 1. Let V be a cube and
l~(v) -~ I(N, Nv), v > 0.
Then under the conditions of the theorem
~+~/N(v) =]x,(v+5)-2/~(v) +]~(v-5)>t--C5', 5>0, (6)
where the constant C can depend only on the region S. (It is, of course, assumed that v + 5 and v-5 are
contained in S. )
Indeed, suppose the lemma is proven. Since the limiting free energy per particle, f(v), is a convex
function [1], there exists for it the left-hand derivative fl'(V)' which is a nondecreasing function, i.e.
]l'(V,) ~ ]l'(V~), if Vt < V2. (7)
On the other hand, (6) means that the function flay) + Cv2/2, and therefore f(v) + Cv2/2 is concave.
Then it too has a left-hand derivative, f/'(v) + Cv that does not decrease, i.e.
h'(v,) + Cv, <~Iz' (v~) +Cv~, if v, < v~.
This inequality in conjunction with (7) proves (3).
We now deal with the formal side of the proof of Lemma 1. By definition
/,(v) = N-' In Sp e-~H(~'~~
where the Hamittonian of the system H(N, V) is defined in a Hilbert space of functions of the corresponding
symmetry of the differential operation
- ?_., § Z*('+,- +>
t i<$
and null boundary conditions on the surface of the cube V with respect to each of the variables x i, i = 1, 2
..... N. We go over to the new variables x i = L~ i, where L is the length of the edge of the cube V (L = V1/3).
In these new variables, each of which ranges over a cube V0 of unit dimensions, the free energy is written
in the form
]~(v) = N-' In Sp e-~h(m~),
where h(N, V) is an operator in the Hilbert space of functions corresponding to the symmetry defined by the
operation
N
- A~ y' (8)
i~ 1 i'<j
159

4. and null boundary conditions on the surface of the cube V0 with respect to each of the variables ~i, i = 1, 2
.... , N~ Using the operator identity
e-~(a+")= e-~A --" ~ e-~(a+S)Be-(~-~)~ dx
o
we now find that
9 _N...~lnSpe-~h(u,,~)= _ ~ dV2/dvz
t
+ 2 ~ dt ~ ds<e-~h(N.V~h,e-('-~)h(~'~)h~ethC"')),
o o
(9)
where
In (9) the second term, which contains the integrals, is nonnegative. This can be seen by writing,
for example, the trace in the basis of the eigenfunctions of the operator h(N, V) or by showing that this term
can be transformed to the form
where
hi (~) =~o e-'h'eth dt.
This circumstance enables us to write down the following equality [after calculating dh/dV and d2h/dV2
by means of (8) and going back to the old variables xi]:
d2t-----~N >1- ~v .-~- (tOT(N,, V) q- Wry),
dvz
where T(N, V) is the kinetic energy operator, i.e., -ZAxi, and
WN= Z {2IX~-- X~Itg' (IX,-- Xk[)--IX~-- XkV~" (Ix,-- XnI)}.
i<k
We now take into account the following inequality, which is due to Bogolyubov:
Sp eAB ~ In Sp eA+~- In Sp e~."
Sp e"~
Taking here -fill(N, V) as A and efl(10T + W) as B, we find that
_ _ i
d~lUdv~ >19v+S {/z(v)- [~(v)}, 1" (10)
where fN(V) = N-1 in Spe -/3H(N, V) for V = Nv, ~I(N, V) = H(N, V)-e(10T(N, V) + WN~ = (1-e)T(N, V) + 1~N,
i.e., fN(N, V) is the free energy per particle corresponding to the "Hamiltonian" ~t(N, V).
From (10) with allowance for the identity
v§ v
h~f~(v) = ~ (v+6--u)fN"dg+f (a-v+6)l,"da (11)
v v-5
it follows
A~2fN(v)>~ y (v+6--u)gN(a)du+ (tt--v+6)g~(a)du,
v v-5
(12)
*This inequality can be obtained if one proves by means of a formula analogous to (9) that the function F(t)
= Spe A + tB is concave [i.e., that F"(t) >- 0] and notes that in this case F'(0) <-- F(1)-F(0). In addition, it,
like other useful inequalities in quantum statistical physics, can be deduced from O. Klein's simple in-
equality [1].
Here and below the quantum and the classical forms of the proof are identical (see also [5]).
160

5. where
Further,
pression
t { v
~(v)=~ IN( )- f~(~)}.
since fN(v) and fN(v) are nondecreasing functions of v [1], gN(v) can be estimated below by the ex-
1
9v~ {/~(',) - f~ @2)},
where vt = min v, v2 = max v. By virtue of (2), fN(v2) is bounded above uniformIy with respect to N [1] and
@,~)ES @,~)6s
[since fN(v) --f(v) as N --~o] f(vl) is bounded below uniformly with respect to N. Therefore, for all vES
g~(v) i>-c,
where C for a fixed potential r can depend only on the region S. This inequality in conjunction with (125
obviously leads to (65.
It is readily seen that of the above arguments the inequality (12) requires strict proof; for in its deri-
vation we have used operations of differentiating the trace of a function of an unbounded operator, we have
reversed the order of the operations of integration and taking the trace, and so forth. It is obviously rather
difficult to prove directly that these operations are justified. Therefore, to derive the inequality (12) rigor-
ously, we shall consider some finite-dimensional approximations of the Hamiltonian H(N, V). For this we
require an auxiliary assertion, which it is convenient to formulate separately.
LEMMA 2. Let A be a positive-definite selfadjoint operator with discrete, finitely multiple spectrum
<- ~ ~ .... that extends to infinity; suppose A can be represented in the form A = At + A2, where Al is
an operator of the same kind as A with eigenvalues P'i and orthonormalized eigenfunctions ~0t, andA 2is a
bounded selfadjoint operator such that IIA2[1 < b~, where ~ = min ~i'
Then the sequence
n
--~i,nZ. = 7e
i=i
where kl. n -< k2, n -< ...kn. n are the eigenvalues of the matrix A (n) with elements (A(0i, ~0k), i, k = I, 2
9 n, ismonotonically nondecreasing and as n -- ~otends to
Z= Spe -A = --Ee-%
i=1
Proof. In accordance with [7], the numbers Xi, n are the n-th Ritz approximation of the eigenvalues
32. The Xi, n will converge to Xi as n --oo if the operator A and the system of "test" functions {~0i} satisfy
the following conditions [7]:
1) A -1 is completely continuous;
2) ~0i are linearly independent;
35 the {~0t} form a complete system in the energy space FA of the operator A, i.e., in the completion
of its domain of definition DA in the norm II(0 IIA = (A(o, ~o).
In our case, Conditions 1 and 2 are satisfied because of the unrestricted increase of the Xi and the
orthonormality of the ~0i, respectively. To verify the third condition, suppose otherwise, i.e., that in FA
there exists a nontrividlelement r such that [%, ~oi] = 0, i = 1, 2 ..... where [r (o], the scalar product in
F A, takes the form (~, Aq~ when at least one of the factors, for example, ~0, belongs to DA. But since
q0iEDA, i=l, 2 .....
[% q~] = (% Acp,), i=t, 2.....
or, with allowance for the conditions of the lemma,
~, (% ~) = - (A~r ~),
where ~i are the eigenvalues of A1 corresponding to the eigenvectors gap Multiplying this relation by its
complex conjugate and summing over all i, we obtain, since the {goi} are complex in the original space and
~i ~-~'
161

6. ,~:l[~ll: < ~IA:IIII~pJI=.
Since IIr II> 0, by hypothesis, it follows that ~ <- IIA211, and this contradicts the condition of the lemma.
Thus, ki, n -+ ~i for fixed i as n ~ =o. It is important that this convergence is monotonic, since [7]
~, ~+, ~ ~, ,.
Therefore, in the expression for Zn one can make the passage to the limit in the sum, and then one obtains
Z. The lemma is proven.
Proof of Lemma 1. As we have already pointed out, it all reduces to proving the inequality (12).
Note first that it is sufficient to prove this inequality for bounded, twice differentiable potentials +(r); for
if r has a singularity at the origin, one can construct a sequence of bounded smooth functions ~k(r) that
tend monotonically downwards to ~(r) as k ~o for any r > a. Then, writing down the representation of the
partition function Z(N, V) in terms of a Weiner integral [1], it is easy to show that it depends continuously
on the potential under such a limiting process.
Further, the inequality (12) is not affected if one adds to the Hamiltonian H(N, V) = T(N, V) + UN an
arbitrary constant ~. We choose ~in such a way that the minimal eigenvalue of the operator T(N, V) +
becomes strictly larger than the norm of the now bounded operator UN [it is sufficient to take ~ > sup
xi
UN (x1.... XN). Then Lemma 2 applies to H(N, V) + ~ [the role of the operators A1 and A 2 is played by
T(N, V) + ~ and UN, respectively]. Since T(N, V) is defined by the differential operation z_J~-~ h~, and null
l
boundary conditions, its orthonormalized eigenfunctions in the case of all three statistics can be taken in
the form
~(x,, x ...... zN) = g-Nn~(z, / V '1'..... x~, I V'l'),
X~.~ . . , , Xn C: V:
where • ..... ~N) are the orthonormalized eigenfunctions of the operator T(N, V0) (V0 is a cube of unit
dimensions) in the space of functions of the corresponding symmetry. Therefore, the elements (H(N, V) r
Ok)' i, k=l, 2 .... n, of the matrix H(n)(N, V) can be written in the form
(H (N, V)r *k) = V-'/'(T(N, Vo)~,, )~h)
-- f U~(V-'%, ..... V-%z,)d~,... d~N.
lro
This relation shows that H(n)(N, V) depends on V in the same way as the operator h(N, V) does in (8). Let
f(Nn)(v) be the logarithm of
Spexp {--[~H(")(N, V)}, V=Nv.
Clearly, (9) is perfectly correct when applied to finite-dimensional matrices. Therefore, the arguments
that lead from it to the inequality (10) become rigorous if we replace H(N, V) and ~I(N, V) everywhere in
them by the n-dimensional matrices H(n)(N, V) and ~I(n)(N, V) and iN(v) and iN(v) by f(n)(v) and ~)(v).
We then obtain the inequality
d~f~ '~ (v)
>/ {f<.")(v)- (v)}, (13)dv 2 9v~e
from which by virtue of (11) it follows that
v+hj i2,~(n) f (n) x O0A~ (v)>~ (v+6-~)gN (a)d~+ (u-v+5)gN (~)du.
v v-5
Now let n tend to infinity. On the left, since Lemma 2 applies to H(N, V), we obtain in the limit A<~2fN(V).
Further, if our main assumption (2) is valid, ~I(N, V) after the addition of a suitable constant also satisfies
the conditions of Lemma 2. Therefore, [~)(v), like f~)(v), tends monotonically downwards to f~N(V) as
n ~+o Therefore, one can make the passage to the limit on the right-hand side of (13) as well, and one
then obtains the inequality (12).
162

7. 1o
2.
3.
4.
5.
6.
7.
LITERATURE CITED
D. Ruelle, Statistical Mechanics, New York (1969).
R. L. Dobrushin and R. A. Minlos, Teoriya Veroyatnostei iee Primeneniya, 1__2,595 (1967).
D. Ruelle, Commun. Math. Phys., 1__88,127 (1970).
J. Ginibre, Phys. Rev. Lett., 2_4.4,1427 (1970).
J. Van der Linden, Physica, 3__88,173 (1968).
R. L. Dobrushin, Teoriya Veroyatinostei i ee Primeneniya, 9, 626 (1964).
S. G. Mikhlin, Variational Methods in Mathematical Physics [in Russian], Nauka (1970).
163