Z. Physik 235, 339-- 352 (1970)

Systems with Negative Specific Heat W. THIRRING*

CERN, Geneva, Switzerland

Received November 20, 1969

Some systems for which the binding energy increases more rapidly than linearly with the number of partieles, are shown to exhibit negative specific heat c for some energies. In thermal contact with larger systems, c < 0 creates an instability, and in the canonical ensemble one sees only a phase transition. It is argued that supernovae are, in essence, a phase transition of this origin.

1. A Surprising Theorem that is Simple to Prove, but which is Wrong

I t is a fact known to astrophysicists that if radiation energy is ex- tracted f rom a star whose nuclear fuel is exhausted, the star will contract and heat up. Thus a star acts like a system with negative specific heat**. There are proofs that under reasonably general assumptions the specific heat of interacting particles is positive. In this paper we shall try to clarify this perplexing situation.

The astrophysical arguments all depend on the virial theorem. Since the latter comes only f rom the 1/r behaviour of the potential and not f rom its attractiveness, one actually arrives at the more general Theorem 1.

Theorem 1. According to classical statistical mechanics, every piece of condensed matter has negative specific heat.

Before proving Theorem 1, we have to specify what we mean by condensed matter. Condensed means that the system keeps together, for E < 0 , without being put in a box, and thus we do not have to include the potential of the box in the Hamiltonian.

By matter we mean a system of N electrons and nuclei with only static Coulomb interactions. Thus the Hamiltonian of our system is

HN=K+V, (1)

the kinetic energy K being the usual N 2

* On leave from the University of Vienna, Vienna, Austria. ** The only reference to this fact that I could find is Landau, L. D., Lifchitz, E.:

Statistical physics, 2nd ed., p. 62. Oxford: Pergamon Press, and D. terHaar, Proc. Int. Conf. on Statistical Mechanics, Kyoto 1968.

23*

340 W. Thirring:

and the potential

V= ~ ei ej rtj (3)

Now the proof is trivial. If ( ) denotes the thermal expectation value, the virial theorems and equipartition tell us (k = 1)

( _ K ) = I (V)=(H)=_E= 3N2

Thus, the specific heat dE 3N

c - d T - 2 <0 , Q.E.D.

- - - T . ( 4 )

(5)

Let us first study this absurd result for the simplest case, namely one particle in a Coulomb potential:

p2 e2 H = 2m r (6)

Although one particle is not much of a thermodynamic system, for which the microcanonical ensemble could be used the above reasoning should also apply to it. The volume of the phase space under the energy surface (0 = step function)

f2(E) =5 d3 x d 3 p O(E-H) (7)

is finite for E < 0 and then easily calculated to be ,,~ ( - E ) --~.

This behaviour comes about since in x-space the volume available goes ,,~ (E)- 3, and from momentum space we get a factor (E) ~. Actually this also follows immediately in quantum theory where we have E = -Ry /n 2, and the number of states ~ below a quantum number n goes as n3"~lEl-~. Thus for the entropy S(E) we have

1 dS 3 S ( E ) = - l n ( - E ) , T - d E - 2E' (8)

in agreement with Eq. (5) for N = 1.

However, for a real thermodynamic system we need many particles, and then one soon realizes where the proof of Theorem 1 goes wrong. First of all we shall need a box, otherwise 0 will be infinite since we can always send one particle to infinity and lower the energy of the rest. Even in box

ON= 5 da~rSdaNpO(E-HN)(N!) -1 (9) b o x

Systems with Negative Specific Heat 341

diverges if we have several particles. Carrying out the p integration we have

e . e . \ 3N/2 ON,~fd3Nr E-- Z -*-J| , (10)

Fij /

where the integral goes over the region where E>~eie f l r i j . To have ~ # 0 for E < 0 we need an attractive pair (i,j), but then the integral diverges for small r , j since the integrand then goes as r;) 3me. Thus we have also to modify the potential at small distances to get a meaningful expression for ON for which the formal arguments of the proof apply. Therefore we have to take into account an external virial due to the box, and an internal virial due to a small distance repulsion. In the next section we shall try to estimate the effect of the former for the gravita- tional case (eiej ~ - xmimj) where we have only attractions, and there- fore the best chance for a small effect of the box. In Section 3 the question of the internal virial will be settled for a very truncated form of the potential for which O r can be calculated exactly.

To conclude this section we shall comment on the quantum theoretic problem. There ~N(E)=Number of states below E exists even without cut-off at small distances. The non-relativistic Hamiltonian (l) has for arbitrary N a lower eigenvalue Eo, in contradistinction to the relativistic case where we get a collapse for e~ei~-tcm~rn j and N sufficiently large. A recent analysis of Dyson 1 and collaborators has shown that I E o I ~ N is a rather exceptional situation for particular combinations of statistics and signs of e. In many cases ]Eol increases faster with N. We shall see later that this leads to interesting thermal consequences. Regarding Theorem 1, one might think that it applies rather to the quan- tum theoretic case, since there the problem of the internal virial dis- appears. However, the equipartition Theorem holds only in the classical limit, the virial theorem being generally true. Thus there seems to be no system for which the proof of Theorem 1 holds. Nevertheless we shall see that Theorem 1 reflects an essential feature of statistical mechanics for gravitationally interacting systems.

2. A Non-Rigorous Treatment of the Gravitational Case

In this section we shall estimate ~N for ei e J ~--~: by a method which has some intuitive appeal and may be appropriate for (unshielded) long-range forces. The idea is that the main part of the gravitational potential that a particle feels comes mainly from the bulk of the particles at large distances rather than from its immediate neighbours. Thus we shall divide the volume V of the box in M ~ N cells of equal size, large

1 Dyson, F., Lenard, A. : J. Math. Phys. 8, 423, 1538 (1967).

342 W. Thirring:

enough to contain many particles but small enough so that the potential can be treated as constant inside a cell. Instead of integrating over all particle coordinates daNr, we shall sum over the number of particles n, in the cell around the point x= (~ = 1 ... M). This is exact if the inte- grand is constant for all particle configurations leading to the same occupation numbers. It is a standard combinational problem to trans- form d a Nr(N!)- i into

1__,,,1," i - 1 . (n2)t ,,~=o(nMt) N, Ln=

In this way we obtain (all ms =�89 h = 1)

~(E)=(N !)- I S d3S p daS ,'O (E - ~= lp] - ~> jv(r~, r j)) =( N !)- ~ S d3N

N N " exp {3-~-~ In [E- ~> f(ri, r,)] 2rce/3N} " O (E--'i~> f(r,, r,) ) oo oo

n l m O n2 "~ 0 /12Mr = 0 ~ = 1

�9 exp ln[E-~n~v(x~,x,)nr L ==/i'

M In V} - ~ n= n = M / e . a t = l

(11)

Here we have used Stirling's formula to the accuracy N! =(N/e) N. So far our manipulations rest on the assumption that the potential is con- stant in each cell and n~>> 1, which can be achieved by making N and M sufficiently large. Next we make an approximation that is popular in statistical mechanics, namely represent ~ by a single term for which

tie*

the summand reaches a maximum as function of the n= subject to the M condition ~ n==N:

~ t = 1

B~ va

8S 3N ~3n ~ 2

M 0 2 o n, M -- ~ v(x=, x # ) n # + l n - - = ~ t / T = c o n s t 3N T# = 1 V "

(i2)

Systems with Negative Specific Heat 343

Here we have identified temperature as

1 dS 3N 1 T - dE - 2 M (13)

E - ~ n~ v(x~, x~) np ot>~

Here the E-dependence of the n ~ does not matter since

~ ~So On ~ __~,=~ On~ # O N = 0 (14) ~=1 ~ a E = - 1 ~E = - u e---g �9

The maximum condition (15) is, of course, the barometric formula n=,,~e -v~ V(x) being the potential due to all particles. This pro-

o correspond cedure can be justified rigorously in the limit N-+oo if the n, to an absolute maximum of S. We shall not attempt to find conditions on v, which assures a maximum, since in our case it is not true anyway. However, one might note that not only in the trivial case v =0 but also for v ( x , y ) = ( r e / N ) ( x - y ) 2 for which the many-body problem can be solved (12) leads to the exact result. The ease of interest for us is v(x, y) = - r e / I x - y ] . We know already that then the expression for O diverges and we shall keep a cut-off at small distances in mind for the case of trouble. Furthermore we shall pass to the continuum limit

M n=--~-=p(x,), ~, M a ,=,---,T-fd x

M

N = ~, n = = I d a x p ( x ) , S = - i d a x p o ( x ) l n - - a = l

po(X) (zc T) ~ e ~

(15)

for which Eqs. (12) assume the familiar form of the equations of static equilibrium of a star consisting of an isothermal ideal gas

Po (x) = e (~/T) s d3 x' rpo (x')/Ix- x'll const. (16)

Guided by intuition we shall look for spherically symmetric solutions and hence use for V the unit sphere. Their Eq. (16) reads (r=[x[)

Po (r) = Po (1) exp d r' r' z Po (r') ( 1 / r - 1) + j" d r' r' Po (r') (1 - r') r

(17)

The differential version of Eq. (17)

d p o = 4 ~z re ~. , T dr - f i - p o ( r ) j d r r '2po(r ') (18)

0

344 W. Thirring:

allows us to express the quantities of interest by the asymptotic form of p:

T p~(1) N = V(1) = - N

tc po(1) ' 1

4rt T p o ( 1 ) - 3N T=4rc T S dr r 3 dp~ o dr

~ =-~cSdr'r'24rCpo(r ') drr24~po(r)=V (19)

0

po(X) S = - S d 3 x po (x) In (~ T) ~ e§

3N 2 7 N t c 2 l n r c T - ~ N - N l n p o ( 1 ) + ~ + 8 ~ P o ( 1 ).

We are now prepared to approach the central issue, namely whether the effect of the external virial will upset our conclusion that the system as a whole has negative e. From Eq. (19) we note

3 N 3N E =--�88 T + V= - - - T+4rc Tp0(1 ) . (20)

2

Here the last term represents the contribution of the external virial, and the question is whether its temperature derivate will overcome the - 3 N/2 we had obtained at the beginning. We shall see that this is not the case; on the contrary, it also contributes negatively. This comes from the property of a star to heat up when it shrinks, and hence Po (1) decreases with increasing temperature. To see this one has to use the well-known solution of Eq. (18) corresponding to a p strongly con- centrated at the origin. One finds that p.,~ 1/r 2 is a solution for T O =Nx/2, Eo = - N 2 x / 4 , and for small deviations from these values the solution is

N { l+ T - T ~ [l+2(r(-l+iVv)/2+r(-1-'VV/2))]} (21) P~

With Eq. (22) we then obtain [for I ( T - To)/T o 1 ~ 1 ]

N z rc 7 N ( T - To) E= 4

(22) r3 nNtr N 1 7 T - T . ]

S = N I _ / ~ l n ~ - l n - ~ - ~ - - ~ 2 2 T~]-- " "

Thus we have actually clr= 7 (23) To = - - ~ - N ,

Systems with Negative Specific Heat 345

0.4

0.2

0

t.L -0.2

-0.4

-0.6

-0.8

7-2~ tn(3-'~)- 3~.~ F =3/2 In ~ -

II+ H I I P I I I T I I I M I I I I I I I H I I I ] I ] FIM 11 M I I I I I

- 2 . 0 - 1.0 0 1.0 2 .0 -g

Fig. I. F as function of 7 according to formula (24)

being even more negative than according to Theorem 1. Now dearly this result cannot be exact because we have not used the cut-off without which there is no S. The fact is that S[po] is not a maximum. Looking at the second derivative p2S/tpo(x)fpo(X ) one finds by expansion in spherical harmonics that for l + 0 it is actually a negative definite kernel but not for l=0 . This can be seen directly by inserting p(r)=(N/47r). ( 3 - 7)r -r, V<3, into S for E = -NZtc/4. One calculates

f 3 7cN~ N 3 . l n ( 3 _ ? ) _ ? ~ " S(7)=N i~ ln~_ ln_4_~+_~ln 7 - 2 7 (24) 5 - 2 7

S,r is actually zero for ? =2, as it has to be since p,,~r -2 is a solution of Eq. (18). But S,?rl?= 2 =N/3. Thus we do not have an absolute maxi- mum but a relative minimum. What happens is that S is unbounded and goes to oo for 7 =2.5 (Fig. 1). This shows that entropy favours a strong concentration at the origin, and we shall encounter this collapse on further occasions. Our findings are parallel to the classical results of the instability of an isothermal ideal star. The origin is clearly the need for a cut-off to get a bounded S. There seem to be ways of modifying the 1/r potential in order to get stability without introducing too much internal virial to make c positive. We shall not discuss this here, since in the next section we shall construct a system with negative c in a much simpler fashion such that the evaluation of S involves no problems.

346 W. Thirdng:

3. An Exact Solution for a Somewhat Artificial Version of a Star

In this section we shall analyse a system that incorporates the es- sential features of the previous one, which lead to c<0. We shall be guided by the two conditions 2 which guarantee that for the micro- canonical S(E) one has

a 2 S { ~ 2 ~ 2 S ~ - ~ 0E 2 <0 and therefore c - - ~ -~-2-] > 0 .

They are roughly that at large distances the forces are not repulsive and that E,,~N. The former condition is satisfied for the gravitational case, and thus the failure of the second must be responsible for c<0. This leads us to the following model for a star. Inside an interaction volume Vo, each particle has an attractive interaction with the other particles inside the volume and outside Vo the particles are free. With the step- function

1 if x e Vo Ov~ 0 if x~Vo (25)

a non-local potential having this feature is (v > 0 is constant)

v (Xk, X j) = -- 2 V Ovo (Xk) Ovo (X j). (26)

In this way the total potential energy is -vN2o, where No is the number of particles in Vo. The evaluation of ~ is now a simple combina- tional problem, and we find (if the volume outside Vo is e F. Vo)

(E) = ~ I d aNp daN x 0 [ E - Z p2 _ Z V(x,, x j)] i i>j

~3N]2 - S d aN X (E + V ~ Ov,(X~) Ovi(Xj)) 3N[2 (27)

N!(3N/2)! r ~'i V N ~3N/2 N (E+No 2 v)3N/2 er(N-No) N

-- (3N/2)! ~ -- ~ eS(E'N'N~ No = N.,in No ! ( N - No) ! So = N.,,~

Thus the expression for ~ is again a sum but in this case over a single variable. Now it is easy to see when S as a function of No has a maximum. Indeed (with Stifling)

OS (E, N, No) 3N No v , No F =- 0 (28) ONo - E-+~o v - m N - N o

has for E<O only one solution since

~2S 3 N v E - 3 N N 2 v N dNo~ - (E+No2v) 2 No(N_No) <0, 0 < N o < N , E < 0 . (29)

2 Linden, J. van der: Physica 32, 642 (1966).

Systems with Negative Specific Heat 347

To discuss Eq. (28) we shall introduce "intensive" quantities ct, ~, 0 where, however, the scaling law is different than usual:

E=N2v(z-1), T=-~NvO, No=N(1-~). (30)

From Eqs. (27) and (28) we get a parameter representation for 0 as function of 5:

0 = e - 2 ~ + ~ 2 > 0 , (31)

e = 2 ~ _ ~ z q - 3 (1 -~ ) F + l n ( 1 --~)/ . " (32)

If we take a large F ("atmosphere much larger than star") there will be a sizeable region with I ln (1 -a ) /c t l~ F. Then we get

0 = 3 ( 1 -e/2) , 0 ~= 3 ' - 2 F < 0 ( 3 3 )

and thus again a negative specific heat. On the other hand, near the minimum of the energy, e ~ 0, cr 0, we have

O=e-2e -3#, 0 ~ 1 (34)

and a normal behaviour. This is expected since in this limit all particles are in Vo and the interaction is just a constant. For E > 0, ~ > 1 nothing guarantees that Eq. (32) determines ~ uniquely, and actually it is not true for larger F. This is shown by the graph of O(E), which has an over-hang for e> 1. Of course, in this case one has to choose the that gives the larger S, and thus bridge the overhang by a vertical line (Fig. 2). This is verified explicitly by calculating (27) for N=200 on a computer (Fig. 3). It is interesting to note that the result for N=10 is already close to and for N=25 practically identical with the asymp- totic curve. This means that at this point we have a peculiar phase transition where for constant E the temperature and N O changes suddenly. Fig. 2 can be described as follows. For large E most particles are outside Vo and the system behaves normally. On extracting energy, the system first cools down until suddenly a finite fraction of the particles fall into Vo. At this moment T jumps up and keeps increasing with decreasing E. Finally, when most particles are in V0, c again becomes positive.

One will have noticed that Eq. (28) is just the usual equilibrium condition of an ideal gas and a similar system with our additional energy - v N z. One can generalize this for systems with energy -vNL

348 W. Thirring:

O

0.90

0.60

0.30

0o00 0.00 0.30 0.60 0.90 1.20 1.50

,~=2e

Fig. 2. Temperature versus energy according to Eqs. (31) and (32)

ooo i - : OoO /

o

0.00 0.30 0.80 0.90 1.70 ~'=2,F=4.5

Fig. 3. Computer evaluation of Eq. (27)

If we minimize the sum of the entropies of such a system and a normal ideal gas

S = S1 (El, N1) + S2 (E2, N2)

= N i [~ In (E 1 + N~ v) - { In N 1 + In V1] (35)

+ N2 [-} ln E 2 - ~ ln N2 + ln V2],

subject to Ni+N2=N, EI+E2=E, we obtain for the corresponding "intensive" quantities

E = v N r ( e - 1), N2=c~g, 0= 3 T Vz (36) 2Nr-Jv , F = In -~ - 1

the relations:

~(1-~) ~-I =2__in i-~.+ 2__F_F ~ - 1+(1 -a )~ , 3 e 3 (37)

0 = e - - l + ( 1 - a ) r.

For 7 =2 this reduces to Eqs. (31) and (32). For 7 = 1 we have the prob- lem of particles that can fall into an external potential well, a system which has positive c. However, for larger 7's we get at certain energies

Systems with Negative Specific Heat 349

0.64

0.48

o 0.32

0.16

0 I l I I I

0.00 0.30 0.60 0.90 1.20 1.50 s

Fig. 4. Temperature versus energy according to Eq. (27)

c < 0 (Fig. 4). This shows that systems whose energy increases more than linearly with N in particle exchange with normal systems will lead to negative c.

4. How These Strange Systems Behave in Thermal Contact So far we have studied the systems with c < 0 using the microcanonical

ensemble assuming that the usual ergodicity arguments that go along with it also apply. Like for ordinary systems one might hope that adding a few "grains of dust" may render them sufficiently ergodic in case they fail to be so from the beginning. Since 8 2 SloPE2< 0, the transition to the canonical ensemble requires detailed study. Let us first put our Sys- tem I with c < 0 in thermal contact (exchange of E, not of N) with another system, System 2. The usual expression for the variation of the total entropy S with

1 ~?S~ 1 ~z _ _ = - T i ~ ~ r, ' c , ae, !

s (E) = S, (e , ) + S2 (E - E,) (3S)

1 1 (5Ea) 2 ( C ~ T 2 + C _ ~ 2 ) 5 S = S E a ( T 1 T2 ) 2 . . . ~ ~ .

tells us the following. Since our 7"/ are positive, we gain entropy for T, ~=/'2 by transferring energy from the hotter to the colder system. For Ta =/ '2 we have a stable situation if ( 1 / c , ) + ( 1 / c 2 ) > 0 . Thus if both c's are < 0 we never get a stable equilibrium, and for c2 > 0 only if [ c a [ > c z . This can be understood as follows. If c a <0, c z <0 and one system is slightly hotter it will transfer energy to the other. In doing so it will heat

350 W. Thirring:

S

I S 11

I ~ ~

f /I

TS :~-~-~ E E

dE

Fig. 5. The phase transition for the region of negative specific heat

up even more, and more energy will be transferred. In this way one gets further and further away from TI=T2 and equilibrium can only be reached if one of the c's again becomes >0. This also explains the instability we found in Section 2 for the pure gravitational case. For c2 > - cl > 0 and, say, 7"1 > T2, energy goes from 1 to 2. Now both temperatures increase, but 7"1 faster than / '2 since I c~l< c2. Thus again no equilibrium is reached. Only for cz <1cl I, 7"2 will change faster than T1, and then an equilibrium is established. For the canonical ensemble, System 2 would be the heat bath and therefore c2 > I cl 1. Thus there will never be an equilibrium as long as c~ < 0 and the systems will exchange energy until Ex is such that cl >0. Hence, in Figs. 2 and 4 the part with e < 0 will be bridged by horizontal lines. This means that given T by System 2, the system will jump from the lowest energy where 7"1 = T to the highest energy where 7"1 =T. The jump occurs when the free energy on the lower side equals the free energy F on the upper side. This is evident by plotting S(E) (Fig. 5) where the part with the wrong convexity is bridged by a straight line since then the upper branch gives the lower F. This explains why in the canonical ensemble there are no negative c' s;

(i ) F = - T l n dES(E)e -~/r (39)

always has the right convexity such that

_ 02F c = - 7' ~ > 0. (40)

What happens in the canonical ensemble is that the region of c < 0 is jumped over by a phase transition of the first kind. This can be seen for the

Systems with Negative Specific Heat 351

system of w where the canonical partition function can also be cal- culated along the same lines.

Thus, to define physically the temperature for this system one cannot use a heat-bath, but one has to use a small thermometer. Then, according to (38), a maximum of S can be reached for cl <0, (0<c2~1c l I) and the energy distribution of system 2 will indicate (OY21/OE~)- ~ as tempera- ture.

5. What has this got to Do with Reality

We have noted in the beginning that stars as a whole act like systems with c < 0. Of course, our considerations cannot be directly applied to them since they are not isothermal, and have inhomogenous chemical composition, internal energy sources, etc. Furthermore, quantum effects will become important. This complicated situation can only be handled by a computer. Nevertheless our considerations may supply a simplified model for the dramatic events occurring in the history of a star. For instance, at a stage where no more nuclear fuel is available which would burn at this temperature, the core contracts and becomes hotter, giving its energy to the outer part which expands and becomes colder. This corresponds just to the heat exchange between two systems with c < 0, described in the last section. The hotter system is now the core and the cooler the outer part of the star. This process seems to occur several times in the lifetime of a star in the formation of red giants or super- novae. These events, although far from being equilibrium phenomena, reflect the instability of systems with negative specific heat*. Another system that should reflect these features is a galaxy. Here the stars are the particles and the dense centre may represent the collapsing phase. However, our considerations shed no light on the time scale a, 4 governing these phase transitions. For supernovae where the energy is carried quickly by neutrinos they are fast, but for galaxies where mainly the Newton potential transfers the energy they will be very slow.

Appendix

For free particles one sees immediately from Eq. (15) that p ( x ) = const and therefore iV/11. This gives the classical

* In addition to this thermodynamic instability there is a dynamic instability for 7<4/3(p,-'p~'). For ?=> 4/3 the system collapses if heat is extracted, for y<4/3 it collapses anyway.

3 Prigogine, I., Severne, G.: Physica 32, 1379 (1966). 4 Chandrasekhar, S. : Principles of stellar dynamics. New York: Dover Press 1960.

352 W. Thirring: Systems with Negative Specific Heat

For N

/s 2 V=-~- ~ (x~- x j)

we first note that this is for N ~ c ~ equivalent to

N

v= Z g. i = 1

The latter is the former plus an harmonic force on the centre of mass, but one degree of freedom out of N does not matter. Thus we anticipate the entropy of a 3 N dimensional harmonic oscillator

- ~ (zcE)21----NlnN+4N. (A.2) S= In ~ x

The barometric formula (14) is now solved by

tc po(x)=e-~X2/r. N (---T-~x ) ,

which gives for the entropy

3 S(E,N)=jd3xpo(X) [-~-ln~ 5 rc 2 3 Trc ] T+-i+-f- x -lnN +~- ln--x---j = (A.2)

The author would like to thank Dr. I. Wacek for her patient carrying out of the computer calculations for this work, and to Dr. A. Martin and Dr. S. Epstein for their help in studying t~2S/rp(x) 6p(x'). Furthermore, many colleagues at CERN helped with stimulating discussions.

Note added in proof. Professor Sciama has kindly pointed out to me that similar considerations have been published previously by D. Lynden-Bell and Roger Wood, Monthly Not. Astr. Soc. 138, p,495--525 (1968). Indeed, except for the exactly soluble model, chapter 3 of this paper, the content of this paper is essentially identical with the one by Lynden-Bell and Wood.

Prof. Dr. W. Thirring Theoretical Group CERN CH-1211 Genf 23, Schweiz