Thermodynamic ensembles and gravitation

Class. Quantum Grav. 7 (1990) 1433-1444. Printed in the UK

Thermodynamic ensembles and gravitation J D Brown, G L Comer, E A Martinez, J Melmed, B F Whitingt and J W York Jr Institute of Field Physics, Department of Physics and Astronomy, The University of North Carolina, Chapel Hill, NC 27599-3255, USA

Received 13 February 1990

Abstract. By including gravitation as described by general relativity as a part of a thermodynamic system, we have obtained formal path integral representations of partition functions for various ensembles including that appropriate to the microcanonical ensemble. This is possible because the boundary conditions for certain well posed Euclidean problems in general relativity exactly correspond to boundary conditions of certain well posed problems in thermodynamics. The different ensembles are obtained using the definition of variables conjugate both in the sense of the field theory of general relativity and in the sense of thermodynamics, the boundary data of which can be prescribed geometrically using gravity.

1. Introduction

For a system of fixed size and temperature, the canonical partition function 2, characterises thermodynamic equilibrium in the canonical ensemble. In the thermodynamic limit, to describe the system in another ensemble with different fixed thermodynamic data, one method is simply to perform an appropriate Legendre transformation on the Massieu function [ 13 log 2, to obtain the new one. (Recall that log 2, is minus the inverse temperature times the Helmholtz free energy.) That 2, can be represented as a path integral [2] is well known. The question we shall address is whether partition functions that characterise other ensembles can be obtained as path integrals rather than by the indirect method that employs Legendre transformations of log 2,. We will show that partition functions for various ensembles, including the microcanonical ensemble, can indeed be represented naturally by path integrals provided that general relativity is included in the description of the thermodynamic system.

As a path integral, a partition function depends on quantities that are fixed in the functional integration, that is, on the boundary data chosen for the fields that make up the system. Suppose, for example, that we consider the microcanonical ensemble, in which the total energy is fixed. For matter without gravity, the total energy must be described by the configuration of the matter fields everywhere, not just at the boundary. But with gravity included, the total energy for the complete system is determined entirely by the behaviour of gravitational variables at the boundary, for it is well known that the total energy of both gravitational fields and matter can be expressed in general relativity as a surface integral over gravitational boundary data. (A more general discussion adapted to the viewpoint of the present work is found in [3].) Thus it becomes possible to formulate the partition function (essentially the density of states) for the microcanonical ensemble as a path integral with energy fixed through boundary t Present address: Department of Physics, Williamson Hall, University of Florida, Gainesville, FL 3261 1, USA.

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conditions. Similarly, we find that an appropriate pressure variable can be expressed by gravitational variables at the boundary, which makes possible a path integral in which pressure is fixed.

More generally, with the inclusion of gravity, all stress-energy-momentum variables can be expressed entirely in terms of surface integrals over boundary data [3]. These surface integrals in the present work play the same role as do the ‘constants of motion’ in traditional treatments of statistical mechanics. Thus can the gap between these approaches and path integral formulations be bridged directly. We wish to emphasise that the mediating role of gravity in no way depends on whether the gravitational interaction is actually strong in a given system. On the other hand, this approach did arise from the necessity of obtaining adequate methods for treating gravitational thermodynamics when black holes are considered [4-71.

In the descriptions of thermodynamics with gravity, Feynman’s prescription [2] for the canonical ensemble has a very simple geometrical interpretation [S, 61. In this case the thermodynamic data that define the ensemble, the size and the temperature, are metric boundary data for a Euclidean problem in general relativity. For the manifold M that we consider below to illustrate the ideas, the boundary aM is topologically S’ x S2 with fixed static 3-geometry. Under the assumption of spherical symmetry, which we adopt for simplicity, the boundary 3-geometry is specified by the area A of the S2 and the length p of the S ’ . (An overbar denotes a quantity defined or evaluated on the boundary.) The partition function then depends on the area A and the inverse temperature p at the surface of the ‘container’, and in the path integral these thermodynamic quantities appear as the fixed boundary values of dynamical variables, namely, components of the metric tensor. Thus, the canonical thermodynamic data are also canonical dynamical data. (Generally, in addition to A and p, there will be data corresponding to matter fields. We here ignore this dependence to focus on the essential role of gravity.) It is important to recognise that in the present context area, not volume, is appropriate in specifying the size of the system: the area is fixed by the directly accessible boundary metric while the volume is subject to changes caused by gravita- tionally induced geometric fluctuations that occur in the interior. (These may be negligible in practice, of course.) Furthermore, if the system contains one or more black holes, there is no useful notion of volume definable. Likewise, p is the inverse temperature at the container wall, that is, on the boundary. In the interior, the local temperature will vary because of gravitational red shifting and blue shifting [8]. (Again, this may be negligible.)

In equilibrium systems the ensemble must be time independent or ‘homogeneous in time’, but self-gravitating systems in thermal equilibrium are not spatially homogeneous. The manifolds we consider are closed and periodic in the Euclidean time direction with the boundary being defined by a constant ‘radial’ coordinate. Consequently, in relating data for an action principle to boundary data of a thermodynamic ensemble, we employ a description canonical with respect to radial derivatives, rather than with respect to time derivatives. More precisely, we consider the Hamil- tonian description in which M is foliated radially and the boundary aM is one of the three-dimensional hypersurfaces in the 3 + 1 decomposition.

In terms of the Massieu function, the quantity thermodynamically conjugate to the boundary area A in the canonical ensemble is

(1) p = a log z&, A)/aA which is p times a ‘surface pressure’ (the latter with the dimensions of surface tension).

Thermodynamic ensembles and gravitation 1435

We will call p the ‘canonical pressure’. Similarly, the variable thermodynamically conjugate to p is the energy

lii; = -a log z& A)/ap.

(Observe that the notion of conjugacy defined here using Massieu functions is familiar [l] but is different from the more frequently used notion based on thermodynamic potentials.) We will show that the variables p and J? thermodynamically conjugate to A and p are also the boundary values of the dynamical variables canonically conjugate to area and inverse temperature. That is, A and p are the boundary values of the 3-metric while p and J? are the boundary values of the momenta that are canonically conjugate to the 3-metric with respect to the radial foliation. Thus, a partition function for an ensemble in which energy is specified is given by a path integral with momentum E rather than coordinate p fixed at the boundary. Likewise, an ensemble with canonical pressure fixed is given by a path integral with the momentum p rather than the coordinate A fixed. The different path integrals are constructed using action integrals that differ from one another by addition or subtraction of boundary terms or PI?.

From the above discussion it is clear that there are four types of ensembles that can be described directly by path integrals. These are the canonical ensemble ( A and p are specified), the canonical pressure ensemble ( p and p), the microcanonical ensemble ( A and I?), and the microcanonical pressure ensemble ( p and J ? ) [ 9 ] . Because thermodynamic quantities such as entropy do not arise as boundary values of any of the canonical variables, ensembles such as that for the enthalpy are not describable directly by path integrals in our formulation. Also note that the grand canonical ensemble differs from the canonical ensemble only in its treatment of matter variables, which for simplicity we have not included in our discussion.

General relativity is thus seen to be important for thermodynamics not only because all matter gravitates (often an unimportant effect in practice), but principally because it is needed to account properly for the different boundary conditions in the path integrals appropriate for different ensembles. Thus, by the inclusion of gravity, traditional statistical thermodynamics is fully promoted to the realm of action principles and path integrals. Furthermore, it is well known that even in Newtonian systems in which gravitation is important, different ensembles can have inequivalent criteria of stability. It is therefore all the more important to examine the path integral formulation in order to understand more clearly relations between dynamic and thermodynamic stability of strongly gravitating matter [ 101.

The variables A, E, p, and E are defined purely by the gravitational field, but they nevertheless represent the area, inverse temperature, canonical pressure, and energy of the complete system. The matter fields are always coupled to gravity and thereby leave their imprint on the geometry. Therefore, to demonstrate the relations of dynamic and thermodynamic variables, it suffices to consider only pure general relativity. We shall illustrate the various ensembles by considering the gravitational field in the black-hole topological sector. This seemingly exotic case in fact provides mathemati- cally a simple example because the classical Euclidean action for black holes already contains non-trivial thermodynamics. The essential ideas can therefore be shown without any difficult computations, in a context where gravity is obviously important.

In section 2 we discuss the relationships between changes of thermodynamic data using standard Legendre transformations and changes of boundary data in the path integral representations of partition functions. The mathematical description of general


relativity with black hole topology is presented in section 3, following [5,6], but with notation altered for later convenience. In section 4 we present the canonical form of the action for spherically symmetric geometries with black-hole topology, using a radial foliation. Starting with the canonical action of section 4, we proceed in section 5 to derive other actions, each with different boundary conditions, whose corresponding path integrals yield partition functions for the various ensembles. In section 6 we discuss the ‘0-loop’ approximation to the partition functions, in which the Massieu functions are given by an appropriate action evaluated at an extrema1 geometry. Some conclusions are contained in section 7 .

2. Partition functions as path integrals

From the canonical partition function Z,-(p, A), the partition functions for other ensembles can be obtained by using suitable integral transforms. For definiteness, consider the example of the microcanonical partition function (density of states) written as a transform of Z,: ,.

zMC(E, A) = J dp(p)Z,(p, A) exp PE. (3)

We shall discuss this and similar expressions at a formal level; that is, for the integration over inverse temperature, we will not specify the measure d p ( p ) , the range of integration, or the integration contour in the complex plane. (See [ l l , 121 for a discussion of some of the issues concerning integral transforms, though of course not in the context of path integrals.) Partition functions depending on canonical pressure p rather than on area can also be generated by appropriate transforms of Z, similar to (3).

The integral transform (3) leads to the usual Legendre transformation relating the ensembles when the former is evaluated in a steepest descent (or stationary phase) approximation. This method becomes more nearly accurate as a thermodynamic limit is approached. (When gravity is included, one must be aware that the traditional thermodynamic limit does not exist because of gravitational collapse. However, physi- cally one only requires that the system can in principle become large enough for fluctuations to become negligible, but not so large that gravitational collapse-or recollapse-would occur.) Assuming that the measure is not exponential, the value of the integral is dominated by the behaviour of the integrand near the value of p corresponding to an extremum, that is, when

E+alogZ,(p, A)/ap=o. (4)

This is the usual thermodynamic definition (2) of energy E in the canonical ensemble. Denoting the value of inverse temperature that satisfies (4) by P ( E , A), we see that the microcanonical partition function (3) satisfies

log zMC( E, A) = EP( E, A) +log Z, (F(E , A) , A ) + corrections. ( 5 )

The correction terms arise from the integration in (3) near the extremum. As the thermodynamic limit is approached, these corrections generally become unimportant and ( 5 ) then expresses the Legendre transformation.

Now let us turn to the issue of changes of boundary data in the actions appropriate for path integral representations of partition functions. For this purpose, assume that Z&, A) is expressed as a functional integral of exp(-lc), where IC is an appropriate


action. Then the right-hand side of (3) can itself be regarded as a functional integral corresponding to the action IMC = IC - FE. In section 4 we will show that 27r times the energy is the boundary value of a momentum variable E conjugate to a metric variable p, with respect to the radial foliation. Thus, the action IMc differs from I , by the subtraction of the boundary term PI?. This is precisely the change necessary to yield an action integral that remains stationary at a classical solution when variations are carried out with the momentum E fixed at its value on the boundary. Accordingly, in a path integral expression for ZMc(E, A), the functional integration is taken over all field configurations with the momentum E and the coordinate A fixed to the values 27rE and A on the boundary.

In section 4, we will also identify minus the boundary value of the momentum conjugate to A with the canonical pressure p. Then the canonical pressure ensemble will be seen to be obtained by fixing the momentum p conjugate to A on the boundary and using the action Icp = IC +FA. A similar conclusion holds for the microcanonical pressure ensemble.

3. Topology, metric and action

In this and the following sections, we will illustrate the general relationships discussed in the previous two sections by considering the pure gravitational field in the black-hole topological sector [ 6 ] .

The metrics we consider are of the form

A ds2=p2dT2+a2dy2+-(d82+sin2 47r 8 dq2) ( 6 )

where p, a and A = 47rr2 are functions of the radial coordinate y only. Thus, the metrics ( 6 ) are static and spherically symmetric. Also, T is periodic with period 27r and y E [0,1]; the variables 8 and q are the usual coordinates of the unit sphere. The corresponding 4-geometries M are all chosen to have the product topology R 2 x S2 with boundary S’ x S2 and Euler number x = 2. The boundary leaf of the radial foliation of M is at y = 1 and has unit normal n p = (0, a-’, 0,O). The proper length of the S’ part of aM is identified with the inverse temperature T I ,

P = Jb’,, p ( 1 ) d T = 2 7rp ( 1 ) (7)

while the value of A on the boundary is the proper area of the S 2 part of aM and is identified with the area A = A( 1) of the ‘container’ of the thermodynamic system. (Throughout, we use the conventions of Misner, Thorne and Wheeler [ 131 and units such that G = c = A = k , = 1.)

The centre y = 0 is not an element of the boundary (which is three dimensional), but rather is a regular 2-surface in M, with non-zero area A(0) 5 47rr:. Thus the centre is distinguished as a degenerate leaf of the radial foliation. The y - T plane near y = 0 must be isometric to a flat disc. The regularity condition for the metrics (6) at y = 0 is

(a-’p’)lv=o= 1 (8)

where a prime denotes alar. This means that the proper radial distance of a point


y = y+ very near the centre is

joy’ a dy = P ( Y + )

where we have used p(0) = 0. The Euler number for (6) is given by

,y = 2 [ a - ’ ~ ‘ ( 1 -(r’)2a-2)]ly=o

= 2[1- ( r‘)2a-2](y=o

( 9 )

where the final expression results from using (8). In order to have x = 2, we see that 2 I 2 [ a - ( U ) ] l y = 0 = O .

As in [ 5 ] , write the Euclidean action as IC = I , - Isubtract, where

1 I --- R a d 4 x + ~ 6, K f i d3x 167~ b 1 -

and K is the trace of the extrinsic curvature K,, of the boundary wit.. induced metric yo. The second term of IC has the value [5,14]

where KO is a constant chosen to make equal to zero the action IC of flat space with the given boundary, that is, S’ x S 2 with induced metric yV. (For discussions of this prescription see [5,14].) The appropriate boundary condition for the canonical action IC is that the metric of aM be fixed.

For the metric (6) one can show that

R G = -2[P’G(a2p)-‘If -2GO,& (14)

where

G: = (2A1A”2) - ’ {A’ ’2 [ (A1)2 (~2A) - ’ - 1677]}’

= ( r f r z ) - ’ { r [ ( r ’ )2a -2 - 111’.

For the y = constant surfaces one finds

K = - (aPA)-’(PA)‘

and

K O = -2(477A-1)1/2.

Inserting (14) through (17) into (12) and (13) and integrating the angles 8 and cp, we find


Extremising IC with respect to a, p and A with the given boundary conditions produces the correct Einstein equations for the metric (6); thus, we have obtained a consistent simplified classical action principle [15]. The path integral constructed from I , is a functional integral over all configurations with p and A held fixed and is identified as the canonical partition function Zc(p, A). (Notice that the first term in (18) vanishes when G: = 0. This fact will be useful in section 6.)

4. Canonical form of the action

We now wish to obtain the action IC of the canonical ensemble in canonical form with respect to the radial foliation. An appropriate form from which to derive the momenta is

I C = & [02wdT jo' dy{-~~(c")-'(A')2-a-'A'~'+(4~A-')'~2~A'

+ 2 ( 4 ~ A ) ' / ~ p ' - 4 ~ a y P }

which is the same as (18) except in two points, which are merely rearrangements. The first is that the boundary term at y = 1 has been incorporated directly into (19). The second is that the fixed regularity conditions (8) and (9) have been used to simplify the final term of (18). We can see that I , does not involve any second derivatives; therefore, the standard canonical relation between field coordinates and momenta is valid. (If one were to write the final term of (18) as a divergence plus a further boundary term at y = 1, one would not have a first order action. Then the manifest simple correspondence between thermodynamic conjugacy and canonical conjugacy in the gravitational action is lost.) We see from (19) that there are two momenta, one conjugate to A and the other to p. There is no momentum conjugate to a because it is the lapse function for the radial foliation. We find

E = t ( 4 ~ A ) ' " - A'/4a (20)

p = $ [ ( 4 ~ A - ' ) ' ' ~ p - (2a/3A)-'(AP2)'] (21)

where E is conjugate to p and p to A. Because the term Isubtract included in IC is a boundary term depending only on the metric of aM, it has effected a very simple canonical transformation on the usual action. From this transformation, the momenta have acquired the first term in (20) and the first term in (21). The final term in (19) has no effect on the momenta. Recall that the inverse temperature was defined by (7). Likewise, the canonical pressure is defined by

1 r 2 r r

Using (20) and (21) in (19), one can show that the canonical form of IC is

1 2n 1 IC = - lo d 7 lo' dy[E/3' +PA' - aH( p, E, A, p ) ] -G lo2w dTA(0)

2 T


where the Hamiltonian density for the radial foliation is

H ( p , E, A, p ) =4{p[$(47rA)1’2-E]+~A-’PE2}. (24)

Variation of IC with respect to A, p, p , E and cy yields, respectively, the following equations of motion

p’ = - C X [ p ( A/41r)-”~ - A-’PE ’3 E’= -cY[A-’E~]

A’=4c~[f(47rA)’’~- E ] (27)

p’ = 4 c ~ ( - p +$EA-’) (28)

H ( P , E, A, P ) = 0. (29)

Equations (27) and (28) are equivalent to the definitions (20) and (21). Equations (25), (26) and (29) are, respectively, the Einstein equations Gi = G: = 0, Gt = 0, and G; = 0. There are no other non-trivial Einstein equations for metrics of the form (6). The boundary conditions are still SAI,,I = SA=O and Spl,=l = O (or Sp=O). Finally, we recall that we required two regularity conditions, (8) and ( 1 l ) , at y = 0. These conditions can be translated, using the definitions (20) and (21), into conditions on the momenta p and E at y = 0. We obtain

p ( 0 ) = -1/4

E ( 0 ) = 4[41rA(O)]’”= 27rr+

as consequences of working on smooth manifolds with non-trivial topology.

5. Actions for the other ensembles

We can now construct actions having boundary conditions different from those appropriate for I C . To obtain the other actions, one simply adds an appropriate boundary term chosen from &j and to IC . This procedure effects a simple canonical transformation. There are three possibilities. One results in what we will call the canonical pressure action

=L[027d.r 27r [o’dy[Ep’-p’A-aH] (32)

where H is given by (24) and p and J7 are fixed. Another choice is the microcanonical action

= 2T Io2‘ d.r Io1 dy [ - E‘@ + pA’ - aH] - - 81r [02‘d~A(0) (33)


where E and A are fixed. Finally we have an action for a microcanonical pressure ensemble where I? and a are fixed:

=‘ lo2’ d.r 1,’ d y [ - E ’ P - p ’ A - a H ] . 2.n (34)

The action for the canonical pressure ensemble may be of particular interest [ 5 ] . An extremum of IC. occurs when the boundary surface adjusts, like the wall of a bubble, to the ambient temperature and pressure. Thus a fixed boundary such as one has in the canonical ensemble is removed from the prescription. This should be useful when mechanical isolation is not a good approximation to the physical conditions.

6. Reduced and extrema1 actions

In this section, the ensembles described by the actions we have constructed are investigated using what can be called a ‘0-loop’ approximation to the logarithm of a partition function. Following [ 6 ] , we choose to obtain the ‘0-loop’ action in two steps. First, the Hamiltonian constraint Gt = 0 is solved and the solution is incorporated into each of the actions I, producing a ‘reduced action’ I* depending only on the boundary data and a few independent physical parameters [6]. Notice that we refer to the Hamiltonian constraint Gt = 0 on the initial data in the ordinary sense, that is, data on a slice of constant time. Second, we evaluate I* at its extremum with respect t? the physical parameters, the boundary data being held fixed, to produce an action I. The 0-loop approximation is to take log 2 = - I .

First recall briefly the results of [6]. With the action IC in Lagrangian form ( 1 8 ) , we impose the Hamiltonian constraint Gt = 0. (The momentum constraints GP = 0, i = y , 8, cp, are satisfied trivially by the metrics ( 6 ) . ) The solution of the constraint combined with the topological condition (10 ) shows that

-

where A . = A ( 0 ) = 4 7 ~ : . Because the proper length of the SI part of dM is B, one finds

I ; = B E ( A , A , ) - $ A , (36)

where

E(A, A,) = ( A / ~ v ) ’ / ’ [ 1 - ( 1 - ( A o / A ) ’ / 2 ) ’ / 2 ] (37)

is the energy [ 5 ] and A 0 / 4 is the well known black-hole entropy. Z? depends on the boundary or thermodynamic data, p and A, and on the physical (metric) paramter Ao.

Now we would like to recover (36) by reducing the canonical form of IC given by (23). One simply uses (27), which is a definition, and (26), which is equivalent to G: = 0, to find

E =f (4 .?rA) ’ / ’ [1 - (1 - d / A ” ’ ) ” ’ ] . (38)

From (31) we find the constant d = AA/’. Comparing (38) with (20) then yields (35). Thus I$ is recovered.


Using the same procedure to reduce the other actions, we find

I : , = ~ E ( A , A,) -$A,+& (39)

15, = -‘A 4 0 (40)

I $ c p = - :A0 + Ap. (41)

and

For IZp, p and p are fixed thermodynamic data while A and A. are independent physical variables. For I X C (37) shows that A. depends uniquely on the thermodynamic data E and A, so it has no independent variables. Likewise, in ILCp there is one independent variable, either A. or A, depending on how one chooses to solve (37).

The partition functions for the canonical pressure, microcanonical, and microcanonical pressure ensembles can be constructed by integrating, with respect to the free parameters, the exponentials of minus the corresponding actions. For Z,, this was done in [6], yielding

Zc(B, A) = dP(A0) exp(-IE) (42)

where dp(Ao) is a suitable measure. The final step of our 0-loop approximation is to approximate the integrals for Zc, Zcp, ZMc, and ZMcp by the value of the integrand at an extremum. Then we obtain an approximate Massieu function in the form -log z = I* (extremum) = i

For the canonical ensemble, the resulting Massieu function is -log Zc = p F = &. , where F = F ( p , A) is the Helmholtz free energy. At a stationary point of IC with respect to Ao, the Hawking inverse temperature p is obtained [5,6]:

p= ( 4 ~ & ) ” ~ [ 1 -(Ao/A)”2]”2 (43)

from which one expresses A. as a function of the thermodynamic data p and A [5]. One finds

f c = p ~ ( ~ , A o ( ~ , p ) ) - $ A , ( ~ , ~ ) (44)

where ,!?(A, A,) has the form (37) . We see that IC has the expected form pE(p, A)- S(p, a), where S is the entropy of the black hole corresponding to the extremum of 1: with respect to Ao.

The reader is referred to [6,15] for discussions of the fact that an extremum of I : is not a spacetime; it is an equivalence class of spacetimes among whose members is the classical Schwarzschild spacetime. It is remarkable that all of these spacetimes have the Hawking temperature (43) and the Bekenstein-Hawking entropy $Ao. By contrast, an extremum of IC in the form (18 ) or (23) , that is, before the ‘reduction’ that set Gg = 0, is a classical Schwarzschild geometry.

A similar evaluation of I : , at an extemum yields

fc, = P J W ( P , PI, AO(P, P ) ) -tAo(P, 8) +p&P, B ) (45)

where the extremum conditions with respect to A. and A yield the previous expression (43) for @, and p = p7, where 7 is given by [ 5 ]


with A and there is only one [16].

parameters and one has

where S = $ A o is the black hole entropy. The final case is

where the previous expressions for 8 and p enable one to express A = A( p, 8) and A, = A,(p, E). Just as was the case with f,, each of these reduced actions evaluated at an extremum represents an infinity of spacetimes, each having the thermodynamic or boundary data given for the particular ensemble to which it corresponds.

We close this section by remarking that the accuracy of the 0-loop approximation to a partition function has as a necessary condition that the extremum of the corresponding action be a minimum. The canonical ensemble is considered in this light in [5,6], where it is shown that there are two extrema. One is a local minimum and the other a local maximum when I$ is varied with respect to A. in the black hole topological sector. In reference to the canonical pressure ensemble [16], one finds that for all p and 8, the action I $ , has a negative mode when expanded about its single extremum, which is therefore a saddle point with respect to variations of A, and A. This saddle point reflects a thermodynamic instability. Hence, an accurate approximation to Z,, cannot rely on the black hole sector alone. In contrast, the action I&, relevant to the microcanonical pressure ensemble has a unique extremum that is a minimum [16]. Finally, we remark that the 0-loop approximation in the microcanonical ensemble gives us directly the dominant factor in the density of states [17], namely ZMc= exp(-&.) = exp(-fMc) = exp(Ao/4), where Ao/4 is the black hole entropy.

4rP2 and A, = 47rr:. In this case the existence of an extremum requires 48 S 1

In the case of the microcanonical ensemble as mentioned before, there are no free

f M C = -s(A, 8) (47)

fMcp= -s(p, E ) + p A ( p , E ) (48)

7. Conclusions

From the results of the previous sections, we see that changes in boundary conditions for the gravitational action correspond to Legendre transformations (for canonical thermodynamical variables) among 0-loop Massieu functions. The fact that actions are the fundamental objects in path integral formulations of statistical thermodynamics implies then that Massieu functions are the fundamental objects in thermodynamics, rather than the more familiar thermodynamic potentials (which are energies) [ 181. This, indeed, is not merely a formal matter, for in thermodynamics including gravity, it has been shown that generalised Massieu functions (actions) are simply additive while the thermodynamic potential energies are not [19].

The most familiar path integral representations of partition functions use as thermodynamic boundary data those which are appropriate for the canonical ensemble. In this article, however, we have obtained a formalism for partition functions obtained directly from path integrals which can include also three other general types of ensembles. In order to accomplish this, we have seen that it was necessary to include gravity, which enables one to express energy-like quantities or their conjugate metrical quantities as boundary data or surface integrals. Thus a profound and hitherto unnoticed consequence of the principle of equivalence is that it points towards a more nearly complete unification of quantum and statistical methods through the path integral.


We are aware that in the great majority of problems of statistical thermodynamics, the gravitational interaction is entirely unimportant. Yet both gravity and thermodynamics are thought to have universal scope. One describes the global structure of spacetime and the other is the seat of the presumably universal tendency toward the increase of entropy. It is therefore essential that a full unification should occur. In this paper we have attempted to take a step in this direction and have illustrated some of the possibilities in approximate form.

Acknowledgments

The authors thank G Horwitz for helpful discussions, and D N Page for pointing out that there should be no T dependence in the metric function U = p’ in [6]. They have received research support from the National Science Foundation of the USA. Support has also been received by JWY from the USA-Israel Binational Science Foundation.

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Thermodynamic ensembles and gravitation

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