# Quantum Capacity and Vacuum Compressibility of Spacetime: Thermal Fields

Hing-Tong Cho<EMAIL_ADDRESS>Department of Physics, Tamkang University,
Tamsui, New Taipei City 251301, Taiwan, ROC  Jen-Tsung Hsiang
<EMAIL_ADDRESS>Center for High Energy and High Field Physics, National
Central University, Taoyuan 320317, Taiwan, ROC  Bei-Lok Hu<EMAIL_ADDRESS>Maryland Center for Fundamental Physics and Joint Quantum Institute,
University of Maryland, College Park, Maryland 20742-4111 U.S.A.

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###### Abstract

An important yet perplexing result from work in the 90s and 00s is the near-
unity value of the ratio of fluctuations in the vacuum energy density of
quantum fields to the mean in a collection of generic spacetimes. This was
done by way of calculating the noise kernels which are the correlators of the
stress-energy tensor of quantum fields. In this paper we revisit this issue
via a quantum thermodynamics approach, by calculating two quintessential
thermodynamic quantities: the heat capacity and the quantum compressibility of
some model geometries filled with a quantum field at high and low
temperatures. This is because heat capacity at constant volume gives a measure
of the fluctuations of the energy density to the mean. When this ratio
approaches or exceeds unity, the validity of the canonical distribution is
called into question. Likewise, a system’s compressibility at constant
pressure is a criterion for the validity of grand canonical ensemble. We
derive the free energy density and, from it, obtain the expressions for these
two thermodynamic quantities for thermal and quantum fields in 2d Casimir
space, 2d Einstein cylinder and 4d ($S^{1}\times S^{3}$ ) Einstein universe.
To examine the dependence on the dimensionality of space, for completeness, we
have also derived these thermodynamic quantities for the Einstein universes
with even-spatial dimensions: $S^{1}\times S^{2}$ and $S^{1}\times S^{4}$.
With this array of spacetimes we can investigate the thermodynamic stability
of quantum matter fields in them and make some qualitative observations on the
compatibility condition for the co-existence between quantum fields and
spacetimes, a fundamental issue in the quantum and gravitation conundrum.

###### Contents

  1. 1 Introduction
    1. 1.1 Energy and Pressure Densities of Quantum Fields and Fluctuations
    2. 1.2 Physical contexts of Quantum Capacitance and Negative Compressibility
    3. 1.3 Methods, Findings and Organization
  2. 2 2d: Thermal Casimir and Einstein Cylinder
    1. 2.1 Low temperature expansion: Approaching $R^{1}\times S^{1}$
    2. 2.2 High temperature expansion: Approaching $S^{1}\times R^{1}$
  3. 3 4d: Einstein universe $S^{1}\times S^{3}$
    1. 3.1 Low temperature expansion
    2. 3.2 High temperature expansion
  4. 4 Even spatial dimensions: $S^{1}\times S^{2}$ and $S^{1}\times S^{4}$
    1. 4.1 Low temperature expansion
    2. 4.2 High temperature expansion
  5. 5 Conclusions and discussions

##  1 Introduction

Three major elements are embedded in the theme explored in this paper: quantum
fields, spacetime, and thermodynamics. The first two describe how quantum
matter is affected by spacetime, its geometry and topology, and how it steers
the dynamics of spacetime, although the latter issue involving backreaction is
not explored here. The relation between matter and spacetime dynamics is of
course underwritten by Einstein’s general relativity theory. With quantum
matter as source, one needs a theory of quantum fields in curved spacetime [1,
2], which is a test field limit of semiclassical gravity theory [3], where the
dynamics of the matter field and the spacetime are treated self-consistently.
The first and third elements refer to the thermodynamics of quantum fields, a
familiar subject, by way of thermal field theory. In this paper we shall only
consider quantum fields under equilibrium conditions, which can exist for
static spacetimes, and be treated by finite temperature quantum field theory.
Later, when we tackle cosmological issues, we shall call upon the more
challenging subject of nonequilibrium quantum field theory [4] and
nonequilibrium quantum thermodynamics [5]. The second and third elements bear
on the thermodynamics of spacetime, which is also an old topic, ranging from
the thermodynamic properties of classical matter, such as gravitating systems
having negative heat capacity [6, 7] to black hole mechanics [8] and
thermodynamics [9] and, when quantum physics is taken into the consideration,
the famous Bekenstein-Hawking entropy [10] and its deep physical meaning.

All told, these are the issues one needs to consider behind the grander views
of spacetime thermodynamics [11], general relativity as geometro-hydrodynamics
[12], and emergent gravity [13, 14, 15], where the large scale structure and
dynamics of spacetime can be phrased in thermodynamic or hydrodynamic terms.
What we want to accomplish in this and a sequel paper [16] is more restricted
in scope, aiming at firing some solid bricks toward building this
superstructure.

###  1.1 Energy and Pressure Densities of Quantum Fields and Fluctuations

One fundamental aspect of special interest to us is fluctuation phenomena in
quantum matter fields and how they influence the dynamics and the
thermodynamics of spacetimes. We describe their importance and plan for two
ways to approach them, by calculating two important thermodynamic quantities,
the heat capacity and the vacuum compressibility here and later, via noise
kernels in quantum field theory in curved spacetime.

#### Importance of Quantum Field Fluctuations Phenomena

Quantum field fluctuations phenomena have both fundamental theoretical and
practical application values. The simplest example of a vacuum polarization
effect is perhaps the Casimir effect [17, 18, 19]: e.g., an attractive force
between two conducting plates, or a repulsive force in a conducting sphere.
When the plates are moved rapidly, particles are created from the vacuum.
Dynamical Casimir effect [20] stems from the amplification of vacuum
fluctuations by changing boundaries [21, 22, 23]. Cosmological particle
creation [24, 25] shares the same physics, the driving agent being the
expanding universe. The theoretical basis is quantum fields in curved
spacetime [1, 2]. Quantum effects in the early universe [26, 27, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38] invoking semiclassical and stochastic gravity
theories [3] are believed to have played a decisive role in shaping our
present universe.

The purpose of our present investigation is to knit a thermodynamic picture of
these quantum field fluctuation phenomena. Our plan is to take a two-prong
approach to explore these issues, one pursued in this and a sequel paper uses
thermal field theory and quantum field theory in curved spacetimes, via free
energy density and partition functions; the other approach invokes the noise
kernel in stochastic gravity [3]. The noise kernel is the vacuum expectation
value of the stress-energy bitensor [39] or the stress-tensor two point
functions (correlators). The 00 component and the ii components give
respectively the fluctuations in the energy and momentum density. An
interesting result from works in the 90s and 00s is the near-unity value of
the ratio of fluctuations in the vacuum energy density to the mean. This
quantity $\Delta$ has been calculated for thermal fields in Minkowski and
Casimir geometries, the Einstein universe, de Sitter and anti-deSitter
spacetimes and more [40, 41, 39, 42, 43, 44, 45]. This is a plain yet
intriguing result. It may hold some deeper meanings in how quantum fields
(matter contents) co-exist with spacetime (geometry and topology). Possible
implications on the condition of our universe is a theme worth further
explorations [46].

In this paper we wish to shed some light on this issue by way of a
thermodynamic approach, by computing two essential thermodynamic quantities,
the heat capacity and the adiabatic compressibility.

#### Thermodynamics of quantum fields reflecting the properties of spacetime

One commonly studied thermodynamic quantity is the heat capacity, well-known
as a measure of the magnitude of energy fluctuations. When it diverges it
spells the breakdown of canonical ensembles in thermodynamics.

In general the heat capacity is negative [6, 7] for gravitating systems (see,
e.g., [47] and references therein). That is why when one applies
thermodynamics in a canonical ensemble setting one needs to add the condition
that the system, e.g, a black hole, needs to be placed in a box or in an AdS
space. We need to separate two effects simultaneously affecting the heat
capacity, one due to quantum fluctuations, the other due to gravity. To
explore gravitating systems without an artificially imposed boundary we need
to use microcanonical ensembles [48] where the starting point would be the
number of accessible states to an isolated system at a certain energy.

The other thermodynamic quantity of equal importance but lesser studied in
field theory is compressibililty, isothermal at finite temperature and
adiabatic compressibility at zero temperature. As is evident from its
definition, it measures how compressible the quantum field is, which depends
on the curvature and topology of the space they live in. In fact when we refer
to, say, the capacitance of a parallel plate capacitor in electrostatics, we
usually just give a formula in terms of geometric measures, such as the area
of the plates and the distance between them. It is implicit that we are
talking about electric charges and fields. It is in this sense that we refer
to the capacity and compressibility of spacetimes in terms of how a quantum
field and its fluctuations behave. Since heat capacity is studied widely we
shall focus more on the latter. The term ‘quantum capacity’ in the title of
this paper refers to the heat capacity of quantum fields, including high and
low temperatures; and when we say ‘vacuum compressibility’ we refer to
adiabatic compressibility which can be defined without any notion of heat.

###  1.2 Physical contexts of Quantum Capacitance and Negative Compressibility

To better appreciate the meaning and significance of these thermodynamic
quantities it is useful to see how quantum capacitance and vacuum
compressibility are defined and used in some more familiar physical systems.
We give three examples here: nuclear “liquid drop” model, 2-dim electron gas
and graphene nanotubes. Nuclear collective model is a good analog because the
Hamiltonian quadrature has the same mathematical form as the metric of the
mixmaster universe [49, 50], a compact anisotropic spacetime, where
compressibility has some intuitive meaning. 2-dim electron gas has similar
thermodynamic behavior as the gravitational field, in that they are both
systems of long range interactions and with negative heat capacities (see,
e.g., [47] and references therein). In graphene we can see how quantum
capacitance enters due to the dependence on the number density. Besides, such
systems have active experiments closely tracking the theoretical analysis.

#### Nuclear compressibility in quantum hadrodynamics

The compressibility of a nucleus is directly related to the surface energy – a
high compressibility implies a stiff surface which will have a large
contribution to the total energy of the deformed nucleus. For example, Price
and Walker [51] developed a self-consistent, relativistic theory of deformed
nuclei based on quantum hadrodynamics and the finite Hartree approximation,
and have applied this theory to the calculation of deformed orbitals in
various light nuclei. They even went into details in explaining why a
relativistic mean field calculation based on quantum hadrodynamics [52] gives
a larger value of compressibility than non-relativistic calculations. Inasmuch
as the nuclear collective model sharing a similar mathematical structure
(Hamiltonian quadratic form) as the mixmaster universe (diagonal Bianchi type
IX metric), the Hartree approximation in seeking self-consistent solutions are
similar to the structure and procedures in semiclassical gravity theory.

#### What does the negative compressibility of a 2-dim electron gas reveal?

For two dimensional electron systems at high magnetic fields, interaction
effects have spectacular transport consequences, e.g., the fractional quantum
Hall effect. The importance of exchange and correlation contributions to the
total energy, and hence the thermodynamics, of interacting electron systems
has long been theoretically appreciated. For example, in the low-density
regime, where interactions dominate the kinetic energy, the exchange energy
alone is sufficient to produce a negative compressibility for the electron gas
– the inverse of the compressibility is given by
$\kappa^{-1}=N^{2}\partial\mu/\partial n$, where $\mu$ is the chemical
potential and $n$ is the number density. Eisenstein et al [53] reported on a
new experimental technique whereby they can directly extract both the sign and
the magnitude of $\kappa$ as a function of electron density $n$. Regions of
negative $\kappa$ are observed at both zero and high magnetic field extreme
quantum limit. Observation of these compressibility features constitutes
strong thermodynamic evidence for existence of the dilute quasiparticle gases
central to theory of the fractional quantum Hall effect. This is one of many
examples of how (macroscopic) thermodynamic quantities may reveal some
important attributes of the underlying (microscopic) constituents and their
interactions.

#### Quantum capacitance of carbon nanotube contains information about its
ground state

We can use the words of Ilani et al [54] to illustrate the importance of
quantum capacitance: “The electronic capacitance of a one-dimensional system
such as a carbon nanotube is a thermodynamic quantity that contains
fundamental information about the ground state. It is composed of an
electrostatic component describing the interactions between electrons and
their correlations, and a kinetic term given by the electronic density of
states.” The measurements in their experiments suggest the existence of a
negative capacitance, which is predicted to exist in one dimension as a result
of interactions between electrons. This lower dimensional system offers a good
example to see how quantum effects enter into the thermodynamic quantities.
“The capacitance of a classical conductor is determined solely by its
geometry. When charged, the electrons distribute in space in a manner that
minimizes their electrostatic energy. Quantum mechanics introduces extra
energies that add new contributions to the familiar classical capacitance
$C_{g}$ determined by the geometry: namely,
$C^{-1}_{tot}=C^{-1}_{g}+C^{-1}_{dos}+C^{-1}_{xc}$ where $dos$ denotes density
of state, and $xc$ denotes exchange and correlations. The first correction
term is caused by the kinetic energy of the electrons. Adding electrons to a
conductor requires finite kinetic energy and therefore this first contribution
$C_{dos}$ reduces the total capacitance. The second contribution results from
the correlated motion of electrons, which generally leads to reduction of
their total electrostatic energy. This adds a negative capacitance term
$C_{xc}$ that increases the total capacitance. In one dimension, the
capacitance plays a special role as it also determines the properties of the
excitations. Described within the Luttinger model, the fundamental excitations
are collective waves of spin or charge. This electrostatic effect is captured
by the compressibility of the electronic gas, or equivalently by its
capacitance. Thus, the central parameter of the Luttinger liquid theory is
directly related to the capacitance by the simple relation that the Luttinger
parameter $g=\sqrt{C_{tot}/C_{dos}}$.” Again we see the deeper significance of
these thermodynamic quantities.

The above cases exemplify how macroscopic quantities can be judiciously used
to reveal the workings of the microscopic constituents, a better understanding
of which can serve as an inspiration for the hydrodynamic and thermodynamics
approaches to probing the microscopic structures of spacetime.

###  1.3 Methods, Findings and Organization

Zeta function method is used in our calculations of the thermodynamic
quantities for a thermal Bose gas (finite temperature quantum scalar field).
The main targets are $C_{V},C_{P}$ related to the energy density fluctuations
at constant volume and pressure, and the isothermal $\kappa_{T}$ and adiabatic
compressibility $\kappa_{S}$ related to the momentum density fluctuations at
finite and zero temperatures. For completeness we also provide the energy
density, entropy and the expansion coefficients $\alpha$. Thermal fields (with
periodic imaginary time) in the following static background spacetimes are
considered: a) 2d Casimir geometry and 2d Einstein Cylinder (periodic in one
spatial dimension); b) 4d Einstein Universe $S^{1}\times S^{3}$ , c) Einstein
universes with even-spatial dimensions $S^{1}\times S^{2}$ and $S^{1}\times
S^{4}$.

The results for the thermodynamic quantities at low and high temperatures are
presented in two tables in the last section. Please refer there for detailed
explanations. In Sec. 2, we derive the thermodynamic quantities in 2d (1-d
space) thermal Casimir and 2d Einstein cylinder. In Sec. 3, in 4d (3-sphere
space) Einstein universe, and in Sec. 4, in 3d (2-sphere) and 5d (4-sphere)
Einstein universes. In Sec. 5 we conclude with a summary and some discussions.
In this paper we focus more on the technical aspects, aiming to get a complete
tally of these thermodynamic quantities. We shall continue to explore their
physical significance to better understand how the thermodynamics of quantum
fields is governed by the underlying spacetime structure and what we can say
about the nature of gravity from the thermodynamics of quantum fields.

##  2 2d: Thermal Casimir and Einstein Cylinder

To illustrate the points we would like to make, on the capacities and the
compressibilities of quantum fields in spaces with various topologies, we
start with the simple case of an Einstein cylinder at finite temperature [55,
56]. The topology is basically $S^{1}\times S^{1}$. (The first entry denotes
time, the second entry denotes spatial dimensions. Since we shall be working
with thermal field in the imaginary time formulation, the first entry will
always generically be $S^{1}$. Only in certain specified limit will $R^{1}$
appear.) The length of the Euclidean time circle is characterized by
$\beta=1/T$ where $T$ is the temperature. The spatial part is in a Casimir
setting where the circle is characterized by the length $L$. In the low
temperature limit, $L/\beta\rightarrow 0$, one has the $R^{1}\times S^{1}$
topology with the leading contribution coming from the Casimir effect. On the
other hand, in the high temperature limit, $\beta/L\rightarrow 0$, one has the
thermal field in two spacetime with the topology $S^{1}\times R^{1}$.

We shall consider a thermal minimally coupled massless scalar field in the
Einstein cylinder with periodic boundary condition. To derive the Helmholtz
free energy we start with the partition function, with the notation of
Phillips and Hu [39],

$\displaystyle Z=e^{W}$ (2.1)

where using the proper-time zeta function method,

$\displaystyle W$ $\displaystyle=$ $\displaystyle-\frac{1}{2}{\rm Tr\
ln}\left(\frac{H}{\mu}\right)$ (2.2) $\displaystyle=$
$\displaystyle\lim_{s\rightarrow
0}\frac{1}{2}\frac{d}{ds}\left[\frac{\mu^{s}}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-1}\,{\rm
Tr}(e^{-tH})\right]$ $\displaystyle=$ $\displaystyle\lim_{s\rightarrow
0}\frac{1}{2}\frac{d}{ds}\left[\frac{\mu^{s}}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-1}\sum_{n}e^{-t\lambda_{n}}\right]$

with the operator

$\displaystyle
H=-\Box=-\frac{\partial^{2}}{\partial\tau^{2}}-\frac{\partial^{2}}{\partial
x^{2}}$ (2.3)

where $\tau$ is the Euclidean time, the eigenvalues

$\displaystyle\lambda_{n_{0},n}=k_{0}^{2}+k^{2}\ \ \ ;\ \ \ k_{0}=\frac{2\pi
n_{0}}{\beta}\ \ \ ;\ \ \ k=\frac{2\pi n}{L}$ (2.4)

and the eigenfunctions

$\displaystyle\phi_{n_{0},n}(x)=\left(\frac{1}{\sqrt{\beta}}e^{ik_{0}\tau}\right)\left(\frac{1}{\sqrt{L}}e^{ikx}\right).$
(2.5)

With the partition function $Z$, we proceed to calculate the free energy $F$.

$\displaystyle Z=e^{-\beta F}\Rightarrow F$ $\displaystyle=$
$\displaystyle-\frac{W}{\beta}$ $\displaystyle=$
$\displaystyle\left(-\frac{1}{2\beta}\right)\lim_{s\rightarrow
0}\frac{d}{ds}\left[\frac{\mu^{s}}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-1}\sum_{n_{0},n}e^{-tk_{0}^{2}}e^{-tk^{2}}\right]$
(2.6)

To proceed we consider the low and the high temperature expansions of this
free energy. Then we can discuss the properties of various thermodynamic
quantities, notably the capacities and the compressibilities that can be
derived from $F$.

###  2.1 Low temperature expansion: Approaching $R^{1}\times S^{1}$

To develop the low temperature expansion, we first rewrite the free energy $F$
using the Poisson summation formula (see, for example, [57]) on the sum over
$n_{0}$.

$\displaystyle\sum_{n_{0}=-\infty}^{\infty}e^{-t(2\pi n_{0}/\beta)^{2}}$
$\displaystyle=$ $\displaystyle\frac{\beta}{2\sqrt{\pi
t}}\sum_{n_{0}=-\infty}^{\infty}e^{-n_{0}^{2}\beta^{2}/4t},$ (2.7)

With this replacement, the free energy can then be expressed as

$\displaystyle F=\left(-\frac{1}{4\sqrt{\pi}}\right)\lim_{s\rightarrow
0}\frac{d}{ds}\left[\frac{\mu^{s}}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-\frac{3}{2}}\sum_{n_{0},n}e^{-n_{0}^{2}\beta^{2}/4t}\,e^{-t\left(\frac{2\pi
n}{L}\right)^{2}}\right]$ (2.8)

To start with, we look at the $n_{0}=n=0$ term. This term is formally
divergent so we regularize it by adding in a mass $m$.

$\displaystyle F|_{n_{0}=n=0}$ $\displaystyle=$
$\displaystyle-\frac{1}{4\sqrt{\pi}}\lim_{m\rightarrow 0,s\rightarrow
0}\frac{d}{ds}\bigg{[}\frac{\mu^{s}}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-\frac{3}{2}}\,e^{-tm^{2}}\bigg{]}$
(2.9) $\displaystyle=$ $\displaystyle-\frac{1}{4\sqrt{\pi}}\lim_{m\rightarrow
0}\left(-2\sqrt{\pi}\,m\right)$ $\displaystyle=$ $\displaystyle 0.$

We shall therefore neglect this term in the following.

For the $n_{0}\neq 0$ and $n=0$ term, we have the finite temperature zero mode
contribution to the free energy.

$\displaystyle F|_{n_{0}\neq 0,n=0}$ $\displaystyle=$
$\displaystyle\left(-\frac{1}{2\sqrt{\pi}}\right)\lim_{s\rightarrow
0}\frac{d}{ds}\left[\frac{\mu^{s}}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-\frac{3}{2}}\sum_{n_{0}=1}^{\infty}\,e^{-n_{0}^{2}\beta^{2}/4t}\right]$
(2.10) $\displaystyle=$ $\displaystyle-\frac{1}{2\sqrt{\pi}}\lim_{s\rightarrow
0}\frac{d}{ds}\left[\frac{\mu^{s}}{\Gamma(s)}2^{1-2s}\left(\beta^{2}\right)^{-\frac{1}{2}+s}\Gamma(\frac{1}{2}-s)\zeta(1-2s)\right]$
$\displaystyle=$ $\displaystyle\frac{1}{2\beta}\,{\rm ln}(\mu\beta^{2}),$

which is proportional to ${\rm ln}\beta/\beta$. Whether this term should be
included in evaluating the free energy has been a controversial issue [58, 59,
60]. In [58], it was argued that it should be excluded. The main reason is
that the entropy derived from this term would go like ${\rm ln}\beta$ which
will blow up in the zero temperature limit, being at variance with the third
law of thermodynamics. On the contrary, the inclusion of the zero mode was
advocated in [59], particularly to preserve the inversion relation between
free energies at low and high temperatures [61]. In a more mundane argument,
the inclusion of zero mode is necessary for the mode function to form a
complete set, which in turns enforces causality. When the zero mode is
missing, it has been shown that the lightcone structure emerges in massless
relativistic field theory can be violated [62, 63]. Here, we shall include
this term for completeness and from past experiences. The zero mode in the
spectrum of an invariant operator governs the infrared behavior of quantum
fields in curved spacetimes (see, e.g., [64]). The infrared behavior of
massless minimally-coupled interacting quantum field in de Sitter universe is
an important problem in cosmology (see, e.g., [65] for a review where earlier
references can be found). In our subsequent discussions we shall explore its
consequences by investigating the properties of the corresponding
thermodynamic quantities derived from it. One further remark is that this zero
mode contribution existent in all compact spaces we have considered is
independent of the details of the spatial geometry. Therefore, as we shall see
in the following, this term will contribute to the free energy in the low
temperature expansion for all spatial configurations with a zero mode in the
eigenspectrum. However, in contrast to the interacting quantum field case
mentioned above, for free fields considered here, there is no zero mode
contribution in the zero temperature free energy expression.

For $n_{0}=0$ and $n\neq 0$,

$\displaystyle F|_{n_{0}=0,n\neq 0}$ $\displaystyle=$
$\displaystyle\left(-\frac{1}{2\sqrt{\pi}}\right)\lim_{s\rightarrow
0}\frac{d}{ds}\left[\frac{\mu^{s}}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-\frac{3}{2}}\sum_{n=1}^{\infty}\,e^{-t\left(\frac{2\pi
n}{L}\right)^{2}}\right]$ (2.11) $\displaystyle=$
$\displaystyle-\frac{\pi}{6L}$

As $n_{0}=0$, we have the zero temperature limit here. The result actually
corresponds to the quantum Casimir effect [17, 18, 19] due to the spatial
circle. This is consistent with the usual derivation of the Casimir energy.
With periodic boundary condition, the allowed frequencies are just
$\omega_{n}=2\pi n/L$. The vacuum energy is thus

$\displaystyle
E=\sum_{n=-\infty}^{\infty}\frac{1}{2}\omega_{n}=\left(\frac{2\pi}{L}\right)\zeta(-1)=-\frac{\pi}{6L}$
(2.12)

which is the same as what we had above.

For $n_{0}\neq 0$, $n\neq 0$, we have

$\displaystyle F|_{n_{0}\neq 0,n\neq 0}$ $\displaystyle=$
$\displaystyle\left(-\frac{1}{\sqrt{\pi}}\right)\lim_{s\rightarrow
0}\frac{d}{ds}\left[\frac{\mu^{s}}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-\frac{3}{2}}\sum_{n_{0},n=1}^{\infty}e^{-n_{0}^{2}\beta^{2}/4t}\,e^{-t\left(\frac{2\pi
n}{L}\right)^{2}}\right]$ (2.13) $\displaystyle=$
$\displaystyle\sum_{n_{0},n=1}^{\infty}\left(-\frac{2}{n_{0}\beta}\right)e^{-2\pi
n_{0}n\beta/L}$ $\displaystyle=$
$\displaystyle-\frac{2}{\beta}e^{-2\pi\beta/L}+\cdots,$

where the ellipsis represents terms which are also exponentially small and
non-analytic like $e^{-2\pi\beta/L}$.

Hence, combining expressions in Eqs. (2.9) to (2.13), the free energy in the
low temperature expansion is given by

$\displaystyle F=-\frac{\pi}{6L}+\frac{1}{2\beta}{\rm
ln}(\mu\beta^{2})+\cdots.$ (2.14)

Indeed, when $\beta\rightarrow\infty$ or $T\rightarrow 0$, we are left with
the first term which is just the Casimir energy.

With the free energy we are able to derive various thermodynamic quantities
including the heat capacities and the compressibilities [66] in the low
temperature limit. First, the entropy

$\displaystyle S=1-\frac{1}{2}{\rm ln}(\mu\beta^{2})+\cdots.$ (2.15)

Other than the exponentially small terms, the contribution to the entropy
comes solely from the zero mode since the first term corresponding to the
Casimir effect in the free energy in Eq. (2.14) is independent of temperature.
As we have mentioned above, the ${\rm ln}\beta$ term diverges as the
temperature $T\rightarrow 0$, apparently violating the third law of
thermodynamics in the traditional settings, under the assumptions of large
spatial volume and high temperatures. This is an important issue which
deserves closer examinations, especially in the context of quantum
thermodynamics in spacetimes with curvature or nontrivial topology. We shall
therefore leave this term as it is and try to explore more its consequences on
the other thermodynamic quantities.

With the entropy $S$, we have the internal energy

$\displaystyle E=F+TS=-\frac{\pi}{6L}+\frac{1}{\beta}+\cdots,$ (2.16)

which is independent of the scale $\mu$. The dominant term of the internal
energy comes from the Casimir effect as given by Eq. (2.12). The subdominant
term, which is linear in $T$, is the zero mode contribution. The corresponding
energy density would then be

$\displaystyle\rho=\frac{E}{L}=-\frac{\pi}{6L^{2}}+\frac{1}{\beta L}+\cdots.$
(2.17)

From the free energy, pressure $P$ is just

$\displaystyle P=-\left(\frac{\partial F}{\partial
L}\right)_{T}=-\frac{\pi}{6L^{2}}+\cdots.$ (2.18)

We thus have negative pressure coming from the Casimir effect. The magnitude
increases as the size $L$ gets smaller. Negative pressure would shrink the
spatial circle and this shrinking force would get larger as the size of circle
gets smaller. This process is very similar to that of a gravitational
collapse. In fact, we shall see in the subsequent discussions, the Casimir
effects of spatial spheres with different dimensions would all induce negative
pressure and same kind of collpases should therefore occur.

From the entropy, we obtain the heat capacity at constant volume which is
given by the second temperature derivative of the Helmholtz free energy:

$\displaystyle C_{V}=-\beta\left(\frac{\partial
S}{\partial\beta}\right)_{L}=1+\cdots$ (2.19)

Hence, other than exponentially small terms, $C_{V}$ is basically a constant.

Furthermore, the second derivative of the free energy with respect to volume
$L$ is related to the isothermal compressibility $\kappa_{T}$,

$\displaystyle\kappa_{T}=-\frac{1}{L}\left(\frac{\partial L}{\partial
P}\right)=-\frac{3L^{2}}{\pi}+\cdots.$ (2.20)

$\kappa_{T}$ is negative due to the negative pressure of the Casimir effect.
This means that when pressure is increased, or the magnitude of the pressure
is decreased, the volume $L$ will increase. This is in contrast to the case of
a normal gas where the volume would decrease when the pressure is increased
with positive compressibility.

There is another second derivative of the free energy

$\displaystyle\frac{\partial^{2}F}{\partial T\partial L}=-\frac{\partial
S}{\partial L}=-\frac{\partial P}{\partial T}$ (2.21)

according to the Maxwell relations. This second derivative is related to the
thermal expansion coefficient defined by

$\displaystyle\alpha=\frac{1}{L}\left(\frac{\partial L}{\partial
T}\right)_{P}.$ (2.22)

By the cyclic relation

$\displaystyle\left(\frac{\partial L}{\partial T}\right)_{P}$ $\displaystyle=$
$\displaystyle-\left(\frac{\partial L}{\partial
P}\right)_{T}\left(\frac{\partial P}{\partial T}\right)_{L}$ (2.23)
$\displaystyle=$ $\displaystyle L\kappa_{T}\left(\frac{\partial P}{\partial
T}\right)_{L}$

Hence, the thermal expansion coefficient can be expressed as

$\displaystyle\alpha=\kappa_{T}\left(\frac{\partial P}{\partial
T}\right)_{L}.$ (2.24)

In our present consideration, with the pressure $P$ in Eq. (2.18) having a
leading term independent of temperature, we have

$\displaystyle\left(\frac{\partial P}{\partial
T}\right)_{L}=-\beta^{2}\left(\frac{\partial P}{\partial\beta}\right)_{L}\sim
e^{-2\pi\beta/L}+\cdots$ (2.25)

which is exponentially small. Thus, with the isothermal compressibility in Eq.
(2.20),

$\displaystyle\alpha\sim e^{-2\pi\beta/L}+\cdots$ (2.26)

and it is also exponentially suppressed.

Using the thermodynamic quantities we have obtained, we can also derive the
heat capacity at constant pressure $C_{P}$. By applying the cyclic and the
Maxwell relations, the following relation can be established [67].

$\displaystyle C_{P}$ $\displaystyle=$ $\displaystyle
C_{V}+\beta^{3}L\kappa_{T}\left(\frac{\partial
P}{\partial\beta}\right)^{2}=1+\cdots.$ (2.27)

Since $\partial P/\partial\beta$ is exponentially small, we can see that
$C_{P}\sim C_{V}$ up to order $e^{-2\pi\beta/L}$.

Finally one can also obtain the adiabatic compressibility

$\displaystyle\kappa_{S}=-\frac{1}{L}\left(\frac{\partial L}{\partial
P}\right)_{S}$ (2.28)

Again using the Maxwell and the cyclic relations, we have the identity [67]

$\displaystyle\frac{\kappa_{S}}{\kappa_{T}}=\frac{C_{V}}{C_{P}}\Rightarrow\kappa_{S}=\left(\frac{C_{V}}{C_{P}}\right)\kappa_{T}.$
(2.29)

As we have seen that $C_{P}\sim C_{V}$ up to order $e^{-2\pi\beta/L}$.
Therefore, we also have $\kappa_{S}\sim\kappa_{T}$ other than exponentially
small terms.

$\displaystyle\kappa_{S}=-\frac{1}{L}\left(\frac{\partial L}{\partial
P}\right)=-\frac{3L^{2}}{\pi}+\cdots.$ (2.30)

Note that for a closed system, the thermodynamic processes would be adiabatic.
Here, $\kappa_{S}$ being negative would induce the kind of collapses we
mentioned above for a closed spatial geometry. In the subsequent sections, we
shall see that this is true for the cases of different spatial geometries in
the low temperature expansion.

With the considerations above, it is also possible to establish the
fluctuations of various thermodynamic quantities. Consider a small part of an
equilibrium system. Assume that the small part is still large enough for the
thermodynamic limit to hold. According to the fluctuation theory of Landau and
Lifshitz [67, 68], one can then derive the fluctuations of the various
thermodynamic quantities in this small part. Using the temperature $T$ and the
volume $L$ as independent variables, we have the mean square fluctuation of
the temperature

$\displaystyle\langle(\Delta
T)^{2}\rangle=\frac{T^{2}}{C_{V}}=\frac{1}{\beta^{2}C_{V}},$ (2.31)

while the fluctuation of the volume

$\displaystyle\langle(\Delta
L)^{2}\rangle=LT|\kappa_{T}|=\frac{L|\kappa_{T}|}{\beta}.$ (2.32)

As the fluctuation should be a positive quantity and in our case $\kappa_{T}$
is actually negative, we therefore define the fluctuation to be related to the
absolute value of $\kappa_{T}$ instead.

Then the fluctuation of the internal energy is given by

$\displaystyle\langle(\Delta E)^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{C_{V}}{\beta^{2}}+\frac{L|\kappa_{T}|}{\beta}\left[-\beta\frac{\partial
P}{\partial\beta}-P\right]^{2}$ (2.33) $\displaystyle=$
$\displaystyle\frac{\pi}{12\beta L}+\frac{1}{\beta^{2}}+\cdots$

which is proportional to $T$. Note that as $C_{V}$ is a constant and $\partial
P/\partial\beta$ are exponentially small, the leading contribution comes from
$L|\kappa_{T}|P^{2}/\beta$.

Again according to the Landau-Lifshitz fluctuation theory, the fluctuation of
the pressure is given by

$\displaystyle\langle(\Delta P)^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\beta L|\kappa_{S}|}=\frac{\pi}{3\beta L^{3}}+\cdots$
(2.34)

which is also proportional to $T$.

Moreover, the correlated fluctuation is then

$\displaystyle\langle(\Delta E)(\Delta
P)\rangle=\frac{P}{\beta}=-\frac{\pi}{6\beta L^{2}}+\cdots$ (2.35)

We can see that the leading behaviors of these fluctuations are all
proportional to $T$. In other words, as $T\rightarrow 0$, all the fluctuations
will vanish. It is therefore apparent that the fluctuations derived above are
all of thermal nature. No quantum fluctuations are included in this theory of
fluctuations.

###  2.2 High temperature expansion: Approaching $S^{1}\times R^{1}$

For the high temperature expansion, we take $\beta/L\ll 1$. Actually, we can
also view this as a large $L$ expansion or the infinite space limit. In this
case, we implement the Poisson summation formula on the spatial sum over $n$,

$\displaystyle\sum_{n=-\infty}^{\infty}e^{-t(2\pi n/L)^{2}}$ $\displaystyle=$
$\displaystyle\frac{L}{2\sqrt{\pi
t}}\sum_{n=-\infty}^{\infty}e^{-n^{2}L^{2}/4t}$ (2.36)

Then the free energy becomes

$\displaystyle F=\left(-\frac{L}{4\sqrt{\pi}\beta}\right)\lim_{s\rightarrow
0}\frac{d}{ds}\left[\frac{\mu^{s}}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-\frac{3}{2}}\sum_{n_{0},n}e^{-t\left(\frac{2\pi
n_{0}}{\beta}\right)^{2}}e^{-n^{2}L^{2}/4t}\right].$

Again the $n_{0}=0$, $n=0$ term will be neglected as in the low temperature
case. For $n_{0}\neq 0$, $n=0$, we have

$\displaystyle F|_{n_{0}\neq 0,n=0}$ $\displaystyle=$
$\displaystyle\left(-\frac{L}{2\sqrt{\pi}\beta}\right)\lim_{s\rightarrow
0}\frac{d}{ds}\left[\frac{\mu^{s}}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-\frac{3}{2}}\sum_{n_{0}=1}^{\infty}e^{-t\left(\frac{2\pi
n_{0}}{\beta}\right)^{2}}\right]$ (2.38) $\displaystyle=$
$\displaystyle-\frac{\pi L}{6\beta^{2}}$

which represents the flat space limit $L\rightarrow\infty$. Note that after
implementing the Poisson summation formula in Eq. (2.36), the $n=0$ term here
does not correspond to the zero mode we discussed in the previous subsection.

For $n_{0}=0$, $n\neq 0$,

$\displaystyle F|_{n_{0}=0,n\neq 0}$ $\displaystyle=$
$\displaystyle\left(-\frac{L}{2\sqrt{\pi}\beta}\right)\lim_{s\rightarrow
0}\frac{d}{ds}\left[\frac{\mu^{s}}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-\frac{3}{2}}\sum_{n=1}^{\infty}e^{-n^{2}L^{2}/4t}\right]$
(2.39) $\displaystyle=$ $\displaystyle\frac{1}{2\beta}\,{\rm ln}\left(\mu
L^{2}\right).$

Finally, for $n_{0}\neq 0$, $n\neq 0$,

$\displaystyle F|_{n_{0}\neq 0,n\neq 0}$ $\displaystyle=$
$\displaystyle\left(-\frac{L}{\sqrt{\pi}\beta}\right)\lim_{s\rightarrow
0}\frac{d}{ds}\left[\frac{\mu^{s}}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-\frac{3}{2}}\sum_{n_{0},n=1}^{\infty}e^{-t\left(\frac{2\pi
n_{0}}{\beta}\right)^{2}}e^{-n^{2}L^{2}/4t}\right]$ (2.40) $\displaystyle=$
$\displaystyle\sum_{n_{0},n=1}^{\infty}\left(-\frac{2}{n\beta}\right)e^{-2\pi
n_{0}nL/\beta}$ $\displaystyle=$ $\displaystyle-\frac{2}{\beta}\,e^{-2\pi
L/\beta}+\cdots$

which consist of exponentially small terms. Hence, for high temperature, the
free energy

$\displaystyle F=-\frac{\pi L}{6\beta^{2}}+\frac{1}{2\beta}\,{\rm ln}\left(\mu
L^{2}\right)\cdots$ (2.41)

with the ellipsis indicating terms which are exponentially small like
$e^{-2\pi L/\beta}$. The leading term corresponds to the flat space
$L\rightarrow\infty$ finite temperature result.

It is interesting to note that from this free energy at high temperature in
Eq. (2.41), if we exchange $\beta\leftrightarrow L$,

$\displaystyle\beta F|_{high}\rightarrow-\frac{\pi\beta}{6L}+\frac{1}{2}{\rm
ln}(\mu\beta^{2})+\cdots=\beta F|_{low},$ (2.42)

including the exponentially small terms, where $F|_{low}$ is the free energy
at low temperature in Eq. (2.14). $\beta\leftrightarrow L$ means
$\beta/L\leftrightarrow L/\beta$, that is, the free energy in this case
possesses an inversion symmetry between the low and the high temperatures.
This symmetry is related to the Cardy formula [61] and the contribution of the
zero mode is crucial for this relation to hold.

As in the low temperature expansion, various thermodynamic quantities follow
from the free energy in Eq. (2.41). The entropy

$\displaystyle S=\frac{\pi L}{3\beta}-\frac{1}{2}{\rm ln}\left(\mu
L^{2}\right)+\cdots$ (2.43)

which is proportional to $T$ in the leading term.

The internal energy

$\displaystyle E=\frac{\pi L}{6\beta^{2}}+\cdots$ (2.44)

which is independent of the scale $\mu$. Note that the leading term gives the
flat space energy density

$\displaystyle\rho|_{L\rightarrow\infty}=\left(\frac{E}{L}\right)_{L\rightarrow\infty}=\frac{\pi}{6\beta^{2}}.$
(2.45)

The pressure

$\displaystyle P=\frac{\pi}{6\beta^{2}}-\frac{1}{\beta L}+\cdots$ (2.46)

Here the pressure is positive. As $L\rightarrow\infty$, $P=\rho$ as it should
be for an one-dimensional massless relativistic ideal gas.

It is interesting to note that in the low temperature limit, the dominant
contribution to $P$ is the Casimir pressure which is negative. Here in the
high temperature limit, the pressure is positive instead. Therefore, there
must be a temperature at which the pressure vanishes. Whether the resulting
configuration is stable or not depends on the sign of the compressibility.
This is what we shall further explore in the following consideration.

The heat capacity at constant volume

$\displaystyle C_{V}=\frac{\pi L}{3\beta}+\cdots,$ (2.47)

is proportional to $T$ in the leading behavior.

The isothermal compressibility

$\displaystyle\kappa_{T}=-\beta L+\cdots,$ (2.48)

where the ellipsis again represents exponentially small terms. Here, although
the pressure is positive, the isothermal compressibility is negative. That is,
at constant temperature, when the volume $L$ is decreased, the pressure also
decreases. Remember $\kappa_{T}$ is related to $\partial P/\partial L$. Since
the first term in Eq. (2.46) for $P$ is independent of $L$, the leading
contribution of $\kappa_{T}$ comes from the second term. Hence, we have the
peculiar situation in which the sign of $P$, that is, whether the pressure is
positive or negative, is determined by the first term in the high temperature
expansion, while the sign of the isothermal compressibility $\kappa_{T}$ is
determined by the second term. Therefore, $\kappa_{T}$ could be positive or
negative no matter what the sign of $P$ is. This is different from what we
encounter in the low temperature expansion where $P$ is always negative due to
the Casimir effect and the corresponding $\kappa_{T}$ is also negative from
this negative pressure.

With the isothermal compressibility $\kappa_{T}$, we can derive the thermal
expansion coefficient

$\displaystyle\alpha=-\frac{\pi L}{3}+\beta+\cdots,$ (2.49)

which is also negative as $\kappa_{T}$. This has the peculiar behavior that,
at constant pressure, the volume $L$ will decrease with increase in
temperature.

Using the relation in Eq. (2.27), we obtain the heat capacity at constant
pressure

$\displaystyle C_{P}=-\frac{\pi^{2}L^{2}}{9\beta^{2}}+\frac{\pi
L}{\beta}-1+\cdots,$ (2.50)

which is negative even though $C_{V}$ is positive.

Finally, from the identity in Eq. (2.29), we have the adiabatic
compressibility

$\displaystyle\kappa_{S}=\frac{3\beta^{2}}{\pi}\left[1+\frac{9\beta}{\pi
L}+\frac{72\beta^{2}}{\pi^{2}L^{2}}+O(\frac{\beta^{3}}{{L}^{3}})\right],$
(2.51)

which is positive in contrast to a negative $\kappa_{T}$. If we view the
quantum field in the spatial circle as a closed system, then its evolution
satisfies the adiabatic condition. A positive $\kappa_{S}$ means that when the
volume decreases, the pressure will increase in this adiabatic situation.
Therefore, a stable state may be reached inspite of a negative $\kappa_{T}$.

Furthermore, we can also consider the fluctuations of various thermodynamic
quantities. First, the fluctuation of the internal energy

$\displaystyle\langle(\Delta E)^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}L^{2}}{36\beta^{4}}+\frac{\pi
L}{3\beta^{3}}+\cdots,$ (2.52)

which is proportional to the $T^{4}$. Next, the pressure fluctuation is given
by

$\displaystyle\langle(\Delta
P)^{2}\rangle=\frac{\pi}{3\beta^{3}L}-\frac{3}{\beta^{2}L^{2}}+\frac{3}{\pi\beta
L^{3}}+\cdots$ (2.53)

which is proportional to $T^{3}$. Also,

$\displaystyle\langle(\Delta E)(\Delta
P)\rangle=\frac{\pi}{6\beta^{3}}-\frac{1}{\beta^{2}L}+\cdots$ (2.54)

which is again proportional to $T^{3}$. As in the high temperature limit,
$E\sim L/\beta^{2}$ and $P\sim 1/\beta^{2}$, we have the ratios
$\langle(\Delta E)^{2}\rangle/E^{2}$ of the order of 1, and both
$\langle(\Delta P)^{2}\rangle/P^{2}$ and $\langle(\Delta E)(\Delta
P)\rangle/EP$ proportional to $\beta/L$.

##  3 4d: Einstein universe $S^{1}\times S^{3}$

For thermal fields in the Einstein universe, the topology of spacetime is
$S^{1}\times S^{3}$. The metric can be written as

$\displaystyle ds^{2}=d\tau^{2}+a^{2}d\bar{\Omega}_{3}^{2}$ (3.1)

where $\bar{\Omega}_{3}$ is the solid angle of a three sphere, and $a$ is the
“radius” characterizing the size of the sphere. The operator $H$ in Eq. (2.2)
is now given by

$\displaystyle
H=-\frac{\partial^{2}}{\partial\tau^{2}}-\frac{1}{a^{2}}\bar{\Box},$ (3.2)

where $\bar{\Box}$ is the Laplacian on $S^{3}$. The eigenvalue of $\bar{\Box}$
on $S^{3}$ is [69]

$\displaystyle\bar{\lambda}_{n}=-n(n+2),$ (3.3)

with degeneracy

$\displaystyle\bar{D}_{n}=(n+1)^{2}.$ (3.4)

Hence, the eigenvalue of $H$ is

$\displaystyle\lambda=k_{0}^{2}+\left(\frac{1}{a^{2}}\right)n(n+2).$ (3.5)

The corresponding eigenfunctions are

$\displaystyle\phi_{n_{0},n}(x)=\left(\frac{1}{\sqrt{\beta}}e^{ik_{0}\tau}\right)\left(\frac{1}{\sqrt{a^{3}}}Y_{3}(\Omega)\right),$
(3.6)

where $Y_{3}(\Omega)$ is the hyperspherical harmonics on $S^{3}$ [70]. Using
the eigenvalues, one can establish the free energy, similar to Eq. (2.6), for
the Einstein universe as

$\displaystyle F$ $\displaystyle=$
$\displaystyle-\frac{1}{2\beta}\lim_{s\rightarrow
0}\frac{d}{ds}\bigg{[}\frac{\mu^{s}}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-1}\sum_{n_{0}=-\infty}^{\infty}e^{-t\left(\frac{2\pi
n_{0}}{\beta}\right)^{2}}$ (3.7) $\displaystyle\hskip
100.0pt\sum_{n=0}^{\infty}(n+1)^{2}\,e^{-\left(\frac{t}{a^{2}}\right)n(n+2)}\bigg{]}$

As in the analysis for the Einstein cylinder case in the last section, we
shall consider the low and the high temperature expansions separately [71, 72,
73].

###  3.1 Low temperature expansion

In the low temperature expansion, it is appropriate to first rewrite the sum
over $n_{0}$ using the Poisson summation formula as in Eq. (2.7). The free
energy is therefore expressed as

$\displaystyle F$ $\displaystyle=$
$\displaystyle-\frac{1}{4\sqrt{\pi}}\lim_{s\rightarrow
0}\frac{d}{ds}\bigg{[}\frac{\mu^{s}}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-\frac{3}{2}}\sum_{n_{0}=-\infty}^{\infty}e^{-n_{0}^{2}\beta^{2}/4t}$
(3.8) $\displaystyle\hskip
100.0pt\sum_{n=0}^{\infty}(n+1)^{2}\,e^{-\left(\frac{t}{a^{2}}\right)n(n+2)}\bigg{]}.$

To start with, we look at the $n_{0}=n=0$ term. This term is exactly the same
as in Eq. (2.9) which is regularized to zero. We shall therefore neglect it in
the following. For $n_{0}=0$ and $n\neq 0$ in Eq. (3.8), we evaluate the
$t$-integral to give

$\displaystyle F|_{n_{0}=0,n\neq 0}$ $\displaystyle=$
$\displaystyle\lim_{s\rightarrow
0}\frac{d}{ds}\left[\left(-\frac{1}{4\sqrt{\pi}}\right)\left(\frac{\mu^{s}\Gamma(s-\frac{1}{2})}{\Gamma(s)}\right)\sum_{n=1}^{\infty}(n+1)^{2}\left(\frac{a^{2}}{n(n+2)}\right)^{s-\frac{1}{2}}\right]$

Here we concentrate on the sum

$\displaystyle\sum_{n=1}^{\infty}(n+1)^{2}\left(\frac{1}{n(n+2)}\right)^{s-\frac{1}{2}}=\sum_{n=0}^{\infty}(n+2)^{2}\left(\frac{1}{(n+3)(n+1)}\right)^{s-\frac{1}{2}}$
(3.10)

Using the Plana summation formula

$\displaystyle\sum_{n=0}^{\infty}f(n)=\int_{0}^{\infty}dx\,f(x)+\frac{1}{2}f(0)+i\int_{0}^{\infty}dy\left(\frac{f(iy)-f(-iy)}{e^{2\pi
y}-1}\right),$ (3.11)

it is possible to evalute this sum as an analytic function of $s$ [74, 75].

$\displaystyle\sum_{n=1}^{\infty}(n+1)^{2}\left(\frac{1}{n(n+2)}\right)^{s-\frac{1}{2}}=\left(-\frac{1}{16}\right)\frac{1}{s}-0.411461+\cdots$
(3.12)

With these considerations, we have the $n_{0}=0$ part of the free energy,

$\displaystyle F|_{n_{0}=0}=-\frac{0.224909}{a}-\frac{1}{32a}\,{\rm
ln}\left(\mu a^{2}\right)$ (3.13)

which represents the contribution from the Casimir effect. This result is
exactly the same as the effective potential obtained in [74]. The dependence
on $\mu$ in the free energy occurs in all Einstein universes with spatial odd
spheres. The corresponding zeta function at $s=0$ is nonzero and hence
contributes to a ln$\mu$ term.

For $n_{0}\neq 0$ and $n=0$, we evaluate the $t$-integral to give

$\displaystyle F|_{n_{0}\neq 0,n=0}$ $\displaystyle=$
$\displaystyle\left(-\frac{1}{2\sqrt{\pi}}\right)\lim_{s\rightarrow
0}\frac{d}{ds}\left[\left(\frac{\mu^{s}}{2^{2s-1}\Gamma(s)}\right)\Gamma(\frac{1}{2}-s)\zeta(1-2s)\right]$
(3.14) $\displaystyle=$ $\displaystyle\frac{1}{2\beta}\,{\rm
ln}(\mu\beta^{2}).$

This is the contribution from the zero mode as we have discussed in the last
section. This same expression will be present in the low temperature
expansions of the free energy in all the cases we consider in this paper.

For $n_{0}\neq 0$ and $n\neq 0$, we again evaluate the $t$-integral first to
obtain

$\displaystyle F|_{n_{0}\neq 0,n\neq 0}$ $\displaystyle=$
$\displaystyle\sum_{n_{0},n=1}^{\infty}\left(-\frac{(n+1)^{2}}{n_{0}\beta}\right)e^{\sqrt{n(n+2)}\,n_{0}\beta/a}$
(3.15) $\displaystyle=$
$\displaystyle-\frac{4}{\beta}e^{-\sqrt{3}\beta/a}+\cdots$

The $n_{0}\neq 0$ and $n\neq 0$ contribution to the free energy in the low
energy expansion is exponentially small, and combining the various parts we
have

$\displaystyle F=-\frac{0.224909}{a}-\frac{1}{32a}{\rm ln}(\mu
a^{2})+\frac{1}{2\beta}{\rm ln}(\mu\beta^{2})+\cdots$ (3.16)

where the ellipsis represents exponentially small terms.

With the above Helmholtz free energy in the low temperature expansion, we
derive the various thermodynamic quantities. For the entropy

$\displaystyle S=1-\frac{1}{2}\,{\rm ln}(\mu\beta^{2})+\cdots$ (3.17)

which is the entropy of the zero mode with the same expression as in Eq.
(2.15).

For the internal energy

$\displaystyle E=-\frac{0.224909}{a}-\frac{1}{32a}{\rm ln}(\mu
a^{2})+\frac{1}{\beta}+\cdots,$ (3.18)

and the energy density, with the volume of the three sphere $V=4\pi^{2}a^{3}$,

$\displaystyle\rho=\frac{E}{V}=-\frac{0.00570}{a^{4}}-\frac{1}{128\pi^{2}a^{4}}{\rm
ln}(\mu a^{2})+\frac{1}{4\pi^{2}a^{3}\beta}+\cdots.$ (3.19)

The leading behavior in the low temperature expansion comes from the first two
terms. They constitute the Casimir contribution which is $\mu$ dependent.
Moreover, the pressure

$\displaystyle P=-\frac{0.00274258}{a^{4}}-\frac{1}{192\pi^{2}a^{4}}{\rm
ln}(\mu a^{2})+\cdots$ (3.20)

which is negative due to the Casimir terms and is also dependent on $\mu$.

From the entropy $S$, we derive the heat capacity at constant volume

$\displaystyle C_{V}=1+\cdots$ (3.21)

which is basically a constant with the temperature terms exponentially
suppressed. From the pressure $P$, we obtain the isothermal compressibility

$\displaystyle\kappa_{T}=-a^{4}\left[0.00330496+\frac{1}{144\pi^{2}}{\rm
ln}(\mu a^{2})\right]^{-1}+\cdots$ (3.22)

which is negative and dependent on $\mu$. Again, the temperature-dependent
terms are exponentially suppressed. Since the temperature-dependent terms in
$P$ are exponentially small, the thermal expansion coefficient $\alpha$ which
depends on $\partial P/\partial T$ is therefore exponentially small too. For
the same reason, the heat capacity at constant pressure $C_{P}\sim C_{V}$ up
to exponentially small terms. The same applies to the two compressibilities
$\kappa_{S}\sim\kappa_{T}$.

With $C_{V}$ and $\kappa_{T}$, we can derive the fluctuation of the internal
energy

$\displaystyle\langle(\Delta E)^{2}\rangle=\frac{2\pi^{2}}{\beta
a}\left[\frac{(0.00274258+0.000527714\,{\rm ln}(\mu
a^{2}))^{2}}{0.00330496+0.000703619\,{\rm ln}(\mu
a^{2})}\right]+\frac{1}{\beta^{2}}+\cdots$ (3.23)

which is proportional $T$. Note that since $\kappa_{T}$ is negative and the
fluctuation should be a positive quantity, we have taken the absolute value of
$\kappa_{T}$ in the formula for $\langle(\Delta E)^{2}\rangle$. The
fluctuation of $P$ is related to the adiabatic compressibility $\kappa_{S}$.
Again we take its absolute value in the formula.

$\displaystyle\langle(\Delta P)^{2}\rangle=\frac{1}{\beta
a^{7}}\left[0.000167431+0.0000356458\,{\rm ln}(\mu a^{2})\right]+\cdots$
(3.24)

which is also proportional to $T$ and dependent on the renormalization scale.
Moreover, the correlation between the fluctuations of $E$ and $P$,

$\displaystyle\langle(\Delta E)(\Delta P)\rangle=-\frac{1}{\beta
a^{4}}\left[0.00274258+0.000527714\,{\rm ln}(\mu a^{2})\right]+\cdots$ (3.25)

It is interesting to see that here all the thermodynamic quantities and their
fluctuations depend on the renormalization scale $\mu$. This dependence
indicates that the thermodynamics of a quantum field in the Einstein universe
involves divergent quantities and a renormalization procedure is needed to
define the various physically measureable quantities. As we shall see in the
following section, this is not the case with even spatial $S^{2}$ and $S^{4}$
dimensional spheres. Actually, the dependence on $\mu$ comes from the fact
that we have an odd-dimensional sphere ($S^{3}$) here. We would expect the
same kind of dependence to occur for all odd spheres or (when adding in the
time dimension) even-dimensional spacetimes [76].

###  3.2 High temperature expansion

In this subsection, we concentrate on the high temperature expansion for the
Einstein universe with $\beta/a\ll 1$. In this case, we consider the Helmholtz
free energy in Eq. (3.7). Here, we would neglect the $n_{0}=n=0$ term. This
term is independent of the temperature $T$ as well as the size $a$ of the
spatial three sphere. Although it is formally divergent, it should be
subtracted in the renormalization procedure.

For the $n_{0}=0$, $n\neq 0$ contribution to the free energy,

$\displaystyle F_{n_{0}=0,n\neq 0}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\beta}\lim_{s\rightarrow
0}\frac{d}{ds}\bigg{[}\frac{\mu^{s}}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-1}\sum_{n=1}^{\infty}(n+1)^{2}\,e^{-\left(\frac{t}{a^{2}}\right)n(n+2)}\bigg{]}$
(3.26) $\displaystyle=$ $\displaystyle-\frac{1}{2\beta}\lim_{s\rightarrow
0}\frac{d}{ds}\left\\{(\mu
a^{2})^{s}\sum_{n=1}^{\infty}(n+1)^{2}[n(n+2)]^{-s}\right\\}.$

We apply the Plana summation formula, as in Eq. (3.11), to the sum

$\displaystyle\sum_{n=1}^{\infty}(n+1)^{2}[n(n+2)]^{-s}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}(n+2)^{2}[(n+1)(n+3)]^{-s}$ (3.27)
$\displaystyle=$ $\displaystyle-1-1.20563\,s+\cdots$

This shows that the sum as an analytic function of $s$ is well behaved near
$s=0$. The $n_{0}=0$, $n\neq 0$ contribution to the free energy can therefore
be written as

$\displaystyle F_{n_{0}=0,n\neq 0}$ $\displaystyle=$
$\displaystyle\frac{0.60282}{\beta}+\frac{1}{2\beta}{\rm ln}(\mu a^{2}).$
(3.28)

Next, we consider the $n_{0}\neq 0$ part of the free energy

$\displaystyle F|_{n_{0}\neq 0}$ $\displaystyle=$
$\displaystyle-\frac{1}{\beta}\lim_{s\rightarrow
0}\frac{d}{ds}\bigg{[}\frac{\mu^{s}}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-1}\sum_{n_{0}=1}^{\infty}e^{-t\left(\frac{2\pi
n_{0}}{\beta}\right)^{2}}$ (3.29) $\displaystyle\hskip
100.0pt\sum_{n=0}^{\infty}(n+1)^{2}\,e^{-\left(\frac{t}{a^{2}}\right)n(n+2)}\bigg{]}$

We concentrate on the sum over $n$. In the high temperature limit, we are
interested in the asymptotic behavior of this sum [77] when $a$ is large or as
a power series in $t/a$. To develop such a series, we again use the Plana
summation formula in Eq. (3.11).

$\displaystyle\sum_{n=0}^{\infty}(n+1)^{2}\,e^{-\left(\frac{t}{a^{2}}\right)n(n+2)}$
(3.30) $\displaystyle=$
$\displaystyle\left(\frac{a}{\sqrt{t}}\right)^{3}\left[\frac{\sqrt{\pi}}{4}+\frac{\sqrt{\pi}}{4}\left(\frac{t}{a}\right)+\frac{\sqrt{\pi}}{8}\left(\frac{t}{a}\right)^{2}+\cdots\right].$

With this asymptotic expansion, one can evaluate Eq. (3.29) as a power series
in $\beta/a$.

$\displaystyle F|_{n_{0}\neq
0}=-\frac{\pi^{4}a^{3}}{45\beta^{4}}-\frac{\pi^{2}a}{12\beta^{2}}-\frac{1}{32a}\left[2\gamma+{\rm
ln}\left(\frac{\mu\beta^{2}}{16\pi^{2}}\right)\right]+\cdots$ (3.31)

where $\gamma$ is the Euler constant.

With Eqs. (3.28) and (3.31), the free energy in the high temperature expansion
is then given by

$\displaystyle F$ $\displaystyle=$
$\displaystyle-\frac{\pi^{4}a^{3}}{45\beta^{4}}-\frac{\pi^{2}a}{12\beta^{2}}+\frac{0.60282}{\beta}$
(3.32) $\displaystyle\ \ \ \ \ +\frac{1}{2\beta}{\rm ln}(\mu
a^{2})-\frac{1}{32a}\left[2\gamma+{\rm
ln}\left(\frac{\mu\beta^{2}}{16\pi^{2}}\right)\right]+\cdots$

From this free energy, we again derive the various thermodynamic quantities.

For the entropy,

$\displaystyle
S=\frac{4\pi^{4}a^{3}}{45\beta^{3}}+\frac{\pi^{2}a}{6\beta}-\frac{1}{2}{\rm
ln}(\mu a^{2})-0.60282-\frac{\beta}{16a}+\cdots$ (3.33)

which is proportional to $T^{3}$ in the leading behavior. Note that it is
dependent on the renormalization scale $\mu$ but only in the subleading term.
For the internal energy,

$\displaystyle
E=\frac{\pi^{4}a^{3}}{15\beta^{4}}+\frac{\pi^{2}a}{12\beta^{2}}-\frac{1}{32a}\left[2+2\gamma+{\rm
ln}\left(\frac{\mu\beta^{2}}{16\pi^{2}}\right)\right]+\cdots.$ (3.34)

With the volume of the three sphere as $V=2\pi^{2}a^{3}$, the first term of
the internal energy $E$ is $\pi^{2}V/30\beta^{4}$ which is again the Stefan’s
law in flat spatial three dimensions. Indeed, in the infinite space limit,
$a\rightarrow\infty$, the energy density

$\displaystyle\rho|_{a\rightarrow\infty}=\frac{E|_{a\rightarrow\infty}}{V}=\frac{\pi^{2}}{30\beta^{4}}.$
(3.35)

As for the pressure,

$\displaystyle
P=\frac{\pi^{2}}{90\beta^{4}}+\frac{1}{72\beta^{2}a^{2}}-\frac{1}{6\pi^{2}\beta
a^{3}}-\frac{1}{192\pi^{2}a^{4}}\left[2\gamma+{\rm
ln}\left(\frac{\mu\beta^{2}}{16\pi^{2}}\right)\right]+\cdots.$ (3.36)

The first term gives the pressure in the infinite space limit,

$\displaystyle
P|_{a\rightarrow\infty}=\frac{\pi^{2}}{90\beta^{4}}=\frac{\rho|_{a\rightarrow\infty}}{3}$
(3.37)

which is the equation of state for a massless relativistic ideal gas in flat
three spatial dimensions.

For the heat capacity at constant volume

$\displaystyle
C_{V}=\frac{4\pi^{4}a^{3}}{15\beta^{3}}+\frac{\pi^{2}a}{6\beta}+\frac{\beta}{16a}+\cdots$
(3.38)

which is proportional to $T^{3}$ in the leading term and is independent of
$\mu$. Furthermore, the isothermal compressibility $\kappa_{T}$ is

$\displaystyle\kappa_{T}$ $\displaystyle=$ $\displaystyle
108\beta^{2}a^{2}+\left(\frac{1944}{\pi^{2}}\right)\beta^{3}a+\frac{81\beta^{4}}{\pi^{2}}\left[\frac{432}{\pi^{2}}+2\gamma+{\rm
ln}\left(\frac{\mu\beta^{2}}{16\pi^{2}}\right)\right]+\cdots.$

From the isothermal compressibility, one can obtain the thermal expansion
coefficient $\alpha$, the heat capacity at constant pressure $C_{P}$, and the
adiabatic compressibility $\kappa_{S}$. They are, respectively,

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{24\pi^{2}a^{2}}{5\beta}+\frac{432\,a}{5}+\frac{18\beta}{5}\left[\frac{5}{6}+\frac{432}{\pi^{2}}+2\gamma+{\rm
ln}\left(\frac{\mu\beta^{2}}{16\pi^{2}}\right)\right]+\cdots.$ $\displaystyle
C_{P}$ $\displaystyle=$
$\displaystyle\frac{32\pi^{6}a^{5}}{75\beta^{5}}+\frac{192\pi^{4}a^{4}}{25\beta^{4}}+\frac{8\pi^{4}a^{3}}{25\beta^{3}}\left[\frac{5}{2}+\frac{432}{\pi^{2}}+2\gamma+{\rm
ln}\left(\frac{\mu\beta^{2}}{16\pi^{2}}\right)\right]+\cdots.$
$\displaystyle\kappa_{S}$ $\displaystyle=$
$\displaystyle\frac{135\beta^{4}}{2\pi^{2}}-\frac{675\beta^{6}}{8\pi^{4}a^{2}}+\frac{10125\beta^{7}}{8\pi^{6}a^{3}}+\cdots.$
(3.42)

The relationship between the heat capacities is
$C_{P}=C_{V}+\beta^{3}V\kappa_{T}(\partial P/\partial\beta)^{2}$. In the
present high temperature expansion, the second term which is of the order of
$(a/\beta)^{5}$ dominates over the first term which is only of the order
$(a/\beta)^{3}$. Hence, we have $C_{P}\gg C_{V}$. For the same reason, since
$\kappa_{S}=(C_{V}/C_{P})\kappa_{T}$, we have $\kappa_{T}\gg\kappa_{S}$.

Next, we give the fluctuations for various thermodynamic quantities in the
high temperature expansion. For the internal energy fluctuation,

$\displaystyle\langle(\Delta E)^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{6\pi^{6}a^{5}}{25\beta^{7}}+\frac{108\pi^{4}a^{4}}{25\beta^{6}}+\frac{9\pi^{4}}{50\beta^{5}}\left[\frac{70}{27}+\frac{432}{\pi^{2}}+2\gamma+{\rm
ln}\left(\frac{\mu\beta^{2}}{16\pi^{2}}\right)\right]+\cdots.$

For the pressure fluctuation,

$\displaystyle\langle(\Delta
P)^{2}\rangle=\frac{1}{135\beta^{5}a^{3}}+\frac{1}{108\pi^{2}\beta^{3}a^{5}}-\frac{5}{36\pi^{4}\beta^{2}a^{6}}+\cdots.$
(3.44)

Lastly, the correlated fluctuation of $E$ and $P$ is

$\displaystyle\langle(\Delta E)(\Delta
P)\rangle=\frac{\pi^{2}}{90\beta^{5}}+\frac{1}{72\beta^{3}a^{2}}-\frac{1}{6\pi^{2}\beta^{2}a^{3}}+\cdots.$
(3.45)

We see that in this high temperature expansion, $\langle(\Delta E)^{2}\rangle$
is proportional to $T^{7}$, while both $\langle(\Delta P)^{2}\rangle$ and
$\langle(\Delta E)(\Delta P)\rangle$ are proportional to $T^{5}$.

##  4 Even spatial dimensions: $S^{1}\times S^{2}$ and $S^{1}\times S^{4}$

In the last section we have considered the Einstein universe with a spatial
three sphere. The consideration there can be extended to Einstein universes
with the general topology of $S^{1}\times S^{d-1}$ with a spatial
$(d-1)$-sphere. The metric is given by

$\displaystyle ds^{2}=d\tau^{2}+a^{2}d\bar{\Omega}_{d-1}^{2}$ (4.1)

where $\bar{\Omega}_{d-1}$ is the solid angle of the $(d-1)$-sphere. Here the
eigenvalue of the Laplacian $\bar{\Box}$ on $S^{d-1}$ is given by [69]

$\displaystyle\bar{\lambda}_{n}=-n(n+d-2),$ (4.2)

and the degeneracy is

$\displaystyle\bar{D}_{n}=\frac{(2n+d-2)(n+d-3)!}{n!(d-2)!}$ (4.3)

Using $\bar{\lambda}_{n}$ and $\bar{D}_{n}$, one can write the Helmholtz free
energy in this generalized Einstein universe as

$\displaystyle F=-\frac{1}{2\beta}\lim_{s\rightarrow
0}\frac{d}{ds}\bigg{[}\frac{\mu^{s}}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-1}\sum_{n_{0}=-\infty}^{\infty}e^{-t\left(\frac{2\pi
n_{0}}{\beta}\right)^{2}}\sum_{n=0}^{\infty}\bar{D}_{n}e^{-\left(\frac{t}{a^{2}}\right)\bar{\lambda}_{n}}\bigg{]}.$
(4.4)

With analyses similar to those in the last section, one can derive the
corresponding free energy $F$ in both the low temperature and the high
temperature expansions for any value of the dimension $d$. Then, one can
derive the various thermodynamic quantities from the free energy. In the
following subsections, we shall work out the $d=3$ and $d=5$ cases explicitly.
Together with the case $d=4$ in the last section, we can have a better
understanding in how the various thermodynamic quantities depend on the
spacetime dimension [72, 76].

###  4.1 Low temperature expansion

In the low temperature expansion with $a/\beta\ll 1$, we first rewrite the sum
over $n_{0}$ in Eq. (4.4) using the Poisson summation formula, as in Eq. (3.8)
for the 4d Einstein universe case. Then, following similar considerations
there, we obtain the free energies for the $d=3$ and $d=5$ Einstein universe
cases.

$\displaystyle F|_{d=3}$ $\displaystyle=$
$\displaystyle-\frac{0.132548}{a}+\frac{1}{2\beta}{\rm
ln}(\mu\beta^{2})+\cdots,$ $\displaystyle F|_{d=5}$ $\displaystyle=$
$\displaystyle-\frac{0.215872}{a}+\frac{1}{2\beta}{\rm
ln}(\mu\beta^{2})+\cdots,$ (4.5)

where the ellipsis represents terms which are exponentially small. Comparing
with the free energy of $d=4$ Einstein universe in Eq. (3.16), we see that in
the free energies above, the term ${\rm ln}(\mu a^{2})/a$ is missing. This can
be traced back to the sum in Eq. (3.12), where the sum, as an analytic
function of $s$, has a pole at $s=0$. This is characteristic for this kind of
sums on unit odd dimensional spheres. On the other hand, for even spheres like
$S^{2}$ and $S^{4}$, the corresponding sums would be power series in $s$
without pole singularities at $s=0$. For this reason, the free energies above
for the $d=3$ and $d=5$ Einstein universes do not contain terms like ${\rm
ln}(\mu a^{2})/a$. Subsequently, as we shall see below, the compressibilities
$\kappa_{T}$ and $\kappa_{S}$, and the thermal expansion coefficient $\alpha$,
together with the various fluctuations, will not depend on the renormalization
scale $\mu$.

Using these free energies, we derive the entropies

$\displaystyle S|_{d=3}$ $\displaystyle=$ $\displaystyle 1-\frac{1}{2}{\rm
ln}(\mu\beta^{2})+\cdots,$ $\displaystyle S|_{d=5}$ $\displaystyle=$
$\displaystyle 1-\frac{1}{2}{\rm ln}(\mu\beta^{2})+\cdots.$ (4.6)

Both have a dependence on the renormalization scale $\mu$. From the entropies,
we have the heat capacities at constant volume,

$\displaystyle C_{V}|_{d=3}$ $\displaystyle=$ $\displaystyle 1+\cdots,$
$\displaystyle C_{V}|_{d=5}$ $\displaystyle=$ $\displaystyle 1+\cdots.$ (4.7)

which are constant except for exponentially small terms. The corresponding
quantities of the Einstein universe in the last section have the same form as
the ones above. Therefore, we would expect that this is true for all the
Einstein universes with general $(d-1)$ dimensional spatial spheres.

The internal energies

$\displaystyle E|_{d=3}$ $\displaystyle=$
$\displaystyle-\frac{0.132548}{a}+\frac{1}{\beta}+\cdots,$ $\displaystyle
E|_{d=5}$ $\displaystyle=$
$\displaystyle-\frac{0.215872}{a}+\frac{1}{\beta}+\cdots,$ (4.8)

and the corresponding energy densities

$\displaystyle\rho|_{d=3}$ $\displaystyle=$
$\displaystyle-\frac{0.0105478}{a^{3}}+\frac{1}{4\pi\beta a^{2}}+\cdots,$
(4.9) $\displaystyle\rho|_{d=5}$ $\displaystyle=$
$\displaystyle-\frac{0.00820215}{a^{5}}+\frac{3}{8\pi^{2}\beta a^{5}}+\cdots.$
(4.10)

The first term in both expressions represents the Casimir energy. Other than
the exponentially small terms, there is also one term proportional to $T$. We
expect Einstein universes with odd spacetimes dimensions to have the same
behaviors for $E$ and $\rho$ as above. Moreover, for Einstein universes with
even spacetime dimensions, there should be an extra term proportional to ${\rm
ln}(\mu a^{2})/a$ as shown in Eqs. (3.18) and (3.19) in the last section.

For the pressures in these cases,

$\displaystyle P|_{d=3}$ $\displaystyle=$
$\displaystyle-\frac{0.00527392}{a^{3}}+\cdots,$ $\displaystyle P|_{d=5}$
$\displaystyle=$ $\displaystyle-\frac{0.00205054}{a^{5}}+\cdots.$ (4.11)

The Casimir pressures in both cases are negative. This is true for all the
cases we have considered in the previous sections in the low temperature
expansion. From the expressions for the pressures, one can derive the
isothermal compressibilities

$\displaystyle\kappa_{T}|_{d=3}$ $\displaystyle=$
$\displaystyle-126.408\,a^{3}+\cdots,$ $\displaystyle\kappa_{T}|_{d=5}$
$\displaystyle=$ $\displaystyle-390.141\,a^{5}+\cdots.$ (4.12)

Both isothermal compressibilities are negative due to the negative pressure.
This is also true for all the isothermal compressibilities in all the cases we
have considered. However, unlike $\kappa_{T}$ in the 4d Einstein universe but
similar to the Einstein cylinder case, the isothermal compressibilities here
are not dependent on $\mu$. Since the pressures $P$ are dominated by the
Casimir effect which is independent of temperature, the thermal expansion
coefficients are exponentially small in these case. Also, for the same reason,
we have $C_{P}\sim C_{V}$ and $\kappa_{S}\sim\kappa_{T}$ up to exponentially
small terms.

Finally, we layout the expressions for the various fluctuations. For the
internal energy,

$\displaystyle\langle(\Delta E)^{2}\rangle|_{d=3}$ $\displaystyle=$
$\displaystyle 0.0441827\left(\frac{1}{\beta
a}\right)+\frac{1}{\beta^{2}}+\cdots,$ $\displaystyle\langle(\Delta
E)^{2}\rangle|_{d=5}$ $\displaystyle=$ $\displaystyle
0.0431744\left(\frac{1}{\beta a}\right)+\frac{1}{\beta^{2}}+\cdots.$ (4.13)

For the pressure,

$\displaystyle\langle(\Delta P)^{2}\rangle|_{d=3}$ $\displaystyle=$
$\displaystyle 0.000629528\left(\frac{1}{\beta a^{5}}\right)+\cdots,$
$\displaystyle\langle(\Delta P)^{2}\rangle|_{d=5}$ $\displaystyle=$
$\displaystyle 0.0000973889\left(\frac{1}{\beta a^{9}}\right)+\cdots,$ (4.14)

The fluctuations are proportional to $T$. This is true for all the cases we
have considered. In addition, the correlated fluctuations of $E$ and $P$,

$\displaystyle\langle(\Delta E)(\Delta P)\rangle|_{d=3}$ $\displaystyle=$
$\displaystyle-0.00527392\left(\frac{1}{\beta a^{3}}\right)+\cdots,$
$\displaystyle\langle(\Delta E)(\Delta P)\rangle|_{d=5}$ $\displaystyle=$
$\displaystyle-0.00205054\left(\frac{1}{\beta a^{5}}\right)+\cdots,$ (4.15)

which are also proportional to $T$.

Since the energies and the pressures in the low temperature limit are all
dominated by the Casimir effect, their leading behaviors are independent of
$T$. Therefore, the ratios $\langle(\Delta E)^{2}\rangle/E^{2}$,
$\langle(\Delta P)^{2}\rangle/P^{2}$, and $\langle(\Delta E)(\Delta
P)\rangle/EP$ go like $a/\beta$ or $aT$ in the low temperature limit.
Fluctuations of all Einstein universes in the low temperature expansion have
similar behavior, except that for even spacetimes the fluctuations will have
dependences on the renormalization scale $\mu$, while here for odd spacetimes
they will not.

###  4.2 High temperature expansion

In this subsection we develop the high temperature expansion with $\beta/a\ll
1$. For the Helmholtz free energies in Eq. (4.4), we again use the same
procedure as in the 4d Einstein universe case in Sec. 3 to obtain

$\displaystyle F|_{d=3}$ $\displaystyle=$
$\displaystyle-\frac{2\zeta(3)a^{2}}{\beta^{3}}+\left(\frac{1}{\beta}\right)\left[0.580842+\frac{1}{3}{\rm
ln}(\mu a^{2})+\frac{1}{6}{\rm ln}(\mu\beta^{2})\right]$ $\displaystyle\hskip
30.0pt-\frac{\beta}{360a^{2}}-\frac{\beta^{3}}{113400a^{4}}+\cdots,$
$\displaystyle F|_{d=5}$ $\displaystyle=$
$\displaystyle-\frac{2\zeta(5)a^{4}}{\beta^{5}}-\frac{2\zeta(3)a^{2}}{3\beta^{3}}$
(4.16) $\displaystyle\hskip
30.0pt+\left(\frac{1}{\beta}\right)\left[0.276064+0.338889\,{\rm ln}(\mu
a^{2})+\frac{29}{180}{\rm ln}(\mu\beta^{2})\right]$ $\displaystyle\hskip
30.0pt-\frac{37\beta}{4536a^{2}}+\cdots.$

The corresponding entropies are

$\displaystyle S|_{d=3}$ $\displaystyle=$
$\displaystyle\frac{6\zeta(3)a^{2}}{\beta^{2}}-\left[0.247509+\frac{1}{3}{\rm
ln}(\mu a^{2})+\frac{1}{6}{\rm ln}(\mu\beta^{2})\right]$ $\displaystyle\hskip
30.0pt-\frac{\beta^{2}}{360a^{2}}-\frac{\beta^{4}}{37800a^{4}}+\cdots,$
$\displaystyle S|_{d=5}$ $\displaystyle=$
$\displaystyle\frac{10\zeta(5)a^{4}}{\beta^{4}}+\frac{2\zeta(3)a^{2}}{\beta^{2}}$
$\displaystyle+\left[0.0461582-0.338889\,{\rm ln}(\mu
a^{2})-\frac{29}{180}{\rm
ln}(\mu\beta^{2})\right]-\frac{37\beta^{2}}{4536a^{2}}+\cdots.$

The internal energies are

$\displaystyle E|_{d=3}$ $\displaystyle=$
$\displaystyle\frac{4\zeta(3)a^{2}}{\beta^{3}}+\frac{0.333333}{\beta}-\frac{\beta}{180a^{2}}-\frac{\beta^{3}}{28350a^{4}}+\cdots,$
$\displaystyle E|_{d=5}$ $\displaystyle=$
$\displaystyle\frac{8\zeta(5)a^{4}}{\beta^{5}}+\frac{4\zeta(3)a^{2}}{3\beta^{3}}+\frac{0.322222}{\beta}-\frac{37\beta}{2268a^{2}}+\cdots.$
(4.18)

The first terms again give the flat space limit of the internal energy. Hence,
as $a\rightarrow\infty$, we have the energy density

$\displaystyle\rho|_{d=3,a\rightarrow\infty}$ $\displaystyle=$
$\displaystyle\frac{E|_{d=3,a\rightarrow\infty}}{4\pi
a^{2}}=\frac{\zeta(3)}{\pi\beta^{3}},$
$\displaystyle\rho|_{d=5,a\rightarrow\infty}$ $\displaystyle=$
$\displaystyle\frac{E|_{d=5,a\rightarrow\infty}}{\left(\frac{8}{3}\pi^{2}a^{4}\right)}=\frac{3\zeta(5)}{\pi^{2}\beta^{5}}.$
(4.19)

For the pressures

$\displaystyle P|_{d=3}$ $\displaystyle=$
$\displaystyle\frac{\zeta(3)}{2\pi\beta^{3}}-\frac{1}{12\pi\beta
a^{2}}-\frac{\beta}{1440\pi a^{4}}-\frac{\beta^{3}}{226800\pi a^{6}}+\cdots,$
$\displaystyle P|_{d=5}$ $\displaystyle=$
$\displaystyle\frac{3\zeta(5)}{4\pi^{2}\beta^{5}}+\frac{\zeta(3)}{8\pi^{2}\beta^{3}a^{2}}-\frac{0.00643812}{\beta
a^{4}}-\frac{37\beta}{24192\pi^{2}a^{6}}+\cdots.$

The first terms give the flat space limit. As $a\rightarrow\infty$, we see
that $P$ will be proportional to the energy density $\rho$ in Eq. (4.19).

$\displaystyle
P|_{d=3,a\rightarrow\infty}=\frac{\zeta(3)}{2\pi\beta^{3}}=\frac{1}{2}\rho|_{d=3,a\rightarrow\infty},$
$\displaystyle
P|_{d=5,a\rightarrow\infty}=\frac{3\zeta(5)}{4\pi^{2}\beta^{5}}=\frac{1}{4}\rho|_{d=5,a\rightarrow\infty}.$
(4.21)

This is consistent with the equation of state of a massless relativistic ideal
gas, $P=\rho/n$, in a $n$-dimensional flat space. From the entropies above, we
can also work out the heat capacities at constant volume.

$\displaystyle C_{V}|_{d=3}$ $\displaystyle=$
$\displaystyle\frac{12\zeta(3)a^{2}}{\beta^{2}}+\frac{1}{3}+\frac{\beta^{2}}{180a^{2}}+\frac{\beta^{4}}{9450a^{4}}+\cdots,$
$\displaystyle C_{V}|_{d=5}$ $\displaystyle=$
$\displaystyle\frac{40\zeta(5)a^{4}}{\beta^{4}}+\frac{4\zeta(3)a^{2}}{\beta^{2}}+\frac{29}{90}+\frac{37\beta^{2}}{2268a^{2}}+\cdots.$
(4.22)

From the pressures, we can derive the isothermal compressibilities,

$\displaystyle\kappa_{T}|_{d=3}$ $\displaystyle=$ $\displaystyle-12\pi\beta
a^{2}+\frac{\pi\beta^{3}}{5}-\frac{\pi\beta^{5}}{700a^{2}}+\cdots,$
$\displaystyle\kappa_{T}|_{d=5}$ $\displaystyle=$
$\displaystyle\frac{16\pi^{2}\beta^{3}a^{2}}{\zeta(3)}+111.109\beta^{5}+\frac{97.9843\beta^{7}}{a^{2}}+\cdots.$
(4.23)

This result is a little bit surprising as we see that $\kappa_{T}|_{d=3}$ is
negative in the high temperature limit, while the isothermal compressibilities
for $d=4$ and $5$ are both positive in the same limit. We have discussed this
point briefly in the last section on the isothermal compressiblity of the
$d=4$ Einstein universe in the high temperature expansion. Here, if we look at
the expressions for the pressure $P$ above in Eq. (LABEL:EinPHigh), we see
that the leading terms of $P$ are independent of $a$. Since $\kappa_{T}$ is
proportional to the inverse of $\partial P/\partial a$, the sign of
$\kappa_{T}$ would depend on the sign of the second terms in Eq.
(LABEL:EinPHigh). Interestingly, for $d=3$, the second term of $P$ is
negative, while for $d=5$, this term is positive. In fact, although we have
not detailed here, one can show that this term is negative only for $d=3$, and
it is positive for all other dimensions $d\geq 4$. Hence, we have the peculiar
result that $\kappa_{T}$ is negative only for the $d=3$ Einstein universe, and
positive for all other Einstein universes.

Since the thermal expansion coefficient $\alpha$ is proportional to
$\kappa_{T}$, we have a negative $\alpha$ for $d=3$ and positive $\alpha$ for
dimensions $d\geq 4$. Indeed, we have

$\displaystyle\alpha|_{d=3}$ $\displaystyle=$
$\displaystyle-\frac{18\zeta(3)a^{2}}{\beta}+1.36062\beta-\frac{0.0275758\beta^{3}}{a^{2}}+\cdots,$
$\displaystyle\alpha|_{d=5}$ $\displaystyle=$
$\displaystyle\frac{60\zeta(5)a^{2}}{\zeta(3)\beta}+49.7752\beta+\frac{42.8332\beta^{3}}{a^{2}}+\cdots.$
(4.24)

This is also true for the heat capacities at constant pressure,

$\displaystyle C_{P}|_{d=3}$ $\displaystyle=$
$\displaystyle-\frac{108\zeta(3)^{2}a^{4}}{\beta^{4}}+\frac{31.4503a^{2}}{\beta^{2}}-0.379195+\cdots,$
$\displaystyle C_{P}|_{d=5}$ $\displaystyle=$
$\displaystyle\frac{600\zeta(5)^{2}a^{6}}{\zeta(3)\beta^{6}}+\frac{619.826a^{4}}{\beta^{4}}+\frac{500.021a^{2}}{\beta^{2}}+\cdots.$
(4.25)

Note that the relationship between $C_{P}$ and $C_{V}$ is
$C_{P}=C_{V}+\beta^{2}V\kappa_{T}(\partial P/\partial\beta)^{2}$. Here, in the
high temperature expansion, the second term, which is proportional to
$\kappa$, dominates over the first one. Hence, the sign of $C_{P}$ is
determined by $\kappa_{T}$, and we have $C_{P}$ negative for $d=3$ and
positive for all other dimensions $d\geq 4$.

It is interesting to see that, for the adiabatic compressibilities,

$\displaystyle\kappa_{S}|_{d=3}$ $\displaystyle=$
$\displaystyle\frac{4\pi\beta^{3}}{3\zeta(3)}+\frac{0.724734\beta^{5}}{a^{2}}+\frac{0.138007\beta^{7}}{a^{4}}+\cdots,$
$\displaystyle\kappa_{S}|_{d=5}$ $\displaystyle=$
$\displaystyle\frac{16\pi^{2}\beta^{5}}{15\zeta(5)}-\frac{1.96158\beta^{7}}{a^{2}}+\frac{1.45329\beta^{9}}{a^{4}}+\cdots.$
(4.26)

they are both positive. This is due to the relation
$\kappa_{S}=(C_{V}/C_{P})\kappa_{T}$. For $d=3$, both $C_{P}$ and $\kappa_{T}$
are negative but the ratio is positive. Therefore, $\kappa_{S}$ is positive in
both cases.

Lastly, we also list the fluctuations for $E$ and $P$ for completeness. For
the energy fluctuations,

$\displaystyle\langle(\Delta E)^{2}\rangle|_{d=3}$ $\displaystyle=$
$\displaystyle\frac{69.3572a^{4}}{\beta^{6}}+\frac{13.2687a^{2}}{\beta^{4}}+\frac{0.501864}{\beta^{2}}+\cdots,$
$\displaystyle\langle(\Delta E)^{2}\rangle|_{d=5}$ $\displaystyle=$
$\displaystyle\frac{343.481a^{6}}{\beta^{8}}+\frac{398.348a^{4}}{\beta^{6}}+\frac{320.334a^{2}}{\beta^{4}}+\cdots.$
(4.27)

For the pressure fluctuations,

$\displaystyle\langle(\Delta P)^{2}\rangle|_{d=3}$ $\displaystyle=$
$\displaystyle\frac{0.0228363}{\beta^{4}a^{2}}-\frac{0.00474943}{\beta^{2}a^{4}}+\frac{0.0000833667}{a^{6}}+\cdots,$
$\displaystyle\langle(\Delta P)^{2}\rangle|_{d=5}$ $\displaystyle=$
$\displaystyle\frac{0.00374241}{\beta^{6}a^{4}}+\frac{0.000723064}{\beta^{4}a^{6}}-\frac{0.000396002}{\beta^{2}a^{8}}+\cdots.$

For the correlated fluctuations of $E$ and $P$,

$\displaystyle\langle(\Delta E)(\Delta P)\rangle|_{d=3}$ $\displaystyle=$
$\displaystyle\frac{0.191313}{\beta^{4}}-\frac{0.0265258}{\beta^{2}a^{2}}-\frac{0.000221049}{a^{4}}+\cdots,$
$\displaystyle\langle(\Delta E)(\Delta P)\rangle|_{d=5}$ $\displaystyle=$
$\displaystyle\frac{0.0787971}{\beta^{6}}+\frac{0.0152242}{\beta^{4}a^{2}}-\frac{0.00643812}{\beta^{2}a^{4}}+\cdots.$

From these results and also the ones in the last section for the $d=4$
Einstein universe, we have in the high temperature expansion, $\langle(\Delta
E)^{2}\rangle$ proportional to $T^{d+3}$, and both $\langle(\Delta
P)^{2}\rangle$ and $\langle(\Delta E)(\Delta P)\rangle$ proportional to
$T^{d+1}$. We also have the rations $\langle(\Delta E)^{2}\rangle/E^{2}$
proportional to $(\beta/a)^{d-3}$, and both $\langle(\Delta
P)^{2}\rangle/P^{2}$ and $\langle(\Delta E)(\Delta P)\rangle/EP$ proportional
to $(\beta/a)^{d-1}$.

##  5 Conclusions and discussions

In this paper, towards the loftier goal of exploring the relationship between
quantum fields, spacetimes and thermodynamics, we take on a more manageable
task, of finding a thermodynamic description of the fluctuations of the stress
energy of quantum fields in several generic static spacetimes. We have derived
expressions for the heat capacity at constant volume and pressure, as well as
the isothermal and adiabatic compressibility of a thermal scalar quantum field
in a 2d (1 time and 1 space) Casimir space, a 2d Einstein cylinder, and 3d,
4d, 5d (spatial $S^{2},S^{3},S^{4}$) Einstein universes. We now analyze these
results considering these factors: comparison between cases with positive and
negative heat capacity and compressibility, effects of curvature, topology and
dimensionality. Our next paper [16] will deal with the same issues in
dynamical spacetimes relevant to cosmology.

Table 1: Low temperature leading behaviors of various thermodynamic quantities. | $d=2$ | $3$ | $4$ | $5$
---|---|---|---|---
$\rho$ | $-\pi/6L$ | $-0.0105478/a^{3}$ | Eq. (3.19) | $-0.00820215/a^{5}$
$S$ | $1-\frac{1}{2}{\rm ln}(\mu\beta^{2})$ | $1-\frac{1}{2}{\rm ln}(\mu\beta^{2})$ | $1-\frac{1}{2}{\rm ln}(\mu\beta^{2})$ | $1-\frac{1}{2}{\rm ln}(\mu\beta^{2})$
$C_{V}$ | $1$ | 1 | $1$ | 1
$C_{P}$ | $1$ | 1 | $1$ | 1
$P$ | $-\pi/6L^{2}$ | $-0.0053/a^{3}$ | Eq. (3.20) | $-0.0021/a^{5}$
$\kappa_{T}$ | $-3L^{2}/\pi$ | $-126a^{3}$ | Eq. (3.22) | $-390a^{5}$
$\kappa_{S}$ | $-3L^{2}/\pi$ | $-126a^{3}$ | Eq. (3.22) | $-390a^{5}$
$\alpha$ | $\sim e^{-2\pi\beta/L}$ | $\sim e^{-\sqrt{2}\beta/a}$ | $\sim e^{-\sqrt{3}\beta/a}$ | $\sim e^{-2\beta/a}$

Our results are summarized in Tables 1 and 2. In Table 1, the leading
behaviors of various thermodynamic quantities of a minimally coupled massless
scalar field in the low temperature expansion are tabulated. We can see that
the energy density $\rho$, the pressure $P$, the compressibilities
$\kappa_{T}$ and $\kappa_{S}$ are all negative. This is because the leading
behaviors are mostly dominated by the Casimir effect. We therefore have the
universal feature that the compressibilities in the low temperature limit, in
the Einstein cylinder and all the Einstein universes studied here, are
negative. In fact, the magnitude of the negative pressure is inversely
proportional to the size of the spatial geometry, meaning that it would
compress the spatial geometry, and while the circle or the sphere gets
smaller, the shrinking pressure would get larger. The process is similar to an
accelerated gravitational collapse. This is actually the mechanism behind the
so-called spontaneous compacification of the extra dimensions in the Kaluza-
Klein scenario [78, 79].

Since the Casimir effect is solely controlled by the spatial geometries, the
leading behavior of the free energy is independent of temperature $T$ or
$\beta$. The temperature-dependent terms are subdominate. They are related to
the entropy $S$ and the heat capacities $C_{V}$ and $C_{P}$ as derivatives of
the free energy with respect to the temperature. In all cases in the low
temperature expansion, the subleading contribution to the free energy comes
from the zero mode with $F_{ZM}=\frac{1}{2\beta}\,{\rm ln}(\mu\beta^{2})$.
From this we obtain the leading behaviors of $S$, $C_{V}$, and $C_{P}$ as
follows:

$\displaystyle S_{ZM}=1-\frac{1}{2}\,{\rm ln}(\mu\beta^{2})+\cdots$
$\displaystyle(C_{V})_{ZM}=1+\cdots\ \ \ \ \ ;\ \ \ \ \ (C_{P})_{ZM}=1+\cdots$
(5.1)

As we have mentioned before, $S_{ZM}$ apparently violates the third law of
thermodynamics. If the zero mode is ignored in the calculation of the free
energy, the temperature-dependent part of the free energy would come from
exponentially small terms. Then, $S$, $C_{V}$, and $C_{P}$ would all be
exponentially small.

Another interesting feature in Table 1 is that for the $d=4$ Einstein
universe, the various thermodynamic quantities, except $\alpha$, are
explicitly dependent on the renormalization scale $\mu$. In fact, as we have
discussed earlier, this feature is also true for all Einstein universes with
odd dimensional spatial spheres. This can be traced back to the sum like Eq.
(3.12),

$\displaystyle
f(s)=\sum_{n=1}^{\infty}\bar{D}_{n}\left(\frac{1}{\bar{\lambda}_{n}}\right)^{s-\frac{1}{2}},$
(5.2)

where $\bar{\lambda}_{n}$ and $\bar{D}_{n}$ are respectively the eigenvalue
and the degeneracy of the Laplacian on a sphere. For odd spheres, as in Eq.
(3.12), the function $f(s)$ has a pole singularity at $s=0$, and this induces
the renormalization scale dependence of the thermodynamic quantities. For even
spheres, the function $f(s)$ is analytic at $s=0$ without the pole term, and
hence there is no $\mu$ dependence for the thermodynamic quantities [76, 75].

Table 2: High temperature leading behaviors of various thermodynamic quantities. | $d=2$ | $3$ | $4$ | $5$
---|---|---|---|---
$\rho$ | $\pi/6\beta^{2}$ | $\zeta(3)/\pi\beta^{3}$ | $\pi^{2}/30\beta^{4}$ | $3\zeta(5)/\pi^{2}\beta^{5}$
$S$ | $\pi L/3\beta$ | $6\zeta(3)a^{2}/\beta^{2}$ | $4\pi^{4}a^{3}/45\beta^{3}$ | $10\zeta(5)a^{4}/\beta^{4}$
$C_{V}$ | $\pi L/3\beta$ | $12\zeta(3)a^{2}/\beta^{2}$ | $4\pi^{4}a^{3}/15\beta^{3}$ | $40\zeta(5)a^{4}/\beta^{4}$
$C_{P}$ | $-\pi^{2}L^{2}/9\beta^{2}$ | $-108(\zeta(3))^{2}a^{4}/\beta^{3}$ | $32\pi^{6}a^{5}/75\beta^{5}$ | $600(\zeta(5))^{2}a^{6}/\zeta(3)\beta^{5}$
$P$ | $\pi/6\beta^{2}$ | $\zeta(3)/2\pi\beta^{3}$ | $\pi^{2}/90\beta^{4}$ | $3\zeta(5)/4\pi^{2}\beta^{5}$
$\kappa_{T}$ | $-\beta L$ | $-12\pi\beta a^{2}$ | $108\beta^{2}a^{2}$ | $16\pi^{2}\beta^{3}a^{2}/\zeta(3)$
$\kappa_{S}$ | $3\beta^{2}/\pi$ | $4\pi\beta^{3}/3\zeta(3)$ | $135\beta^{4}/2\pi^{2}$ | $16\pi^{2}\beta^{5}/15\zeta(5)$
$\alpha$ | $-\pi L/3$ | $-18\zeta(3)a^{2}/\beta$ | $24\pi^{2}a^{2}/5\beta$ | $60\zeta(5)a^{2}/\zeta(3)\beta$

In Table 2, the leading behaviors of the thermodynamic quantities in the high
temperature expansion are tabulated. One can see that the leading behaviors
coincide with those of a massless relativistic ideal thermal gas in
$d$-dimensional flat spacetimes, with the equation of state $P=\rho/(d-1)$.
More specifically, the leading behaviors of $P$ and $\rho$ are both of the
order of $T^{d}$, independent of the size of the spatial sphere.

For $d=2$ and $3$, we have the peculiar results that $\kappa_{T}$, $C_{P}$,
and $\alpha$ are negative although the pressure $P$ itself is positive. This
is related to the subleading terms of $P$ as shown in Eqs. (2.46) and
(LABEL:EinPHigh). We can see that the leading terms are positive but the
subleading terms are negative. Therefore, with $\kappa_{T}$ being related to
the inverse of $\partial P/\partial a$, it is also negative. Due to this
negative sign of $\kappa_{T}$, the thermal expansion coefficient $\alpha$ and
the heat capacity at constant pressure $C_{P}$ would both be negative. Note
that since $\kappa_{S}=(C_{V}/C_{P})\kappa_{T}$ and the signs of $\kappa_{T}$
and $C_{P}$ cancel to give a positive adiabatic compressibility $\kappa_{S}$.
On the other hand, for $d\geq 4$, the subleading term of $P$, as exemplified
in Eq. (3.36) for $d=4$ and Eq. (LABEL:EinPHigh) for $d=5$, is again positive
so we have all the thermodynamic quantities positive for all dimensions $d\geq
4$. One may wonder if this peculiar behavior of a thermal quantum field in
$d=2$ Einstein cylinder and $d=3$ Einstein universe would be related to the
peculiarity of gravity in $1+1$ and $2+1$ spacetime dimensions [80, 81].

In our next paper [16] we shall consider dynamical quantum fields and use the
nonequilibrium free energy density discovered in [5] to explore the quantum
capacity and vacuum compressibility of the Universe.

Acknowledgments HTC is supported in part by the Ministry of Science and
Technology, Taiwan, ROC, under the Grants MOST109-2112-M-032-007 and
MOST110-2112-M-032-009. J.-T. Hsiang is supported by the Ministry of Science
and Technology of Taiwan, R.O.C. under Grant No. MOST 110-2811-M-008-522.

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