Thermodynamics of bosonic systems in adS spacetime

We analyze the Thermodynamics of massless bosonic systems in D-dimensional anti-de Sitter spacetime, considering scalar, electromagnetic and gravitational fields. Their dynamics are described by Poschl-Teller effective potentials and quantized in an unified framework, with the determination of the associated energy spectra. From the microscopic description developed, a macroscopic thermodynamic treatment is proposed, where an effective volume in anti-de Sitter geometry is defined and a suitable thermodynamic limit is considered. Partition functions are constructed for the bosonic gases, allowing the determination of several thermodynamic quantities of interest. With the obtained results, general aspects of the Thermodynamics are explored.


I. INTRODUCTION
Asymptotically anti-de Sitter geometries gained a new relevance with the anti-de Sitter/conformal field theory (adS/CFT) correspondences [1][2][3]. In the best-known scenario, this duality establishes a dictionary between gravitational dynamics in anti-de Sitter spacetime and N = 4 supersymmetric Yang-Mills theory. Generalizations were proposed considering other geometries and field theories, leading to the gauge/gravity correspondences. Applications in a variety of physical settings were developed, from fundamental quantum gravity models to phenomenological condensed matter systems.
Despite the present success of the adS/CFT program, relevant questions remain for the complete understanding of the duality. In particular, there are still open issues concerning the relation between the Thermodynamics of the gravity and field theory sides. Specifically focusing on the gravitational physics, thermodynamic aspects of asymptotically adS spacetimes have received considerable attention in recent years [4][5][6][7][8][9][10][11][12][13][14][15][16].
An important related theme is the behavior of a thermal gas in anti-de Sitter (adS) spacetime, eventually collapsing to form a black hole. Hawking and Page, within an Euclidean path integral approach, considered the Thermodynamics of thermal gases and black holes in adS space [17]. They observed that the Schwarzschild-adS and the thermal adS geometries are different phases of a single physical system.
In the present work, we consider the thermal anti-de Sitter spacetime, that is, the setup treated by Hawking-Page in the no-black hole regime. We analyze the Thermodynamics of bosonic systems in adS geometry within a real time approach. Scalar, electromagnetic and gravitational fields are considered. A fundamental issue in this framework is the perturbative and quantum dynamics in adS background. Results concerning the decomposition of fields of interest in adS are commented in [18][19][20][21][22]. The quantum scalar field in adS spacetime was considered in [23][24][25], and the quantization of the electromagnetic field was studied in [26][27][28][29].
To characterize the Thermodynamics of a bosonic gas in adS geometry, a unified framework for the quantization of the scalar, electromagnetic and gravitational fields is introduced. Energy spectra for the physical modes are derived and partition functions are constructed. In a suitable thermodynamic limit, quantities of interest are calculated and an overall analysis is performed. The present work suggests that adS geometry qualitatively behaves as a confining box (as expected), but with correction terms reflecting the non-trivial geometric background.
The structure of this paper is presented in the following. In section II the adS spacetime and the fields of interest are introduced. In section III the bosonic systems are quantized and the energy spectra determined. In section IV the Thermodynamics of the considered scenarios are analyzed. A qualitative description of the thermodynamic characteristics and instabilities of the considered systems is presented in section V. Final remarks are made in the section VI. We use signature (−, +, · · · , +) and natural units with G = = c = k B = 1 throughout this paper.

A. Background geometry
In this work we consider the D-dimensional anti-de Sitter (adS) spacetime. More precisely, we will denote as adS geometry the universal covering space of the maximally symmetric Lorentzian geometry with negative constant curvature. With this definition, adS geometry does not contain any closed time-like curves and it is a well-behaved solution of vacuum Einstein equations with a negative cosmological constant Λ.
The D-dimensional adS geometry (M, g µν ) can be locally described as a product of a 2-dimensional manifold M 2 and the (D − 2)-dimensional sphere S D−2 . The subspace M 2 is spanned by a time-like coordinate t and a radial space-like coordinate r. The coordinate system based on θ 1 , θ 2 , . . . , θ D−2 describes S D−2 . With this decomposition, the background metric can be written in the form where γ ij is the metric of the (D − 2)-dimensional unit sphere S D−2 , γ ij dθ i dθ j = dθ 1 2 + sin 2 θ 1 dθ 2 2 + · · · + sin 2 θ 1 · · · sin 2 θ D−2 dθ D−2 2 , and the function f (r) is given by The parameter L, the "adS radius", is related to the negative cosmological constant Λ as The choice of the foliation in Eq. (1) is suitable to the investigation of the equations of motion associated to the bosonic fields. A convenient chart is based on the so-called tortoise coordinate x, defined as x ≡ˆd r f (r) = L arctan r L .
For the adS geometry, a simple expression for the function r(x) can be obtained, r = L tan(x/L). In terms of the tortoise coordinate x, the metricg ab on the two-dimensional adS geometry (M 2 ,g ab ) is written as with y 0 = t and y 1 = x. In the following sections, the covariant derivatives on M , M 2 and S D−2 are denoted by ∇ µ , ∇ a and D i , respectively. Anti-de Sitter spacetime has interesting properties. An important feature of this geometry is that its spacial infinity is time-like. Massless particles reach spacial infinity in a finite time, according to a static observer. Assuming specific boundary conditions, usually denoted as "reflexive boundary conditions" [23], massive and massless particles can be confined into adS geometry.
From those considerations, adS spacetime can be interpreted as a box with an effective volume V eff , associated to a given static observer. Consider an inertial observer that sends a massless particle outward. According to this observer, the time t ∞ for the particle to reach spacial infinity is A natural definition for an effective radius (R eff ) is R eff ≡ c t ∞ = cπL/2, where we have introduced back the speed of light c for clarity. Neglecting non-essential proportionality factors, we define an effective volume associated with adS geometry as The notion of an adS effective volume will be important for the implementation of a thermodynamic limit on the field dynamics at the anti-de Sitter spacetime. Further comments on this definition will be presented in section V D.

B. Massless scalar field
The simplest bosonic system considered is a massless scalar field Φ satisfying the Klein-Gordon equation, where is the Laplace-Beltrami operator. We expand Φ in terms of scalar spherical harmonics S kS as In Eq. (10), {k 2 S } are the eigenvalues of S kS [18], given by The integers {m i } label the modes in each subspace spanned by the eigenmodes of S kS . The equation of motion for each mode φ sc ℓ is obtained by straightforward substitution of the expansion (10) in Eq. (9), where V sc (x) is a scalar effective potential, whose explicit expression is The coefficients A sc and B sc for the massless scalar potential are with the multipole index ℓ assuming values in ℓ = 0, 1, 2, . . . .

C. Electromagnetic field
A more complex bosonic perturbation in adS background is the electromagnetic field. Its dynamics is determined by the electromagnetic tensor F µν , subjected to Maxwell's equations, From the classical electromagnetic tensor F µν , the potential A µ is defined as The equations of motion (16) can be decomposed into two parts, decoupling A µ into vector and scalar modes (A el-vc µ and A el-sc µ respectively). Each of these modes, after being expanded in terms of appropriate spherical harmonics, generates a set of second order linear differential equations, characterized by certain effective potentials. In the following, the formalism presented in [18][19][20] will be used.

Electromagnetic vector modes
Electromagnetic vector modes A el-vc µ can be written as an expansion in vector harmonics {V kV i } as where φ el-vc ℓ (x) is the master variable associated to this mode. The function φ el-vc ℓ (r) represents the radial modes of A el-vc µ . The vector spherical harmonics V kV i satisfy the eigenvector equation given by From the expansion (18), it is obtained a differential equation associated to the master variable, a wave equation with the same form of Eq. (12), where the effective potential V el-vc (x) associated to the electromagnetic vector mode is written as The coefficients A el-vc and B el-vc are with the multipole index ℓ = 1, 2, . . ..

Electromagnetic scalar modes
Electromagnetic scalar modes A el-sc µ can be written in terms of two quantities, a vector A a orthogonal to S n and a scalar quantity A. Hence, with a convenient decomposition in spherical harmonics, A el-sc µ can be expressed as From the first Maxwell's equation in (16) one obtains that where the quantity k 2 S is the scalar harmonic eigenvalue presented in Eq. (11). The result in Eq. (24) implies [18] that there is a function φ el-sc ℓ satisfying∇ a φ el-sc It is then introduced a master variableφ el-sc and, in terms ofφ el-sc ℓ , electromagnetic scalar modes are characterized by a wave equation in the form (12) with the effective potential [18] The constants A el-sc and B el-sc are given by: with the multipole index ℓ = 1, 2, . . ..

D. Gravitational perturbation
Quantum properties of the gravitational field can be considered in a field theory effective approach. Gravitational perturbations propagating in adS spacetime can be expanded in terms of harmonic functions on S D−2 . The perturbed Einstein equations are then expressed in terms of a set of gauge invariant quantities [21]. These quantities are combinations of the metric perturbations h µν , which are related to the perturbed spacetime metric g µν as with g (0) µν representing the adS metric. By taking appropriate combinations of gauge invariant variables constructed from h µν , the perturbative equations are reduced to three decoupled sets. They furnish the tensor, vector and scalar perturbations, which will be described in the following with the formalism of [18][19][20].

Gravitational tensor modes
Gravitational tensor perturbations are present in spacetimes with dimension equal or larger than 5. This set of gravitational modes can be represented in terms of tensor spherical harmonics as follows, where T kT ij are transverse traceless harmonic tensors on S D−2 , with the quantities {k 2 T } defined as Using the master variable [18] given by the equation of motion for the gravitational tensor mode assumes the form of Eq. (12), with an effective potential V gr-t written as The coefficients A gr-t and B gr-t are with the multipole index ℓ = 2, 3, . . ..

Gravitational vector modes
Gravitational vector modes can be expanded in terms of vector harmonic functions V kV The quantities V kV ij are vector-type harmonic tensors on S D−2 built from the transverse harmonic vectors V kV i , with the eigenvalue k 2 V presented in Eq. (19). From the functions f ℓa and H V ℓ , a new gauge invariant quantity Z ℓa is defined: A master variable φ gr-vc ℓ is implicitly introduced by the relation where ǫ ab is the Levi-Civita tensor in M 2 . In terms of φ gr-vc ℓ , vector-type gravitational modes satisfy a wave equation with the form of Eq. (12) [19], where the effective potential V gr-vc is The constants A gr-vc and B gr-vc are given by with the multipole index assuming values in ℓ = 2, 3, . . ..

Gravitational scalar modes
Gravitational scalar perturbations can be expanded in terms of scalar harmonic functions {S kS } as The terms S kS i and S kS ij are scalar-type harmonic vectors and tensors on S D−2 respectively, built from the scalar harmonic functions S kS as Gauge invariant quantities can be defined for ℓ ≥ 2 and written in terms of a master variable φ gr-sc ℓ (after a rather involving procedure, as seen for example in [20]). The function φ gr-sc ℓ satisfies a wave equation with the form of Eq. (12) with an effective potential given by V gr-sc , where The coefficients A gr-sc and B gr-sc are given by with the multipole index ℓ = 2, 3, . . ..

III. QUANTIZATION, ENERGY SPECTRA AND DEGENERACIES
As seen in section II, for the bosonic fields considered, the classical dynamics are described by equations of motion having the form with a Pöschl-Teller [30] effective potential V(x), The constants A and B depend on the specific mode considered, according to previous section.
In the approach used for the quantization of the scalar, electromagnetic and gravitational perturbations in adS spacetime, we consider the quantum properties of the potential associated to each mode. The first step in the quantum treatment is to define the one-particle Hilbert space H 1 associated to a given perturbation. We start with a natural domain of functions that are absolutely continuous in the interval [0, πL 2 ], together with their first derivatives. The one-particle Hamiltonian can be introduced asĤ That is, the notion of energy is being defined by static observers following integral curves of ∂/∂t. From the one-particle sector, the Fock space associated with the perturbations is constructed through the usual procedure. Taking into account Eq. (50), the positive and negative energy modes should span a dense subset of H 1 , whereφ(x) are the solutions of the "time-independent" version of the equation of motion (48):Ôφ An important point in the present development is that the deficiency indices of the operatorÔ are not zero for arbitrary values of A and B [20,31]. Therefore, additional boundary conditions must be supplied in order to specify an adjoint extension ofÔ. This issue is related with the fact that, for the quantum treatment of fields in adS spacetime, the geometry is not globally hyperbolic, with its spacial infinity being time-like. As already commented in [23,32,33], additional conditions must be supplied for the well-posedness of the quantization process.
Let us initially consider the case where A ≥ 3/4 and B ≥ 3/4. The general solution of the time-independent equation (52) with the Pöschl-Teller potential is well-known (see for example [31]). It can be written as where u 1 (x) and u 2 (x) are linearly independent solutions of Eq. (52) given by with ζ ǫ p,q = p + q + In Eqs. (54) and (55), F [α, β; γ; z] denotes the Gaussian hypergeometric function. The parameters p and q are expressed in terms of the constants A and B as Constants C 1 and C 2 will be fixed by the boundary conditions. In the case A ≥ 3/4 and B ≥ 3/4, the function u 2 is not square integrable and we must set C 2 = 0 in Eq. (53). The asymptotic behavior of u 1 at spacial infinity is [31] lim x→πL/2 Since ǫ > 0, 2p + 1/2 > 0 and 2q + 1/2 > 0, for q ≥ 3/4 the function u 1 is square integrable only for specific values of ǫ which satisfy [20,31]: Condition (59) The functions u ± are square integrable for −1 ≤ 4A < 3. However, in general, the asymptotic forms of u ± , denoted here by ψ * , are not compatible with a symmetricÔ. For A = −1/4, the requirement that this symmetry is enforced impliesā 1 a 2 =ā 2 a 1 . This last condition can be parametrized as [31] The variable θ introduced in Eq. (61) parametrizes the boundary conditions. Consequently θ determines the selfadjoint extensionsÔ θ ofÔ for −1/4 < A < 3/4. A physical interpretation is that θ is a parameter which controls the amount of particles that can "leak out" at spacial infinity. Indeed, the case θ = ±π/2 (a 2 = 0) fixes ψ * (πL/2) = 0. This condition can be interpreted as a generalized Dirichlet condition [20] and represents the reflexive condition at infinity. On the other hand, θ = 0 (a 1 = 0) can be interpreted as a generalized Neumann condition and represents a scenario where spacial infinity is transparent.
The spectrum ǫ θ of the operatorÔ θ also depends on θ. For θ = ±π/2, it follows that ǫ ±π/2 = ǫ [31]. That is, the expression (59) is obtained again. The case A = −1/4 shows some peculiarities, but can be treated in an analogous manner. The important point here is that the boundary condition ψ * (πL/2) = 0 implies that the operatorÔ is self-adjoint and whose associated spectrum is still given by Eq. (59).
Hence we take as H 1 the set of functions H 1 : R → C defined on the dense domain D (R + ), the space of smooth functions with compact support satisfying reflexive boundary conditions: Table I. Energy spectra for massless bosonic fields in adS geometry with D ≥ 4 and n = 0, 1, 2, . . ..

Field Scalar modes Vector modes Tensor modes
In this way, the operatorÔ is self-adjoint and its spectrum is given by Eq. (59) for any values of A and B relevant in the present work.
With the results obtained for the Pöschl-Teller potential, the spectra of the several perturbations considered can be derived. Relation (59), combined with definitions (57) for p and q, plus the specific forms of the constants A and B, determine the spectra: where mode is a label indicating a particular perturbation, with mode ∈ {sc, el-vc, el-sc, gr-t, gr-vc, gr-sc}, and For clarity's sake, we also present the explicit form of the spectrum associated to each considered mode in table I. From Eq. (63) (or table I), we define the spectrum sequences in such a way that ǫ mode j < ǫ mode j+1 . Hence, the index j labels the energy levels. With the energy spectra determined, their associated degeneracies can be considered. Indeed, for a given perturbation, there are in general several modes associated to a specific energy value ǫ mode j . We denote the degeneracy of the energy ǫ mode All the considered spectra can be written as where α mode are obtained from Eq. (63) (or table I). These constants do not depend on n or ℓ. The degeneracies will be given by the multiplicity of the eigenvalues of the spherical harmonic modes. Following the development in [25], with Γ denoting the usual gamma function. For the scalar and gravitational vector modes, the degeneracy functions are .
Considering the electromagnetic vector and gravitational vector modes, Finally, for the electromagnetic scalar and gravitational scalar modes, IV. THERMODYNAMICS OF BOSONIC SYSTEMS

A. Thermodynamic and quasi-thermodynamic limits
In the treatment introduced in section III, the fields were assumed to be free, not allowing direct coupling among the quantum particles. The adS geometry has a confining potential that acts as a box, with an effective volume V eff defined in Eq. (8). We consider the anti-de Sitter geometry populated by a thermal gas, composed by massless particles with a well-defined energy and subjected to the Bose-Einstein statistics. The number of particles is not conserved and the chemical potential µ is constant and null.
Since the bosonic system is in thermal equilibrium with fixed temperature T and fixed chemical potential (µ = 0), a grand canonical ensemble is used. For a given mode, the associated partition function Z mode is where j labels the energy levels and D mode j is the associated degeneracy, according to expressions (69)-(71). The partition function of a given field is obtained summing the partition functions of the modes related to that field, as will be explicitly done in the next subsections. A thermodynamic quantity X linked to a certain bosonic field will be denoted as X field , where field ∈ {sc, el, gr}.
Macroscopic quantities can be established if it is possible to define a proper thermodynamic limit. Using the effective volume introduced in Eq. (8) and considering that we are employing a grand canonical ensemble, we define the thermodynamic limit [34] as which implies L → ∞ with T fixed and µ = 0.
In order to obtain the thermodynamic limit of the partition functions in Eq. (72), we consider the norm of the sequence ǫ mode j : j = 0, 1, . . . for the bosonic perturbations. For the spectra in Eq. (63), the associated sequence norms have the same value ∆ǫ, Result (74) indicates that a continuum limit for the sequence can be obtained in the limit L → ∞. From the expression of the partition function, its continuum limit is derived rewriting Eq. (72) as a Riemann sum in the form In the thermodynamic limit, ∆ǫ → 0 and D mode (Lǫ) ∼ (Lǫ) D−2 for all perturbations discussed. Since D mode (Lǫ) ln 1 − exp − ǫ T /L D−2 is a bounded continuous function on ǫ, the Riemann sum is approximated by a Riemann integral, and an asymptotic expression for ln Z mode is obtained: Eq. (77) can be seen as an analogous Thomas-Fermi approximation in the adS bosonic systems. It is also relevant to consider not only the thermodynamic limit, but also how this limit is approached. Hence, in the present work, we define a quasi-thermodynamic limit [39] as V eff large but finite, with T fixed and µ = 0 .
This limit is specially relevant in the present work. Unlike the usual Thermodynamics in a Minkowski cavity (where the thermodynamic limit can be associated to the cavity radius going to infinity), the finiteness of the adS radius is naturally compatible with the quasi-thermodynamic limit.
It is convenient to rewrite the partition function (72) in terms of the "reduced energy" K mode The elements of set {K mode j } are integers that do not depend on L or T , obtained in a straightforward way from Eq. (63). We conclude that the partition function Z mode depends on L and T as Let us consider the high-temperature regime of the bosonic systems, that is, LT ≫ 1. In this limit, it is useful to rewrite the integral expression for ln Z mode . Making ǫ = Tǫ, and Strictly speaking, this integral representation for ln Z mode is only relevant for LT ≫ 1. In practice, moderate values of LT are enough to guarantee a reasonable applicability of the result, as discussed in subsection V B. A general expression for the partition function associated to a given perturbation in the large temperature regime, beyond the dominant term (LT ) D−1 , can be obtained. Expanding the degeneracy function (66) aroundǫ = 0, and using the integral representation for the Riemann zeta function ζ, the partition function (82) can be written as For values of LT close to 1, the integral formula in Eq. (82) is not useful. In this case, expression (79) for ln Z can be numerically treated. In the present work, we have conducted an extensive numerical investigation of the scalar, electromagnetic and gravitational partition functions and derived quantities. The results will be reported in the following sections.
We consider next the low-temperature limit. That is, the quasi-thermodynamic limit with a large effective volume and 0 < LT ≪ 1. In this regime, the fundamental mode of any perturbation considered dominates the sum in Eq. (79). An analytic result for the partition function associated to a given mode can be determined, Explicit expressions for the scalar, electromagnetic and gravitational fields will be presented in the next subsections. Once Z mode is obtained, it is straightforward to construct Z field . And with the partition functions, all the associated thermodynamic quantities can be readily calculated. For example, entropy (S field ), internal energy (U field ) and pressure (P field ) are given by: In the next subsections, we will explore specific characteristics of the particular bosonic fields considered.

B. Thermodynamics of the massless scalar field
We initially consider a thermal gas of "scalar photons" in anti-de Sitter spacetime. For the massless scalar field, there is only one mode, implying that the partition function for field = sc is equal to the partition function for mode = sc. The degeneracy function D sc (Lǫ sc ) is given by Eq. (69).
The low-temperature regime of the massless scalar field is characterized by the partition function in Eq. (86) with Hence, in this regime and from this expression, The low-temperature result for ln Z sc is compared to the numerical evaluation of the partition function in figure 1.
The intermediate and high-temperature regime of the scalar gas can be explored with the integral formula (82) for ln Z sc . Considering the dominant and sub-dominant terms (since we are assuming a finite value of LT with LT ≫ 1), Using Eq. (91), the integral in Eq. (82) can be evaluated. The partition function in this limit is The high-temperature expression for ln Z sc is compared to the numerical evaluation of the partition function in figure 1.
It is straightforward to obtain explicit expressions for thermodynamic quantities, once the partition function is determined. For instance, the entropy is the internal energy is and for the pressure, C. Thermodynamics of the electromagnetic field The next bosonic system to be considered is a thermal gas of photons. As seen in subsection II C, the electromagnetic perturbations can be decomposed in two independent modes, the electromagnetic vector and scalar perturbations. Hence, the partition function Z el for a photon gas in D-dimensional adS spacetime is given by where Z el-vc D and Z el-sc D are the partition functions associated to the electromagnetic vector and scalar modes respectively.
In the low-temperature regime, the values for the constants K mode 0 = Lǫ mode 0 associated to the electromagnetic field are In the four-dimensional case, K el-vc 0 = K el-sc 0 and both scalar and vector modes contribute. In higher dimensions, K el-sc 0 < K el-vc 0 and the scalar mode dominates. It follows from Eq. (86) that the leading contribution for the electromagnetic partition function in the limit T → 0 is The low-temperature analytic expression for ln Z el is compared to the numerical evaluation of the partition function in figure 2. From expression (98), some thermodynamic quantities for the adS photon gas in the low-temperature regime are: In the high-temperature regime, the degeneracies of the vector and scalar electromagnetic modes can be written as (103) 2ζ ( (105) With expressions (104) and (105), we obtain the partition function of the photon gas in the high-temperature regime: The expression for ln Z el in Eq. (106) is illustrated in figure 2, where the analytic expression is compared to the numerical evaluation.
With the formula (106), the entropy, internal energy and pressure of the photon gas can be derived: It is finally considered a thermal graviton gas in D-dimensional adS spacetime. As seen in subsection II D, gravitational perturbations can be decoupled in three independent modes (tensor, vector and scalar). In the four-dimensional case, the gravitational tensor mode does not contribute, and partition function for the graviton gas is expressed as ln Z gr = ln Z gr-vc + ln Z gr-sc , for D = 4 . (110) In five or higher dimensions, all three gravitational modes are relevant, and the gravitational field partition function is written as ln Z gr = ln Z gr-t + ln Z gr-vc + ln Z gr-sc , for D > 4 .
Let us consider the low-temperature limit. Carrying out a development very similar to the one presented in the previous subsections, we observe that If D = 4, both scalar and vector modes are relevant in the low-temperature regime. For higher dimensions, the scalar mode dominates. Using Eq. (86), we obtain in the limit T → 0 that Result (113) for ln Z gr in the low-temperature limit is illustrated in figure 3, where the analytic result is compared to the numerical development. Entropy, internal energy and pressure are readily calculated: For the intermediate and high-temperature regimes, considering that D gr-t (LTǫ) = D sc (LTǫ), D gr-vc (LTǫ) = D el-vc (LTǫ) and D gr-sc (LTǫ) = D el-sc (LTǫ), combined with results from previous subsections, the calculation of the associated partition functions is straightforward, Expression (117) for ln Z gr is illustrated in figure 3, where the analytic result is compared to the numerical calculation. From Eq. (117), the entropy, internal energy and pressure associated to the adS graviton gas are calculated:

V. ANALYSIS OF THE THERMODYNAMICS
In this section, we will explore general characteristics of the Thermodynamics associated to bosonic gases in anti-de Sitter spacetime. Structural aspects, numerical results, instabilities and the issue of a proper thermodynamic volume will be explored.

A. Structural aspects
Homogeneity, or in a broader sense quasi-homogeneity (i.e., homogeneity of degrees different of one and zero), is an essential property for a well-defined Thermodynamics [36,37]. While quasi-homogeneity is explicit in the strict thermodynamic limit, it is not so beyond this limit, in the quasi-thermodynamic regime. However, the partition functions can be written as Z field = f (LT ), as seen in Eq. (80). It follows that quasi-homogeneity is recovered demanding that LT should be a homogeneous function of degree zero (otherwise, different powers of LT will give different degrees). We note that it is the same requirement due to scaling arguments for the consistency of the Schwarzschild-anti de Sitter Thermodynamics [15,37,38].
The condition Z field = f (LT ), besides ensuring quasi-homogeneity, also furnishes a consistency result. This restriction on the form of the function Z field implies that where the expressions in (87) were used. Taking into account our definition of V eff in Eq. (8), The equation of state (122) shows that the massless bosonic gases behave as a null fluid, as expected. We conclude that the proposal for the effective volume in Eq. (8) is compatible with the general thermodynamic description for the bosonic systems treated in the present work. More comments on this topic will be presented in subsection V D.
Let us consider the low-temperature limit of a massless bosonic field in adS spacetime. Taking into account results (90), (99) and (114), it follows that the entropy behaves as where K field S is a positive constant which depends on the particular field and dimension considered. Entropy tends to zero as the temperature approaches absolute zero, and therefore the adS bosonic systems respect the third law of Thermodynamics.
In the high-temperature regime, the general qualitative behavior of the adS bosonic systems can be analyzed. With the integral formula (82) for the partition functions and D mode (LTǫ) = (LTǫ) D−2 1 + o 1 LT for large (but finite) LT , we obtain Therefore Result (125) indicates that the bosonic fields in adS spacetime obey the D-dimensional Stefan-Bolzmann law in the thermodynamic limit.

B. Numerical results
We performed an extensive numerical investigation of the partition functions and associated quantities presented in previous section. We observed that the functions ln Z field are monotonically increasing in the dimensionless parameter LT , growing exponentially with low LT and as a power-law with high LT . The intermediate temperature regime smoothly connects the low and high temperature limits. No discontinuities or divergences were observed in these quantities. This result is illustrated in figure 4, where graphs for ln Z gr are presented for several values of D. The qualitative behavior is the same for other fields and dimensions. As expected, the analytic expressions in the low and high-temperature limits are compatible with the numerical results, as illustrated in figures 1, 2 and 3.
An issue is the convergence of the numerical and analytic results produced. Specifically, we focus on the partition functions written as discrete sums, and the integral versions of those quantities obtained by analogous Thomas-Fermi approximations. The integral formulas for ln Z field are good approximations for the partition functions if LT is higher than a given number (an analogous Thomas-Fermi critical temperature). Typically, good concordance is already observed with LT ≈ 5.
More precisely, convergence can be characterized by the behavior of the relative difference ∆ mode between the expressions defined with sums and integrals: Our numerical results indicate that ∆ mode tends to zero as a power-law in the form We illustrate the dependence of ∆ mode with LT in figure 5, considering the scalar gas. Results for the other modes and fields are qualitatively similar. Concerning the internal energy, entropy and pressure, the general behavior of these quantities is largely independent of the field and dimension. The functions U field , S field and P field are positive-definite and tend to zero as T tend to zero. With a constant effective volume, U field and S field are monotonically increasing in T , growing as a nearexponential with low T and as a power-law with high T . With the temperature constant, P field grows monotonically as the effective volume increases. These characteristics are illustrated in figures 6-8.

C. Thermodynamic instabilities
Thermodynamic instabilities are identified by the behavior of quantities such as the thermal capacities and compressibility. Let us initially consider the thermal capacity at constant volume (C field V ), defined as Since the functions U field with constant L are monotonically increasing, C field V is always positive. Therefore the system is stable for any constant volume process. Sample graphs for the thermal capacities are presented in figure 9.
We consider now the isothermal compressibility β field T , defined as  Our numerical results indicate that the pressure P field is a monotonically increasing function in V eff (with constant T ). It follows that β field T is always negative. This means that the system is unstable under isothermic processes (that is, considering only variations in the cosmological constant). The general behavior of β field T is presented in figure 10. The instability generated by β field T < 0 is expected, since the usual photon gas in a cavity also presents this phenomenon [41]. For instance, let us consider the high-temperature regime. In the thermodynamic limit, the isothermal compressibility can be written as where K field β is a positive constant which depends on the dimensionality and of the particular bosonic perturbation. Expression (130) indicates that there is no contribution for the compressibility from the dominant term. That is, the sub-dominant term dictates the behavior of β field T in the high-temperature limit.  We can compare the corrections in the Stefan-Boltzmann law in adS spacetime with the photon gas confined in a cavity (considering Minkowski spacetime). In the high-temperature regime, the energy density of the several bosonic systems considered have the form: Explicit expressions for the constants a field D and b field D can be readily determined. For example, considering the Figure 10. Graphs for β gr T as a function of the V eff with D = 4, 5, 6. In the graphs, T = 1. The qualitative behavior is the same for other fields and dimensions. electromagnetic field in four dimensions, Expression (132) can be compared to the analogous result concerning the photon gas in a cavity, which has the form [42] We observe that in the usual "Minkowskian cavity" case, Eq. (133), the term proportional to T 3 is not present, indicating that there is no contribution proportional to the cavity area in the energy distribution [40]. The conclusions from this particular case can be generalized for other fields and dimensions. The presented results show that the adS bosonic systems have a different behavior of the usual photon gas in a cavity, even in the thermodynamic limit.

D. Effective volume and homogeneity
In section II A, the effective volume in adS spacetime was introduced within a geometric context. The goal here is to show that this choice of thermodynamic volume is a consequence of the homogeneity of the equation of states in the thermodynamic limit.
Let us consider (in the internal energy description) that the bosonic systems in adS background have two independent thermodynamic variables: the entropy S and a still not determined thermodynamic volume V . Besides, as usual, in the free energy F = −T ln Z = U − T S, the independent variables are T and V . We want to show that V is given by the effective volume V eff ≡ L D−1 introduced in section II A. We are assuming that, in the thermodynamic limit, entropy and volume are extensive variables (first order homogeneous), while temperature and pressure are intensive variables (zeroth order homogeneous).
Considering the hypothesis made, the Euler theorem for homogeneous functions applies, giving U field = T S field − P V .
In previous expression, P is the conjugate variable (to be determined) associated to the volume V . Combining Eq. (134) with relations (87), we obtain From section IV, the partition function of a given bosonic field in the thermodynamic limit can be written as where A is a constant which depends on the particular field considered. Using expression (136), At this point, we evoke homogeneity again with the Gibbs-Duheim relation [35] dG field = V dP − S field dT .
In the bosonic systems treated here, formed by non-interacting massless fields, G field = µ = 0. Therefore Expression (139) shows that, when the pressure is written in terms of T , it does not depend on any other variable, that is P = P (T ).
Using relations (137) and (136), an explicit form of the pressure in terms of the entropy can be determined, with a constant k. Solving equation (139), From previous result, where C is a constant. Finally, using Eq. (140) we obtain The volume V in Eq. (143) coincides with our choice of effective volume V eff , up to a non-essential multiplicative constant. We conclude that the definitions of thermodynamic volume and thermodynamic limit considered in the present work are compatible with the homogeneity of the equations of state in the thermodynamic limit.

VI. FINAL REMARKS
In the present work we considered the equilibrium Thermodynamics associated with massless bosonic fields in anti-de Sitter spacetime with reflexive boundary conditions. Specifically, scalar, electromagnetic and gravitational perturbations are treated. The classical dynamics and the quantization of the several matter contents of interest are based on Pöschl-Teller effective potentials. Thermodynamic quantities are calculated for the fields of interest, and an analysis of the Thermodynamics is performed.
An important point in the developed treatment is a suitable definition of effective volume. This effective volume has a natural geometric interpretation, being compatible with the homogeneity of the equations of state in the thermodynamic limit. Imposing Bose-Einstein statistics and proper thermodynamic limits (based on the volume introduced), the Thermodynamics is determined. The proposed definitions of effective volume and adS thermodynamic limits should be relevant in other analyses in anti-de Sitter background.
In the actual calculation of the partition functions and derived quantities, analytic and numerical tools are employed. Specifically, analytic results are available in the low-temperature and high-temperature regimes, while numeric techniques are (in principle) always applicable. The analytic and numeric calculations coincide when both can be performed, corroborating the approach used.
Bosonic Thermodynamics in anti-de Sitter background is, in many ways, similar to the Thermodynamics of more usual physical systems. For instance, both the adS bosonic setup and the photon gas in a Minkowskian cavity have the same low-temperature and high-temperature behavior. Both scenarios have negative isothermal compressibilities, indicating that they are unstable under isothermic processes. Nevertheless, there are distinctions. In the intermediate temperature regime (with LT ≈ 1), the adS Thermodynamics is quantitatively different from the "usual" boson gas confined in a cavity. In the high-temperature limit, the sub-dominant terms do not coincide.
It is worth pointing out that, although we considered only bosonic fields, the method presented here can be generalized to other physical scenarios. For instance, fermionic thermal gases in anti-de Sitter are of great interest. Fermions should be part of the Hawking atmosphere in an adS black hole, and therefore they should be relevant in the Hawking-Page phase transitions. Work along those lines is currently under way.