Existence of a critical point in the phase diagram of the ideal relativistic neutral Bose gas

We explore the phase transitions of the ideal relativistic neutral Bose gas confined in a cubic box, without assuming the thermodynamic limit nor continuous approximation. While the corresponding non-relativistic canonical partition function is essentially a one-variable function depending on a particular combination of temperature and volume, the relativistic canonical partition function is genuinely a two-variable function of them. Based on an exact expression of the canonical partition function, we performed numerical computations for up to hundred thousand particles. We report that if the number of particles is equal to or greater than a critical value, which amounts to 7616, the ideal relativistic neutral Bose gas features a spinodal curve with a critical point. This enables us to depict the phase diagram of the ideal Bose gas. The consequent phase transition is first-order below the critical pressure or second-order at the critical pressure. The exponents corresponding to the singularities are 1/2 and 2/3 respectively. We also verify the recently observed `Widom line' in the supercritical region.


Introduction
Spinodal curve, by definition, consists of points in a phase diagram where the isothermal volume derivative of pressure vanishes [1,2]: The curve corresponds to the border line between thermodynamic stable and unstable regions, ∂ V P (T, V ) < 0 and ∂ V P (T, V ) > 0 respectively [3].
Moreover, if it exists, the spinodal curve amounts to a first-order phase transition under constant pressure. The temperature derivative at fixed pressure acting on an arbitrary physical quantity which is a function of temperature and volume, is given by the chain rule of calculus: On the spinodal curve, the denominator of the second term vanishes, and the temperature derivative of a generic physical quantity along the isobar diverges [4]. This provides a possible mechanism for a finite system to manifest genuine mathematical singularities, without taking the thermodynamic limit [5].
If we fill a rigid box with water to full capacity and heat the box, the temperature will increase but hardly the water evaporates. According to a well-known standard argument against the emergence of a singularity from a finite system that is based on the analytic property of the canonical partition function [3], no discontinuous phase transition should occur. Nevertheless, opening the lid will set the pressure as constant or at 1 atm, and the water will surely start to boil at 100 degree Celsius. The point is that the usual finiteness of physical quantities in a canonical ensemble is for the case of keeping the volume fixed. Once we switch to the alternative constraint of keeping the pressure constant, first order phase transition featuring genuine mathematical singularities may arise from a system possessing finite number of physical degrees. The question is then the existence of the spinodal curve. 1 In our previous work [4], we investigated the thermodynamic instability of the ideal non-relativistic Bose gas confined in a cubic box, based on an exact expression of the corresponding canonical partition function. The result was that if the number of particles is equal to or greater than a certain critical value 1 which turns out to be 7616, the ideal Bose gas subject to Dirichlet boundary condition indeed reveals thermodynamic instability characterized by a pair of spinodal curves and consequently undergoes a first-order phase transition under constant pressure. The two spinodal curves are identified as the supercooling and the superheating of the ideal Bose gas, where k B is the Boltzmann constant, m is the mass of the particle, P is an arbitrary given pressure, and τ * P , τ * * P are dimensionless constants of which the numerical values depend on the the number of particles, N , as follows [4]: where ζ is Riemann zeta function. In standard textbooks, e.g. [3,[6][7][8][9], this temperature is typically obtained by treating the ground state separately and applying the continuous approximation to all the rest of the excited states. From the comparison of Eq.(3) and Eq.(4), it appears that the continuum limit smoothes out some details of the thermodynamic quantum structure.
In the present paper, we generalize our previous work to the relativistic case.
We investigate into the spinodal curve of ideal relativistic quantum gas of neutral bosons confined in a cubic box, subject to Dirichlet boundary condition. While the non-relativistic canonical partition function is essentially a one-variable function depending on a particular combination of temperature and volume, the relativistic canonical partition function is genuinely a two-variable function of them. We show that if the number of particles is equal to or greater than the critical number 7616, the ideal relativistic neutral Bose gas features a spinodal curve with a critical point where the supercooling and the superheating lines converge. Further, we verify the recently observed 'Widom line' [10] emanating from the critical point as the crossover between liquid-like and gas-like behaviour in the supercritical region [11] (see also [12] and [13]). The critical number coincides with that of the non-relativistic system and hence, the present paper confirms our previous numerical results [4].
The main motivation for us to consider the relativistic ideal Bose gas in the present paper as a generalization of the non-relativistic system we studied previously is, by letting the corresponding canonical partition function be a two-variable function, to observe a more intriguing phase diagram such as a critical point and the Widom line. After all, quantum mechanics and relativity are two cornerstones of modern physics. For pioneering earlier works on the ideal relativistic Bose gas via the continuous approximation we refer to Refs. [14][15][16][17]. For recent discussion see [18] and references therein. In the present paper, we restrict ourselves to neutral bosons. The inclusion of anti-bosons, as in [19][20][21], will be treated separately elsewhere [22].
The rest of the paper is organized as follows: In section 2, we present model independent analysis on canonical ensemble.
We first review three different expressions for the canonical partition function of generic non-interacting identical bosonic particles. We then, simply by assuming the existence of a spinodal curve, show that under constant pressure first-order phase transitions feature singularities of exponent 1/2 while second-order phase transitions have the critical exponent 2/3. Section 3 is exclusively devoted to the ideal relativistic neutral Bose gas confined in a cubic box. After analysing non-relativistic and ultra-relativistic limits, we present our numerical results. When the number of particles is equal to or greater than 7616, the ideal relativistic Bose gas reveals a spinodal curve with a critical point. We draw the corresponding phase diagram.
The final section 4 summarizes our results with comments.
2 General analysis 2.1 The canonical partition function: review When a single particle system is completely solvable, each quantum state is uniquely specified by a set of good quantum numbers, which we simply denote here by a vector notation, n. With the corresponding energy eigenvalue E n and β = 1/(k B T ), we define for each positive integer a, where the sum is over all the quantum states. In particular, when a = 1, λ a coincides with the canonical partition function of the single particle.
For an N -body system composed of the non-interacting identical bosonic particles above, we may write the corresponding canonical particle function in three different manners: • As first obtained by Matsubara [23] and Feynman [24], where the sum is over all the partitions of N , given by non-negative integers m a with a = 1, 2, · · · , N satisfying N = N a=1 a m a .
• By a recurrence relation first derived by Landsberg [25], where we set Z 0 = 1.
• From Ref. [4], where Ω N is an almost triangularized N ×N matrix the entries of which are defined by In particular, every diagonal entry is unity.
The first expression (6) implies that, compared to ideal Boltzmann gas, ideal Bose gas has higher probability for the particles to occupy the same quantum state [4]. The last expression (8) is useful for us to see when the canonical partition function reduces to the conventional approximation [26]: which would be only valid if all the particles occupied distinct states, as in high temperature limit. Indeed, in this limit we have λ 1 → ∞. Hence det(Ω N ) → 1 and the reduction (9) holds.
According to the Hardy-Ramanujan's estimation, the number of possible partitions grows exponentially like e π √ 2N/3 /(4 √ 3N ), and this would make any numerical computation based on the expression (6) practically hard for large N . Among the three above, the recurrence relation (7) provides the most efficient scheme of N 2 order computation.

Critical and noncritical exponents: universal results
When the energy eigenvalue of the single particle system E n (5) depends on volume, the canonical partition function depends on temperature and volume, so does the pressure: We consider, at least locally, 2 inverting P (T, V ) to express the temperature as a function of P and V like T (P, V ). Plugging this expression into ∂ V P (T, V ), let us define a quantity Φ as a function of P and V : As we change the volume keeping the pressure constant, Φ indicates whether the thermodynamic instability develops or not on the isobar. With the minus sign in front, now negative Φ corresponds to the thermodynamic instability.
In a similar fashion to (2) we write from which it is straightforward to obtain and relate the first and second volume derivatives of T (P, V ) to Φ(P, V ) as Further, on the spinodal curve ( Now critical point can be defined as follows. At the critical pressure P = P c , the system is generically stable i.e. Φ > 0, except at only one point that is the critical point. This implies that the critical pressure line is tangent to the spinodal curve, such that at the critical point, Φ and ∂ V Φ vanish: Moreover, from (11), (13), (14), these two conditions are equivalent to and also to Eq. (17) corresponds to the usual definition of the critical point as an inflection point in the critical isotherm on a (P, V ) plane [3]. With (15), Eq. (17) implies that the critical point is an extremal point on the spinodal curve.
On the other hand, Eq. (18) implies that the expansion of T (P, V ) near the critical point on the critical isobar starts from the cubic order in V − V c (for related earlier works see [27] and references therein): Clearly this leads to the following critical exponent: Similarly, from (13) and (17), the expansion of Φ on the critical isobar starts Thus, from (2), any physical quantity given by the temperature derivative along the critical isobar diverges with the universal exponent 2/3. This includes the critical exponent of the specific heat per particle under constant pressure: On the other hand, since the specific heat per particle at constant volume is finite for any finite system, the corresponding critical exponent is trivial: Note that our conclusion does not exclude the possibility for C V to develop a peak. We merely point out that the peak will be smoothed out if it is sufficiently zoomed in. Further from (21), the inverse of Φ gives the critical exponent of the isothermal compressibility on the critical isobar: Finally, from (17), we also obtain an isothermal critical exponent at the critical temperature, T = T c : On a generic isobar below the critical pressure, P < P c , the thermodynamic instability given by Φ(P, V ) < 0 appears naturally over an interval in volume and hence in temperature as well. The two ends then correspond to a supercooling point and a superheating point. At these points Φ vanishes 8 and between them Φ is negative, as depicted in FIG. 1 It is crucial to note that the volume assumes triple values between the supercooling and the superheating temperatures at constant pressure less than P c . In the derivation of the critical exponents above we tacitly assumed In fact, the equivalence among (16), (17) and (18) generalizes up to an arbitrary order, say n : For all k = 1, 2, · · · , n, the vanishing of Clearly these satisfy the following "scaling laws" [3,29]: Especially, the case of n = 1 corresponds to the first-order phase transition at noncritical, supercooling and superheating points. Even at these points our analysis shows the existence of the universal exponent: which agrees with the mean field theory result [30] (see also [31]). Around With positive integer valued good quantum numbers: the spatial momentum is quantized as p = π n/L, such that the single particle Boltzmann factor in (5) assumes the form: Here we have introduced, for simplicity of notation and the forthcoming analysis, dimensionless temperature: as well as dimensionless volume: The canonical partition function is then two-variable function of them, Z N (T , V). We further define dimensionless pressure by: and a dimensionless indicator of the thermodynamic instability, as for Φ in (11): Other physical quantities we are particularly interested in the present paper are specific heats per particle at constant volume or under constant pressure: (36) • Non-relativistic limit. The non-relativistic limit corresponds to the limit of low temperature and large volume with T V 2/d held fixed. For large volume, the Boltzmann factor (31) reduces to that of the non-relativistic particle, up to the shift of the energy by mc 2 : Physical quantities like φ, C V , C P become one-variable functions depending on the single variable T V 2/d , or alternatively T P −2/(d+2) [4]. The constant energy shift is irrelevant to them. Furthermore, at extremely low temperature only the ground state energy matters and the canonical partition function reduces to Hence near to absolute zero temperature we obtain and also For large volume, Eq.(39) further leads to V = (N/P) d/(d+2) and this agrees with Ref. [4]. Apparently the volume assumes a finite value at absolute zero temperature, essentially due to the Heisenberg uncertainty principle. It is worthwhile to recall in the large T V 2/d limit of the non-relativistic case [4]. Note that the first relation is equivalent to PV = N T .
• Ultra-relativistic limit. The ultra-relativistic limit corresponds to the limit of high temperature and small volume with T V 1/d held fixed. At sufficiently high temperature, highly excited states dominate and from (9) the canonical partition function reduces to (Z 1 ) N /N !, where the Boltzmann factor also reduces as Consequently, the canonical partition function and the physical quantities like φ, C V , C P become one-variable functions depending on the single variable T V 1/d , or alternatively from (34) and (42) on T P −1/(d+1) .
Moreover, for large T V 1/d we may safely assume the continuous approximation to obtain [32], which implies in the large T V 1/d limit, cf. (41):

Numerical results
Here we present our numerical results on the ideal relativistic neutral Bose gas confined in a cubic box, i.e. d = 3, at generic temperature and volume, performed by a supercomputer (SUN B6048). Our analysis is based on the recurrence relation (7) and the previous work [4] 5 . 5 The numerical computation inevitably requires a momentum cutoff as for n· n, which generically deforms the result at high temperature. We choose mutually different, sufficiently large values of the cutoff, such that we maintain at least for some interval of temperature the high temperature behaviour (44), and report only the cutoff independent results (c.f. Refs. [33,34]).

13
• Emergence of a spinodal curve (FIG. 2). As N grows, φ develops a valley of local minima, which eventually assumes negative values if N ≥ 7616. In FIG. 2b, P c denotes merely the numerical value, P c = 2.015 1967 × 10 −11 , which amounts to the critical pressure in the system with one more particle, N = 7616.
• Specific heat per particle under constant pressure (FIGs. 3, 4, 5). During the first-order phase transition under constant pressure which is less than the critical value, the volume and hence every physical quantity are triple valued between the supercooling and the superheating temperatures. We focus here on C P , the specific heat per particle under constant pressure.
For the explicit behavior of other quantities we refer to our earlier work [4]. The plane decomposes into three parts: phase I with C P ≃ 0, phase II with C P ≃ 2.5k B and phase III with C P ≃ 4k B . When N = 1 the transitions are monotonic and smooth. As N grows, at the borders between I and II as well as between I and III, a range of peaks emerges which will eventually diverge for N ≥ 7616. FIG. 3b has been cut at the height of C P = 5k B , and the actual peak rises up to C P ≃ 5.336 84 × 10 6 k B .  5a); below the superheating point, C P ∼ ±(T * * − T ) −1/2 (FIG. 5b); and around the critical point, C P ∼ |T − T c | −2/3 (FIG. 5c). The numerical data are for N = 7616, P = 5.000 170 056 40 × 10 −12 (FIG. 5a,b) or P c = 2.015 1967 × 10 −11 (FIG. 5c). The error bars originate from the numerical uncertainty beyond these digits. The straight lines correspond to the theoretical slopes, −1/2 (FIG. 5a,b) or −2/3 (FIG. 5c).
• Phase diagram of the ideal relativistic neutral Bose gas for N = 10 5 . (FIG. 6). Our numerical results of the critical point are listed below for some selected N :  Comparison with the non-relativistic results in Ref. [4] indicates that the critical point of N = 7616 is in the non-relativistic region while those of N = 10 4 and N = 10 5 are not.

19
In summary, as seen from FIG. 3 and FIG. 6, the ideal relativistic neutral Bose gas divides the (T , V/N ) phase diagram into three parts.
• Phase I: condensate with C p ≃ 0 (40), • Phase II: non-relativistic gas with C p ≃ 2.5k B (41), • Phase III: ultra-relativistic gas with C p ≃ 4k B (44), Equivalently the (P, T ) phase diagram splits into three parts. The resulting equation of state from the spinodal curve of the relativistic ideal gas resembles the Van der Waals equation of state which is derived by assuming both repulsive potential at short distance and attractive potential at long distance. It is well known that the effective statistical interaction of ideal Bose gas is attractive [3]. Our result seems to suggest a more intriguing structure. The fact that the volume is finite at absolute zero temperature seems also to indicate a repulsive interaction at short distance related to the trapping potentials, may also lead to qualitatively similar or even more intriguing phase diagrams.