Bose-Einstein Condensation of Quantum Hard-Spheres as a Deposition Phase Transition and New Relations Between Bosonic and Fermionic Pressures
Kyrill A. Bugaev, Oleksii I. Ivanytskyi, Boris E. Grinyuk, Ivan P. Yakimenko
aa r X i v : . [ nu c l - t h ] S e p Bose-Einstein Condensation of Quantum Hard-Spheres as a Deposition PhaseTransition and New Relations Between Bosonic and Fermionic Pressures
Kyrill A. Bugaev , , Oleksii I. Ivanytskyi , , Boris E. Grinyuk and Ivan P. Yakimenko Bogolyubov Institute for Theoretical Physics, Metrologichna str. 14-B, Kyiv 03680, Ukraine Department of Physics, Taras Shevchenko National University of Kyiv, 03022 Kyiv, Ukraine CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal and Department of Physics, Chemistry and Biology (IFM),Link¨oping University, SE-58183 Link¨oping, Sweden
We investigate the phase transition of Bose-Einstein particles with the hard-core repulsion inthe grand canonical ensemble within the Van der Waals approximation. It is shown that thepressure of non-relativistic Bose-Einstein particles is mathematically equivalent to the pressure ofsimplified version of the statistical multifragmentation model of nuclei with the vanishing surfacetension coefficient and the Fisher exponent τ F = , which for such parameters has the 1-st orderphase transition. The found similarity of these equations of state allows us to show that withinthe present approach the high density phase of Bose-Einstein particles is a classical macro-clusterwith vanishing entropy at any temperature which, similarly to the classical hard spheres, is akind of solid state. To show this we establish new relations which allow us to identically repre-sent the pressure of Fermi-Dirac particles in terms of pressures of Bose-Einstein particles of two sorts.Keywords: quantum gases, Van der Waals, equation of state, Bose-Einstein condensation, depositionphase transition I. INTRODUCTION
The phenomenon of Bose-Einstein (BE) condensation is, probably, one of the most striking manifestation of col-lective quantum effects [1, 2]. Due to its great importance the phase transition (PT) of BE condensation in the idealgas is discussed in all textbooks on statistical mechanics. In the wast majority of these textbooks it is written thatthe BE condensation of ideal gas is the 3-rd order phase transition (see, for instance, [1]), although in the famousbook [2] (see the section 12.3 for details) it is argued that the BE condensation is the 1-st order PT between liquidand gas. The main question we answer here is what kind of PT is the BE condensation in the quantum system withthe simplest interaction, namely with the hard-core repulsion? In all textbooks it is written that the BE condensateis the group of particles with zero momentum. However, the question is what is it? Is it a liquid or a solid?In what follows we demonstrate that the pressure of the non-relativistic BE particles with the hard-core interactiontaken in the Van der Waals (VdW) approximation can be identically reduced to the one of the simplified version ofstatistical multifragmentation model (sSMM) [3] with a vanishing surface tension of the constituents (see below). Thisexactly solvable model was formulated in [4] and solved exactly in [5–8], while its new and more realistic generalizationcan be found in [9]. Although the sSMM [4–9] lacks the Coulomb interaction between the nuclei and the asymmetryenergy of nuclei, its exact analytical solution established both in the thermodynamic limit [5, 6, 9] and for finitevolumes [7, 8] is able to qualitatively describe the main properties of the nuclear liquid-gas PT.The mathematical similarity between the VdW EoS of BE hard spheres and the sSMM allows us to show that thehigh density phase of BE particles with hard-core repulsion is a classical macro-cluster which, similarly, to the classicalhard-spheres is a solid state [10, 11] and not a liquid as it was argued in K. Huang book [2]. In our analysis we alsoanalyzed the pressure of Fermi-Dirac (FD) particles with the hard-core repulsion which in many respects is similar tothe one of sSMM, although it does not have the 1-st order PT. This analysis allows us to find out some new relationsbetween the pressures of BE and FD particles with the hard-core repulsion, which help us to demonstrate that themacro-cluster of BE particles is, indeed, a classical object. The found relations allow us to clearly demonstrate underwhat conditions the FD particles with the hard-core repulsion can have the first order phase transition.The work is organized as follows. In Sect. II we analyze the pressure of BE and FD particles with the hard-corerepulsion in the VdW approximation in a form convenient for the grand canonical ensemble. Sect. III is devoted todiscussion of the properties of the macro-cluster with the help of the BE-FD decomposition identities which identicallyrepresent the pressure of FD particles in terms of two BE pressures. Our conclusions are given in Sect. IV.
II. BE CONDENSATION AS THE 1-ST ORDER PHASE TRANSITION
The equation of state (EoS) of hard-spheres with BE or FD statistics in the grand canonical ensemble variablesunder the Van der Waals approximation for the hard-core repulsion can be obtained either analyzing the free energyof the Van der Waals gas in canonical ensemble [12, 13] or more rigorously from the quantum partition function inthe grand canonical ensemble [14]. In the grand canonical variables it has the form p ± = p id ± ( T, ν ) ≡ ± T g Z d k (2 π ~ ) ln (cid:20) ± exp (cid:20) ν − e ( k ) T (cid:21)(cid:21) , where ν ≡ µ − p ± , (1)where the lower sign is for the BE statistics, while the upper sign is for the FD one. Here T is temperature ofthe system, µ is its chemical potential, ν is an effective chemical potential, g is the number of spin-isospin states(degeneracy factor), m is the mass of particle, V = πR is the “eigen volume“ of particle, and R is the half of theminimal interaction range of the hard-core potential U ( r ) of a one component system (with a single hard-core radius) U ( r ) = (cid:26) , | r | > R, ∞ , | r | ≤ R. (2)The potential U ( r ) acts in a simplest possible way: (i) if two particles 1 and 2, for definiteness, do not interact, i.e. thedistance between them | r | > R is larger, than two hard-core radii R , then U ( r ) = 0 and, therefore, their total energyis the sum of their single-particle (kinetic) energies e and e ; (ii) if these two particles interact, then | r | = 2 R and U ( | r | = 2 R ) = ∞ , but such configurations do not contribute to partition (and all thermodynamic functions), sincethey are suppressed by the statistical operator exp h − ˆ H hc T i due to an infinite potential energy (here ˆ H hc denotes theHamiltonian of the system). As a result, the total energy of the particles with the hard-core repulsion equals to thesum of their single-particle (kinetic) energies and this allows one to find the pressure (1) directly from the quantumpartition function. In other words, the particles with the hard-core interaction behave as an ideal quantum gas.This is a important property of this EoS which leads to a well-known practical consequence, namely that the energyper particle coincides with the one of the ideal gas. Due to this property the sophisticated equations of state with thehard-core repulsion, known as the hadron resonance gas model, are very successfully used to describe the multiplicitiesof hadrons [15–17] and light (anti-, hyper)nuclei [18, 19] which are measured in the high energy nuclear collisions andto get a reliable thermodynamic information about next to the last stage of such collisions.For further analysis it is convenient to introduce the auxiliary functions F ± ( p ) ≡ T K max X l =1 ( ∓ ( l +1) l n id (cid:20) Tl , ν ( p ) (cid:21) , ⇒ Eq . (1) becomes p ± = F ± (p ± ) , (3) n id (cid:20) Tl , ν (cid:21) = Z g d k (2 π ~ ) e l [ ν − √ m ~k ] T ≃ Z g d k (2 π ~ ) e l (cid:20) ν − m − k m (cid:21) T = g (cid:20) m T π l ~ (cid:21) e l [ ν − m ] T , (4)where the particle number density of Boltzmann point-like particles with temperature T and chemical potential ν isdenoted as n id [ T, ν ] and the upper limit of sum in Eq. (3) is K max → ∞ . To avoid the unnecessary complexity inour derivations through out this work we regard the limit K max = 2 K + 1 → ∞ strictly in this sense. For the BEstatistics (sign − in Eqs. (3) and (4)) it is not important, but it is very important for the case of FD statistics (sign+ in Eqs. (3) and (4)) .The function F ± in (3) is, apparently, obtained by expanding the ln-function in Eq. (1). For large values of l ≫ lm ≫ T is valid for any non-vanishing mass m and, therefore, in this case one can use the non-relativisticapproximation in the left hand side momentum integral in Eq. (4) and get the right hand side expression (4). However,for convenience we will use such an approximation for any l ≥
1, assuming that considered temperatures are verylow compared to the particle mass, i.e. m ≫ T . Moreover, in what follows we will always use the non-relativisticapproximation for particle energy, unless it is specified explicitly.To make a direct comparison with the sSMM [4–8] we explicitly write Eq. (1) for the BE statistics ( a = − p − = T g (cid:20) m T π ~ (cid:21) K max X k =1 ( − a ) ( k +1) k exp (cid:20) k ( µ − m − V p − ) T (cid:21) , (5)using Eq. (3) and the right hand side Eq. (4). Comparing Eq. (5) with Eq. (15) from Ref. [5], one can see thatthe pressure of BE hard spheres is mathematically absolutely equivalent to the sSMM with the “volume“ 4 kV of k -nucleon nuclei, with the vanishing surface tension of all nuclei and with the Fisher exponent τ F = (or for theindex τ ≡ τ F + = 4 in terms of Refs. [5, 6]).Due to the mathematical similarly to the sSMM, using the exact solution of sSMM [5, 6, 8] one can immediatelyconclude that Eq. (5) describes two phases: the gaseous phase p g = p − ( T, ) for the low densities defined by theinequality µ < µ c ( T ), and high density phase pressure p s = ( µ − m )4 V for µ > µ c ( T ). According to the Gibbs criterionthe PT occurs, if the pressures of two phases are equal, i.e. p g ( T, µ c ) = p s ( T, µ c ). This equation defines the phaseequilibrium curve µ = µ c ( T ) of the 1-st order PT.At the PT curve µ = µ c ( T ) the effective chemical potential becomes ν c = µ c − V p g ( T, µ c ) = µ c − V p s ( T, µ c ) ≡ m . (6)Using this result one can identically rewrite the pressure at PT curve as p c − = T g (cid:20) m T π ~ (cid:21) K max X k =1 k = |{z} K max →∞ T g Γ (cid:2) (cid:3) (cid:20) m T π ~ (cid:21) ∞ Z t e t − dt, (7)where we used the integral representation of the Riemann ζ (cid:2) (cid:3) -function [20]. Here Γ( n + 1) = n ! is the usual gamma-function. Taking t = ωT in the integral in Eq. (7), one recovers the traditional representation of pressure as an integralover the particle energy ω [1].Although the critical pressure (7) coincides with the one obtained usually for the point-like particles [1], the particlenumber density of gas n − is modified due to the presence of hard-core interaction. Using the particle number densityof the gas of point-like particles n id − ( T, ν ) one can write n id − ( T, ν ) ≡ ∂p id − ( T, ν ) ∂ν = g (cid:20) m T π ~ (cid:21) K max X k =1 k exp (cid:20) k ( ν − m ) T (cid:21) , (8) n − ( T, ν ) ≡ ∂p id − ( T, ν ) ∂µ = n id − ( T, ν )1 + 4 V n id − ( T, ν ) . (9)From Eq. (9) one can see that at the PT curve the particle number density of the gas is smaller than the particlenumber density of the dense phase, since n − ( T, ν c ) = n id − ( T, ν c )1 + 4 V n id − ( T, ν c ) < n s ≡ ∂p s ∂µ = 14 V , (10)and, hence, for any finite temperature T the particle number density of point-like particles n id − ( T, ν c ) is finite too.Therefore, the particle number density of gaseous phase is smaller than the one of the high density phase as indicatedby the inequality (10). As a result, the BE PT is of the 1-st order.Substituting into Eqs. (8) and (9) the value ν = ν c one can get the temperature of BE condensation as T BEc = 2 π ~ m " gζ (cid:2) (cid:3) · n − − V n − . (11)Note that for large values of the excluded volume V and high particle number densities n → V the hard-corerepulsion may essentially increase the value of the PT temperature and make it more realistic compared to thetraditional estimate obtained for the point-like particles [1], i.e. if one takes the limit V → g → g eff and the one of excluded volume 4 V → V eff which correspond to a more realisticEoS than the VdW EoS and which is able to reproduce the pressure of quantum particles beyond the second virialcoefficient approximation at least in some (even in a narrow) range of thermodynamic parameters.Since we are also interested in analyzing the case of FD particles, we would like to obtain the above result using adifferent approach, namely without referring to the sSMM results of Refs. [5–7]. First we consider the limit µ = → ∞ in Eq. (5) for very large, but finite values of K max . Apparently, this limit should correspond to the dense phase of ourEoS. Then in this limit V p s /T ≫ µ > m + 4 V p s the leading terms of Eq. (5) for a = − " p s K max T φ ( T ) ≃ K max [ µ − ( m + 4 V p s )] . (12)Here the thermal density of the gas of classical hard spheres is denoted as φ ( T ) = g (cid:2) m T π ~ (cid:3) . In deriving Eq. (5) wehave chosen the large values of chemical potential µ > µ c , which are not allowed in the thermodynamic limit, but forfinite systems they can be used [7, 8]. Now from Eq. (12) one can see that for K max → ∞ the logarithmic correctiondisappears and the pressure of dense phase p s = µ − m V acquires a familiar form.In order to show that the EoS (5) for a = − D p − ≡ T ∂p − ∂ρ id − .Hereafter to avoid a confusion we will distinguish the particle number density of point-like particles as the functiongiven by the right hand side of Eq. (8) and the same quantity as the independent variable ρ id − . The derivative D p − is more convenient to employ for the spinodal instability point of the gas than the derivative ∂p − ∂n − , since its expressionis simpler. Note that vanishing of the spinodal instability point of the gas taken at the given isotherm signals aboutthe 1-st order PT [1]. Indeed, the expression for D p − D p − ≡ T ∂p − ∂ν ∂ν∂ρ id − = (cid:20) K max P k =1 1 k exp h k ( ν − m ) T i(cid:21) (cid:20) K max P k =1 1 k exp h k ( ν − m ) T i(cid:21) − , (13)shows that, if the effective chemical potential ν = µ − V p − approaches the value ν = m , then the derivative ∂ρ id − ∂ν ≡ ∂n id − ∂ν → ∞ diverges for K max → ∞ and, hence, in this limit D p − = 0. Thus, we have found that the spinodalinstability point of the gas of BE hard spheres coincides with the PT curve.Now we turn to the analysis of the FD particles with the hard-core repulsion. For ν ≤ m the pressure of suchparticles p + and its ν -derivative can be explicitly written as p + = T g (cid:20) m T π ~ (cid:21) K max X k =1 ( − ( k +1) k exp (cid:20) k ( ν − m ) T (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν = µ − V p + , (14) n id + ≡ ∂p + ∂ν = g (cid:20) m T π ~ (cid:21) K max X k =1 ( − ( k +1) k exp (cid:20) k ( ν − m ) T (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν = µ − V p + . (15)A similarity with the sSMM can be more clearly seen for ν = µ − V p + → m , if in the sum (14) one adds the eventerms to the preceding odd ones p + = T g (cid:20) m T π ~ (cid:21) K max − X k ∈ odd exp h k ( ν − m ) T i k " − k ( k + 1) k exp (cid:20) ( ν − m ) T (cid:21) + exp h K max ( ν − m ) T i K max ≃ (16) ≃ T g (cid:20) m T π ~ (cid:21) K max − X k ∈ odd k exp (cid:20) k ( ν − m ) T − k (cid:21) + exp h K max ( ν − m ) T i K max , (17)where we expanded the binomial ( k + 1) keeping two leading terms and approximated the ratio k / ( k + 1) ≃ exp[ − k ]. Evidently, this approximation is suited for k ≫
1, but for qualitative analysis it is very convenient, sincein the vicinity of PT the main role is played by the largest cluster. Eq. (17) shows that, apart from the term with k = K max , in the left vicinity of the point ν → m − τ F = anda vanishing value of surface tension coefficient.Apparently, from Eqs. (14) and (17) one can also derive Eq. (12) and establish the pressure of dense phase p s = µ − m V similarly to the case of BE particles. However, the derivative ∂ρ id + ∂ν ≡ ∂ p + ∂ν = gT (cid:20) m T π ~ (cid:21) K max X k =1 ( − ( k +1) k exp (cid:20) k ( ν − m ) T (cid:21) , (18)with respect to the effective chemical potential ν = µ − V p + is finite for ν = m , since, in contrast to the caseof BE particles, the sum staying in Eq. (18) converges in the limit K max → ∞ . Indeed, with the help of integralrepresentation of the Riemann ζ -function [20] for ν = m one finds ∞ X k =1 ( − ( k +1) k = 1Γ (cid:2) (cid:3) ∞ Z t − e t + 1 dt ≃ . , (19)and, therefore, the derivative D p + ≡ T ∂p + ∂ρ id + does not vanish for ν = m and, hence, there is no 1-st order PT in thiscase.In our opinion this is a very simple and good example that the presence of a macro-cluster with the finite probabilityin a finite system is a necessary, but not a sufficient condition of the 1-st order PT existence in such a system. Webelieve this is an important message to be taken into account by the authors of Refs. [21, 22] who consider thepresence and gradual disappearance of the macro-cluster as a signal of the 1-st order nuclear liquid-gas PT in finitesystems. The whole point is that in finite systems the macro-cluster of maximal size can appear as the metastablestate of finite probability not only for the 1-st order PT, but also for the 2-nd order PT or even for the cross-over[7, 8]. The present analysis once more shows one that for vanishing surface tension coefficient the value of the Fisherexponent τ F defines the PT order [8].One can readily check that all the results on PT existence remain valid, if one uses the relativistic expression forparticle energy, i.e. if one makes a replacement m + k m → √ m + k . However, in this case the BE condensationdoes not look mathematically identical to the sSMM and, hence, the corresponding analysis is not made here. III. DECOMPOSITION IDENTITY BETWEEN BOSONIC AND FERMIONIC PRESSURES
Apart from the formal difference between the EoS of the BE and FD particles we would like to understand (i)whether our interpretation of the appearance of classical macro-clusters is correct, and (ii) under what circumstancesthe appearance of macro-cluster can be associated with the 1-st order PT in the system of FD particles. Indeed, anabsence of the 1-st order PT in the EoS of FD particles with the hard-core repulsion may question the validity ofour hypothesis about the classical macro-clusters existence and, therefore, one may think that BE condensation leadsto an appearance of quantum macro-cluster with BE statistics, while the quantum macro-cluster with FD statisticscannot be formed due to some reason, namely due to the Pauli blocking principle.To demonstrate the validity of our hypothesis we consider a peculiar mathematical identity between the BE andFD pressures which we call a BE-FD decomposition identity − T g Z d k (2 π ~ ) ln h − exp h ν −√ m + k T ii| {z } p B ( νT ,m,g ) ≡≡ T g Z d k (2 π ~ ) ln h h ν −√ m + k T ii| {z } p F ( νT ,m,g ) −
18 ln h − exp h ν −√ m + k T ii| {z } p B ( νT , m, − g ) , (20)which will help us to understand the appearance of a classical macro-cluster for BE and FD statistics. The fact thatnow we do not use the non-relativistic approximation to the particle energy is not important.The BE-FD decomposition identity (20) can be obtained in the following sequence of steps: first we note thatln h − exp h ν −√ m +4 k T ii ≡ ln h h ν −√ m + k T ii +ln h − exp h ν −√ m + k T ii . (21)Next one can integrate Eq. (21) over d k with the degeneracy factor g and change a particle momentum on the lefthand side of Eq. (21) as 2 k → k and in the momentum integral to get a multiplier . Finally, interchanging thepositions of integrals for lighter and heavier bosons one arrives at Eq. (20).Eq. (20) shows one that for the given values of T and ν the pressure of ideal gas of bosons (the upper line ofEq. (20)) of mass m and degeneracy g can be identically decomposed into the sum of two terms. The first pressurecorresponds to the ideal gas of fermions with same mass and degeneracy (the first term on the right hand side ofEq. (20)), while the second pressure describes the bosons with the double mass and double charge (and the doubleexcluded volume V , if ν = µ − V p accounts for the effects of hard-core repulsion as above), but with the reduceddegeneracy g . Then the heavy bosons may be interpreted as “pairs“ of fermions.Applying the BE-FD decomposition identity (20) ( n −
1) times to the pressure p B (cid:0) ν B T , m, − g (cid:1) of “pairs“, onecan identically extract the contribution of bosonic macro-cluster ( n ≫
1) with the mass 2 n m , the charge 2 n and thedegeneracy 2 − n g from the pressure of bosons of mass m , charge 1 and degeneracy g and get the following usefulrelation p B (cid:16) ν B T , m, g (cid:17) ≡ p B (cid:16) n ν B T , n m, − n g (cid:17) + n − P k =0 p F (cid:16) k ν B T , k m, − k g (cid:17) , (22)where p F (cid:16) k ν B T , k m, − k g (cid:17) denotes the pressure of auxiliary fermions with the mass 2 k m , the charge 2 k and de-generacy 2 − k g . For low temperatures T ≪ m one can safely use the non-relativistic approximation for the energy ofparticle. Applying the identity (22) to the gas pressure of bosons p − of the EoS considered in the preceding section,i.e. for ν B ≤ ν c , one can immediately conclude that for ν B < m the effective chemical potential of the bosonicmacro-cluster on the right hand side of Eq. (22) is ( ν B − m )2 n → −∞ for n ≫ ν B < m . It is evident, that the bosonic macro-cluster on the right hand side of Eq. (22) does notexist for ν B = m as well, since for n ≫ − n g → ν B < m as well. Therefore, in the whole gaseous phase and at the condensation curve of the EoS of BEparticles with the hard-core repulsion considered in the preceding section the bosonic macro-cluster is absent, i.e. for ν B ≤ m one finds p B (cid:16) ν B T , m, g (cid:17) = ∞ P k =0 p F (cid:16) k ν B T , k m, − k g (cid:17) , (23)that the pressure of BE particles can be identically written as an infinite sum of the pressures of FD particles withcertain masses, charges and degeneracies. In the preceding section it was shown that the pressure of FD particles withthe hard-core repulsion does not have the 1-st order PT and, thus, in the thermodynamic limit there is no fermionicmacro-cluster for each pressure staying on the right hand side of Eq. (23). However, the pressure of BE particlesstaying on the left hand side of Eq. (23) demonstrates the 1-st order PT of the BE condensation. Therefore, theonly possible explanation out of this apparent contradiction is that the BE condensation leads to an appearance ofthe classical macro-cluster which is the sum of individual classical macro-clusters generated by the set of fermionicpressures that are staying on the right hand side of Eq. (23).Now it is appropriate to discuss the properties of the dense phase of BE hard spheres within the VdW approximation.Since the pressure of dense phase p s = µ − m V does not depend on the temperature explicitly, then the entropy density ofdense phase s s = ∂p s ∂T = 0 is zero at any temperature, while the particle number density of this phase is n s = ∂p s ∂µ = V .Furthermore, from the thermodynamic identity ε s = T s s + µn s − p s = m V (24)one can see that the energy density ε s of the dense phase, indeed, corresponds to the particles at rest which have thehighest possible density within the adopted approximation. Therefore, similarly to the case of classical hard spheres itis more appropriate to call this phase as the solid state [10, 11] (since there is not attraction among the particles andthe surface tension coefficient is zero). Furthermore, it seems it is more appropriate to consider the BE condensationof particles with the hard-core repulsion as the deposition PT from a gas to a solid. Of course, one has to rememberthat, on the other hand, it is a condensate of hard spheres with a vanishing momentum.A mathematical similarity with the exact solution of sSMM allows one to reliably interpret the BE condensationof hard spheres as the 1-st order PT in which the gas condenses into a classical macro-cluster of the size 4 V K max with K max → ∞ in the thermodynamic limit. Hence, at the PT curve there should exist the phase boundary. As itwas shown above, formally, a macro-cluster corresponds to the term k = K max in the expression for pressure p − inEq. (5). Therefore, formally a macro-cluster can be considered as a single classical particle which is at rest. FromEq. (5) one can see that its statistical weight is the Boltzmann one. Such an interpretation is similar to the sSMM[5, 6] with the difference that in the sSMM a macro-cluster is a droplet of liquid which has the non-vanishing surfacetension below the critical temperature [3, 5, 6, 9] and a finite entropy which vanishes at T = 0 only.Moreover, considering the EoS (5) with the effective values of g → g eff and 4 V → V eff which allow one atleast in the narrow range of thermodynamic parameters to reproduce the realistic EoS of quantum particles at highdensities close 0 . /V − . /V and sufficiently high temperature T for which the effects of quantum statistics arenot important, one should still have the BE condensation PT on the one hand. On the other hand, this should bethe region of the deposition phase transitions for the classical hard spheres [10, 11]. Thus, we again should concludethat at high temperature T the BE condensation of quantum hard spheres should match with the deposition PT ofthe classical hard spheres.Coming back to the ideal gas of BE particles one should consider the limit V → V our conclusions about the deposition PT remain valid and, therefore, the wholeargumentation of K. Huang in Ref. [2] about the BE condensation as the 1-st order PT is correct. Only the K. Huanginterpretation of this PT as a gas-liquid one seems to be inconsistent with the modern interpretation of the PT ofhard spheres.At the moment it is not clear, if it is just a coincidence that at low pressures the real gases of mono- and diatomicmolecules, except for the helium-4 for pressures below 25 atm., indeed, demonstrate the deposition PT under cooling.Maybe a more realistic EoS of quantum particles can resolve this problem.It is remarkable that the BE-FD decomposition identity (20) allows one to establish another important interpre-tation. The right hand side of the identity (20) corresponds to pressure of a mixture of the ideal gases of fermionsand their pairs (which are the bosons) with the same degeneracy, but with the double mass and double charge, whichare taken with the wight 1 /
8. The left hand side of the identity (20) shows that such a mixture should experiencethe 1-st order PT of BE condensation. From the famous work of L. N. Cooper [23] it is known that the pairing offermions can, indeed, happen under not very restrictive conditions leading to the BE condensation of fermionic pairsand the BE-FD decomposition identity (20) illustrates such a possibility for a mixture discussed above. However, forthe appearing of Cooper pairs the fermions must have an attraction, which is absent in the EoS discussed here.It is evident that the identity (21) is valid for any dimension D = 1 , , , ... . Introducing the pressures of BEparticles (sign − ) and FD particles (sign +) of mass m that have the chemical potential ν and temperature Tp D ± ( T, ν, m ) ≡ ± T g Z d D k (2 π ~ ) D ln (cid:20) ± exp (cid:20) ν − e m ( k ) T (cid:21)(cid:21) , where e m (k) ≡ p m + k , (25)one can generalize the BE-FD decomposition identity (20) to the dimension D for the fractional mass and chargevalues p D − ( T, ν, m ) ≡ D p D − (cid:0) T, ν , m (cid:1) − D p D + (cid:0) T, ν , m (cid:1) . (26)For chargeless and massless particles, i.e. for ν = 0 and m = 0, the BE-FD decomposition identity (26) gives usthe following relation between the BE and FD momentum integrals p D − ( T, , ≡ D D − p D + ( T, , ⇒ ∞ Z x D dxe x − D D − ∞ Z x D dxe x + 1 , (27)which for D = 3 leads to a well-known identity ∞ Z x dxe x − ∞ Z x dxe x + 1 = π , (28)Note, however, that the right equation (27) follows from the left one after integrating the pressures of massless andchargeless particles over the angles, first, and, then, after integrating them over dk D by parts.Applying the identity (26) to its right hand side n times, one obtains another identity p D − ( T, ν, m ) ≡ Dn p D − (cid:0) T, ν n , m n (cid:1) − n P k =1 Dk p D + (cid:0) T, ν k , m k (cid:1) . (29)For n ≫ ln (cid:2) max( νT ; mT ) (cid:3) with the help of identity (27) one can establish an approximative relation p D − ( T, ν, m ) ≃ D ( n +1) D − p D + (cid:0) T, ν n , m n (cid:1) − n P k =1 Dk p D + (cid:0) T, ν k , m k (cid:1) , (30)which again relates the pressures of BE and FD particles. Note that Eqs. (25), (26), (29) and (30) are valid for theparticles with the hard-core repulsion, i.e. for ν = µ − ( D − V D p D − ( T, ν, m ), where the eigenvolume of particles inthe D -dimensional space is denoted as V D . IV. CONCLUSIONS
In this work we recapitulate the VdW equation of state of BE particles with the hard-core repulsion in the grandcanonical ensemble. Our analysis shows that the pressure of non-relativistic BE particles is mathematically equivalentto the one of the exactly solvable model with the 1-st order PT known as the sSMM. The EoS of BE particlescorresponds to the sSMM with the vanishing surface tension coefficient and the Fisher exponent τ F = . Such asimilarity allows us to show that within the present approach the high density phase of BE particles is a classicalmacro-cluster with vanishing entropy at any temperature which, similarly to the classical hard spheres, is a kind ofsolid state. Considering the limit of very small eigenvolume of BE particles we argue that the ideal gas of BE particleshas the 1-st order PT as it was suggested by K. Huang in his famous textbook [2] a long time ago.To explicitly demonstrate that a macro-cluster with the BE statistics does not exist in this EoS we investigatesome peculiar relations between the pressures of BE and FD particles, the BE-FD decomposition identities, showingthat under some conditions the pressure of FD particles can be identically rewritten in terms of two BE pressures.Moreover, we establish an exact representation of the pressure of BE particles of mass m , charge 1 and degeneracy g as a series of pressures of FD particles with the masses 2 k m , the charge 2 k and degeneracy 2 − k g , where k arepositive natural numbers. These new relations help us to correctly interpret the properties of a high density phase ofBE particles with hard-core repulsion.In fact, here we establish a principally new look at the problem of BE condensation. Of course, the consideredmodel is oversimplified, but now one can use all the achievements of the SMM [3–6, 9] and introduce the surfacepart σ ( T ) k of the free energy of k -particle clusters (here σ ( T ) is the temperature dependent coefficient of surfacetension). Such a modification will make model more realistic, since the surface part of free energy partly accounts forthe short range attraction among the constituents like it is done in the full SMM. Note that in this case, however, themodified right hand side of Eq. (5) cannot be already reduced to the pressure of point-like particles p − ( T, ν ) with theshifted chemical potential ν = µ − V p − . Acknowledgements.
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