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CBPF CENTRO BRASILEIRO DE PESQUISAS FÍSICAS Notas de Física
CBPF-NF-038/86 D-DIMENSIONAL IDEAL GAS IfT PARASTATISTICS: THERMODYNAMIC
PROPERTIES
by
M.C. de Sousa Vieira and C. Tsallis
RIO DE JANEIRO 1986 NOTAS DE FlSlCA é uma prc-publicaçio de trabalho original em Física
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CBPF-NF-033/36 D-DIMENSIONAL IDEAL GAS IN PARASTATISTICS:THERMODYNAMIC
PROPERTIES
by
M.C. de Sousa Vieira and C. Tsallis
Centro Brasileiro de Pesquisas Físicas - CBPF/CNPq Rua Dr. Xavier Sigaud, 150 22290 - Rio de Janeiro, RJ - Brasil CBPF-NF-038/86
ADSTRACT
We consider a parastatistics ideal gas with energy spectrum e « | lc |ot (a > 0) or even more general in a d-dimensional box with volume V (periodic boundary conditions), the number N of the gas particles being well determined (real particles), or not (quasi, particles). We calculate the main thermcdynamic quantities (chanical potential, internal .energy, .specific heat. C, equation of states, latent heat, average numbers of particles) for arbitrary d,ct,T (temperature) and p (maximal number of particles per state allowed in the parastatistics). The main asymptotic regimes are worked out explicitely. In par- ticular, the Bose-Einstein condensation for fixed '..density v N/V appears as a non uniform convergence in the p->-« limit, in complete ana- logy with the standard critical phenomena which appear in interac_ ting systems in the N •*•*> limit. The system behaves -i essentially like a Fermi-Dirac one for aLJL finite values of p, and reveals a Bose-Einstein behavior only in the p •+» limit. For : instance, at low temperatures C«T if j ' <*> and C.T if p •*». Finally the Sommerfeld integral and its expansion are generalized to an arbitrary finite p.
Key-words: Parastatistics; Ideal gas; Bose-Einstein condensation; Çommerfeld integral. 1 CBPF-NF-038/86
1 INTRODUCTION
Since the pioneer work by Gentile'-1-! many different studies
have been done which interpolate between Bose-Einstein and
Fermi-Dirac: statistics (see for instance P.ef. [XjKPossible applica- tions have been searched in Field Theory and Elonentary Particles Phy-
sics *- -• as well as in Condensed Matter Physics (molecular exciton;. and magnons'--' , quantum Hall ef feet'-^•') . The ideal gas
in parastatistics naturally constitutes a privileged refer-
ence system which can be used as unperturbed starting point to study more complex systems. Also it will be shown that many exact (or asymptotically exact) analytical expressions can be obtained which could serve as testing material for vaxious ap- proximation methods. The purpose of the present paper is the establishment of the main therrnodynamic properties; of such gas at fixed volume in d-dimensions (d>0), and- for a quite general energy . d a/2 spectrum e =( J a.k? ) (a. >0 and a >0; a =1 corresponds to Vj«l J *' J photons, and to short wave-vector acoustic phonons, among others; a = 2 corresponds to non relativistic free particles and to short wave-vector acoustic magnons in Heisenberg ferromagnets, among others).
In Section 2 we introduce the specific ideal gas model and obtain the associated density of states; in Sections 3 and 4 we calculate the main thermodynamic quantities respec- tively corresponding to fixed (real particles) and unfixed
(quasi-particles) total number of gas particles; finally we conclude in Section 5. In Appendix we generalize to • para- statistics. ;thô Camierfold integral and its standard expansion. CBPF-NF-038/86
-2-
2 PARASTATTSTICS IDEAL GAS - DENSITY OF STATES
We consider an ideal gas of N particles (or quasi-particles) in a d-dimensional box. Each particle behaves as a planar wave with energy spectrum given by
' a \ (cx/2
where a. >0 V., a >0 and k. is the j-th component of the wave vector }«. The particular case a. =a, V., yields c « ka (k = |k|) . Periodic boundary conditions are considered on the box which is assumed to be an orthogonal hyperparallelepiped with side d lengths ÍL.} and volume V = .it. L.. The possible wave vectors are given by
k. = p- n. - (n. =0,±l,±2,...yj) (2)
j which inserted into Eq. (1) yield
d (2n)2a. I ')' J n? - 1 ' (3) j-1 e17"^? J This is the equation of an hyperellipsoid whose volume provides the number of states 4>(e) whose energy is equal to or lower thane. Consequently
with —3 —
a/d (41) Vd/a
The density of states p(e) is therefore given by
d/a-1 p(c) = àt (5) 0 0
Notice that vie treat in (N -*«>, v-»-00, N/V-*constant, e ->0). The parastatistics thermal equilibrium average number f(c-) of particles per state is given by'—' p+1 f(e) = (6) e^-^-l e e-y) , or alternatively by. .(p-j)P(e-v) I 3 f(c) = (61) P L where $ 21/^DT is the inverse temperature, u is the chemical potential, and p is the maximal number of particles allowed per state (p=l reproduces the Fermi-Dirac statistics and p ->•« the Bose-Einstein one); see Fig. 1. CBPF-NF-038/B6 -4- 3 FIXED TOTAL NUMBER OF PARTICLES (REAL PARTICLES) 3.1 - Chemical potential and ground state population We consider here the total number N of gas particles as well determined. If we respectively note N and N the popula- tions of the ground state and the excited states, then N = NQ(T) + Ne(T) (7) We shall now calculate the chemical potential u(T,p), which is completely determined by N = jde p(e)f(e) . (8) Let us first consider theT-^O limit, 'in which case f(e) is the step function indicated in Fig. 1. We then have N =N and H- j dep(e)p= o o eo where W0(p) = u(O,p) and we have used Eq. (5) (with p(e) 5 0 for e < 0). Consequently we have that yo(D CBPF-NF-038/86 -5- where »o(1> ' Vf) % or even ict/d (12) where we used Eq. (41) to rake explicit the dependence of y on the concentration N/V. It is convenient to introduce the fol- lowing reduced quantities: t = kBT/y0(l), (13) p = n/po(l), (14) and x = 6/no(l) (15) Equation (8) can therefore be rewritten as follows: p-t-1 d/o-1 (16) t-f dxx (p+l)(x-y)/t ^ e which makes obvious that/ in energy units of v>0(D/ the chemical potential (and in fact all the thermodynamic quantities we shall be interested in) depends on d and a only through the ratio d/a (see also Ref. p}) • Through a further transformation y Sx/t we obtain CBPF-NF-038/86 -6- t"d/a= f°dy yd/ct'1 which provides t as an explicit function of M/t,p and d/a. The results are indicated in Fig. 2. As can be observed therein, the thermal dependence of ü, for fixed d/a, flattens in the low tem- perature region for increasing p, and eventually exhibits, in the» p-*» limit and d/a>l, a "plateau" at jl = 0 (the width of the pla- teau increases with increasing and not too high d/a -1). This is of course the Bose-Einstein condensation, and we see in the . pre- sent framework that it appears in -the p •»•» limit through a non uniform convergence completely similar to those which are ob- served, in the N ->» limit, for interacting systems presenting phase transitions. No Bose-Einstein condensation at finite tenperatures exists for à/à i 1. In particulicr for ã/a = 1, this .is easy to verify as the integral of Rj. (17) is straightforwardly solved, and we obtain the following de- pendence of t on í /ti t = 1 = i . (10) t in I .I*'. This expression leads, in the p •*•<*> limit (and taking into ac- count that y < 0 for t > 0), to the standard d = a= 2 result, i namely V - tind -e"1/c) (19) CBPF-NF-038/86 -7- where we verify that y vanishes at no other place than at t =0. Let us now go back to the general expression indicated in Eq. (17), and denote by t* the (finite) temperature at which V vanishes for fixed p and d/ct. It follows that P+l 1 (20) The integral can be expressed (see page 325 of Q.Ò]) in terms of the Riemann zeta and gamma fúntions ç and T, and we obtain ) -ct/d 1 - C(d/a)r(d/a+l)J (21) (p+l)d/a-1 These results are illustrated in Fig. 3; • note that t (d/ct) £ lim t*ip,ã/a). precisely is the Bose-Einstein condensa c po- tion critical (reduced) temperature (the phase transition only exists for p •+•»), and it is given by -O/d (22) hence -0/d k(d/o)r(d/o (22') \ (d/a-l) if a/a -1 (22") 5 a if d/a d ' A few more steps will lead us to the thermal dependence of the (reduced) population NQ/N of the ground state below the critical temperature (above the critical temperature, NQ/N vanishes in CBPF-NF-038/86 -8- the thermodynamic limit N -»•<»). We have that N e de p(e)f(e;y=O) ~ = 1 d d let 1 - a hence /a (t J a/d a/d P 1 + ) \~ (24) The solution of this equation clearly has the form (25) The substitution of this expression into both sides of Eq. (24) enables, by simple identification, the knowledge of io } up to any desired order. To the lowest correction we obtain CBPF-NF-038/86 -9- 2ct/d (26) Note the changement of curvature which occurs at d/a=l. As a matter.of fact, for d/cx=l, the departure of y from 1/p is slower than any power of t (and not only t2). For this case we obtain,- from Eq. (18), (t-M),d/a=l.p<«) (27) (27') -te "1/c (t -0,d/a =l,p ••«) For the high-temperature asymptotic behavior we can present Eq, (17) as follows: 4-1 - i-i d j -es + ..'.) CO 2. 4-a 1 a e '•fdy y + e J dy y + ... hence CBPP-NF-038/86 -10- t-d/a d/a d/a 1 2 (p+l) - hence 2?d/a rr/dU Finally, by using z =JI/t, we obtain , /, v .-d/a+-d/a+11 -~pd/a+l -S ten t - tenri- + I) —|^—^^—r- + a \o / 2d/a rUr/d + .\ (t -> «,Vd/a) (28) Note that, as expected, the high-temperature leading terms dó not depend on p (Maxwell -Boltzmann limit). 3.2 - Internal energy, specific heat, equation of states and latent heat The internal energy is given by ü « í de ep(e)f(e) (29) o hence CBPF-NF-038/86 -11- (30) with u(t) nâ ta dy p+1 (31) e(p+i)(y-P/t) _x urn a t where we have used Eqs. (5), (9) and (11) . This expression, to gether with y(t) determined in Section 3.1, canpletely determines the thermal dependence of u and consequently U/N. The specific heat C is given by C - 2H (32) C " dT Therefore the specific heat c per particle is given by •• B where we have used Eq. (30) . The results are presented in Fig. 4. Let us now discuss the low-temperature behavior of c. For finite p we can expand the integral of Eq. (31) as indicated in the Appendix, and then use Eqs. (26) and (27); we obtain (t *0) (34) This is an interesting result, as it shows that asymptotically C«T ftoi all finite, value.6 oh p and d/a,.thus generalizing the well known result for quasi-free electrons in a metal (d=3,o=2 andp-1), CBPF-NF-038/86 -12- In the p •*• <»> limit ,. Bq. (31) becomes l d/a u(t) = £ ta dy -Z (t (36) hence d/a c = Ér(â + 2)c(^+l)t (37) a A a / \a ' for the case d/a > 1 _ d/d , A Ç(f4 4 t Nd/a (37') where we have used Eq. (22'). The C * T 'a law we have obtained generalizes the Debye law for acoustic i*ionons (d =3, a = 1) as well 3/2 as the T 3/2 llaaw for magnong ir the standard Heisenberg ferro- magnet (d»3,o CBPF-NF-038/86 -13- Let us now fccun the high-temperature behavior of c. The integral of Eq. (31) can be treated in the same way we did for Eg. (17); we obtain d' 1 i , d/a - 1 _ p(pd/a - D l p pd/a [r(d/«+i)] t J (t * «>) . (38) '•Itote that in the Fermi-Dirac(Bose-Einstein)statistics/ the specific ' heat approaches the classical d/a value from below (above) when d/cc > 1, and the opposite happens when d/a <1; for d/a =1, the approach occurs from below for all values of p. Consequently C vs.T presents, for d/a >1, a maximum for p hJLqh enough (the maximum becomes a cusp in the p •*•>» limit); for d/a <1, it presents a maximum for p low enough. Let us now deduce, the equation of sta.tes. The grand-canoni- cal partition function-£ associated with the wavevector jc is given by 5,= !•.**...• .* =^ 1 - e The total partition function z equals v zg, consequently it K p(e)£n. o We know from Thermodynamics that the pressure P satisfies 6PV = in; hence CBPF-NF-038/86 -14- BPV = i de pUUn o (41) where we have used Eq. (5). On the other hand, also from Thermo dynamics, we have that = _ | fixed By 3P 'fixed d/a-1 -Í 1 - e And by using again Eq. (43) we oJtain p _ a U (42) P " d V which is exact for all value.* 0(J p. Eq. (4 2) transforms the analysis of the pressure into that of U, which we have already done. Let us finally calculate the latent heat L (per particle) associated with the nose-Einstein condensation (first order) phase transition. The Clapeyron equation states that (43> where v (v ) is the volume per particle in the notonal [conduced) n c phase. But in the N >> 1 limit v vanishes'- -• and v equals V/N, therefore CBPF-NF-038/86 -15- L o N a [d (U/N)1 = T dL dT JT d Utit where we have used Eq. (42) and the definitions of u and t. By replacing Eq. (36) in this expression we obtain .d Ia and, by using Eq. (22*), we finally obtain Clr (44) ~(d/a-l) if d/a- (44«) d/a +1 if d/o -»•• The dependence of L on d/o is represented in Fig. 5. 4 UNFIXED TOTAL NUMBER OF PARTICLES (QUASI-PARTICLES) In the present situation N is unfixed, therefore u van- ishes for all temperatures. The average runber -16- hence where we have used Eq. (5). It follows that for d/a > 1 C(d/a)T(d/a+l) d/a 1 — (kBT) (46) /2 (P+D d/a-1 V rCd/2+l)(í a. J-l J where we have used Eq. (41). Let us now focus the internal energy..It is given by dee p U) (47) -r e—-] hence d/a+1 U (k T) p -t-1 B dyy4/a V d/a ey-l hence d/a+l C(d/a+l)r(d/a+1)d/a 1 - T) (46) (p.1)""-1 (k d d/2 l II • 2 JI r(d/24-l)(jn1ajJ d/a+l .where we liave used again Eq. (4').The present U/V «T law gen- eralizes the well known black-body T" Stefan-Boltzmann 3aw (d=- %, a = l, p•+<*>). The generalized radiation pressure can be obtained by replacing Eq. (48) into Eq. (42) which still holds. CBPF-NF-038/C6 -17- CONCLUSION In an unified framework, we treat the parastatistics of an ideal confined (fixed volume) d-dimensional gas of N particles (or quasi-particler) whose energy spectrum is given by Eq. (1) (which contains the important isotropic case e « |k|a(a >0)) ; each state can be occupied at most by p particles. W.3 have studied, for arbitrary d,a,p,T, and for both fixed and unfixed N cases, various thermodynamic quantities (chemical potential, average populations, internal energy, specific heat, pressure and latent heat). In what follows we summarize the main results. (Í) The density of states satisfies p(e) ae . This form will imply thai, the thermal dependence of all equilibrium thermodynamic quantities, and for all p, will depend on d and a on through the ratio d/a. (ii) The Bose- Cinstein condem-at í on appears, for d/a > 1, as a non uni- form convergence in the p -><» limit, in complete analogy with phase transitions in interacting systems which appear as non uniform convergences in the thermodynamic limit N -*». We obtain the explicit dependence of the critical 1 temperature Tc on d/a (Eq. (22 )): it is T£ « (d/a -1) for d/a % 1, and T «5 a/d for d/a >> 1. The macroscopic popula- tion of the ground state satisfies N /N =1 - (T/T )d/a * 0 C for T explicit function of d/a (Eq. 44 )); it satisfies L/T« l/2(d/a-l) for d/a^l, and L/T ^d/a+1 for d/a >>1. The picture whiefch emerges (here and in the paragraphs (iv) and (v) which fol low) is that the p •*•<*> limit is deeply different from any CSPF-NF-038/86 -18- other case (i.e. 0 (iii) For fixed N, finite p and low temperature, the chemical potential presents a quadratic departure from the T = 0 value (Eq. (26)); the curvature changes its sign at d/a=l. In the high temperature regime the first two (or three for p>l) dominant terms (of the chemical potential) do not depend on p. (iv) For fixed N, finite p, arbitrary d/a and low temperature, the specific heat satisfies C «* T thus generalizing the standard result for quasi-free electrons in a coriductor (d = 3, a = 2, p =1) . Ii, the p-*» limit we obtain, for both fixed and unfixed N cases, C«T thus generalizing the ' Debye-law for acoustic phonons in a,crystal (d =3,a=l)and 3/2 the T -law for magnons in a Heisenberg ferromagnet (d = 3,0=2). .(v) For fixed N, arbitrary p and d/a, and high • temperature, C approaches the classical value (d/a)NkB< For p low (high) enough, C aproaches this value from below (above) when d/a >1, and the opposite happens when d/a <1; con- sequently C presents, for d/a >1, a maximum for p high enough (oup in the p •*•" limit) and presents, for d/a <1/ a maximum for p low enough. CBPF-NF-038/86 -19- (vi) For arbitrary p, d/ot,T and for both fixed and unfixed N cases, the pressure P and the internal energy U are re- lated through U = (d/a)PV (V Svolurae) , thus generalizing the standard result U = (3/2)PV for d = 3, a = 2 ai.d p=l,«. (vii) For unfixed N, arbitrary p, d/a arid Tf the density of the internal energy (proportional to the power irradiated per unit area of the confining box) satisfies U/V«T (Eq. (48)), thus generalizing the T' Stefan-3oltanann law (d =3,a =l,p Also, since the 1968 paper by Gunton and Buckingham1—F9l' (see also C123 and references therein), the Bose-Einstein condensation is known to be related to the criticality of the spherical model (ii-vector model with n •>•<») . Consistently, the fact that .Tc should vanish in the d/a •• 1 limit can be inferred from this re lationship. Finally, as a closing remark let us recall 'thac several studies '- " 6J are already available in which the bosonic limit (p •+») is analysed under a variety of conditions (for both spinless and magnetized bosons)«By following along the lines of the present paper, it could be interesting to extend those results to arbitrary values of p. CBPF-NF-038/86 -20- We acknowledge with pleasure critical reading of the manuscript by W. Theumann, as well as useful remarks S.R.A, Salinas and A. Theumann. One of us (MCSV) acknowledges very fruit ful discussions with A.M. Mariz as well as a Fellowship from CNPq/Brazilf the other one (CT) has benefited, at the early stages of this work, from fruitful interaction with O.L.N. Telles Menezes. CBPF-NF-038/66 -21- APPENDIX-GENERALIZED SOMMERFELD INTEGRAL Here we generalize to an arbitrary finite p the standard Scmnerfeld integral for the FD statistics,and its low temperature expan- sion. We consider the integral I(z) =1 dyH(y)f(y;z) * (A.I) —00 where To characterize the function H(y) we introduce K(y) = I7 dy'Hiy1) (A.3) —oo K(y) is assumed to satisfy the following requirements: (i) lim Kiy) = 0 , (A.4) hence H(y) - ^f '. (A.5) (ii) lim f(y;z)K(y) = 0 ; (A.6) y-»«o (iii) K(y) is analytic at y = z . (A.7) These conditions arc almost always satisfied in the physical CBPF-NF-038/86 -22- situations as typical functions H(y) identically vanish for y < 0, a- symptofcically behave as a power-law in the y •*•» limit, and are soft functions at y = z (H(y) might present singularities, but they are normally integrable and located at places such as y - 0, and not at y =z) . Integrating (A.I) by part? we obtain K(y)f (y;z) - 1—06 df (y; z) - J (A.8) where we have used conditions (A.4) and (A.6) and the fact that, lim f(y;z) = p. By expanding K(y) at y =z we obtain where we have used the facts that - p and that df/dy is an even function of (y - z) and therefore all odd terms in the sum over n vanish. By using the connection between K(y) and H(y) we can rewrite Eg. (A.9) as follows: p f H,y,dy (A.10) CBPF-NF-038/86 -23- with 1 2 d p+1 a - rkx * r , L_ •m " (2m) ! J axx dx I (p+l)x . x , 1 dxx 2m (2m): 2m-1 Í sinh2x (2TT)2m|B (A.11) 2m-l 2 m where in the last step we have used Q,o3 (page 352) / B2m = being the Bernoulli numbers. By using the fact that |B2ml (2m)! ç(2m)/(22m"ITi2in) we can finally express « as follows: am = 2ç(2m) 1 i- (A. 12) ° L (P+1) and consequently dyH(y)f (y;z) - pif H(y)dy + 2 I L (p-i)2^1. 2m-1 - PJ H(y)dy + i- H'(z) L p+iJ 1 -• H"1 (z) + ... (A.13) CBPP-NF-038/36 -24- which is the generalization we were looking for. Quite frequently H(y) is :.def ined as follows: rO -if y < 0 H(y) (A.14) AyT if y >, 0 In such case,Eq. (A.13) can be .rewritten as follows: j dyH(y)f(y;z) o Az Y+l jS0 Finally if we identify y = c/kgT and z=y/kBT, Eq, (A.13) takes the form JdeH(e)f(c) - pf H(e)de I m-1 —go (A.16) + o (¥)' CBPF-NF-038/86 -25- where in the estimation of the rest o (k T/y) we have as- (n sumed the quite frequent fact that H > (y) is of the order of H(u)/yn. CBPF-NF-038/86 -26- CAPTION FOR FIGURES Fig. 1 - Parastatistics with maximal occupation p: average po- pulation per state as a function of the energy c mea- sured from the chemical potential y, for finite and vanishing temperatures. The point (e-u,f) = (0,p/2) is a center of symmetry. Fig. 2 - Thermal .dependence of the (reduced) chemical potential for typical values of p and d/ct. Notice that for all p <« the behavior is qualitatively the same (in the sense that p is analytic for all finite values of t) no matter the value of d/a, and can be characterized by the Ferrai-Dirac (p=l) behavior, wheras :the p -»• • limit (Bose - Einstein) presents two different regimes/ one occuring for d/a £ 1 (no Bose-Einstein condensation), the other one occurlnf for d/o > 1 (Bose-Einstein condensation ;due to non-analyticity of jJ appearing at a &JLnJLt Fig. 3 - Reduced temperature at which u vanishes as a function of d/a and p. The p *• •» limit yields the d/a-dependence of the (reduced) critical temperature corresponding to the Bose-Einstein condensation. Fig. 4 - Thermal dependence of the (reduced) specific heat for typical values of p and d/a (a-e). For the p •*•<» limit we have represented in (f) the d/a -dependence of the height of the cu&p which appears in c vs. t for d/a >1. Fig. 5 - The d/a-dependence of the latent heat L per particle (in units of k_T) associated with the Bose-Einstein conden- sation (first-order phase transition). CBPF-NF-036/86 -27- CBPF-NF-033/C6 -28-. ft (b) noz CBPF-NF-038/86 -29- FIO. 2 CBPF-NF-033/86 -30- (e) t-t-l FI6.2 CBPF-NF-038/86 -31- (oi 0 I 10 d/a CBPF-NF-038/S6 -32- (a) 0.5- P'10 p'5 p'2 FI0.4 CBPF-NF-038/36 -33- re; p'5 s'i .-i Z t id) -34- 30 (f) 10 0 12 4 6 ifa FIO. 4 CBPF-NF-038/86 -35- «o (D CBl'F-NF-038/86 -36- REFERENCES G. Gentile, N. Cimento 12» 493 (1940) and !L9, 109 (1942). S.I. Ben-Abraham, Am. J. Phys. 3_8, 1335 (1970). [3] A.J. Kálnay, Int. J. Theor. 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