PROPERTIES OF FRACTIONAL EXCLUSION STATISTICS IN INTERACTING PARTICLE SYSTEMS

Transcription

1 PROPERTIES OF FRACTIONAL EXCLUSION STATISTICS IN INTERACTING PARTICLE SYSTEMS DRAGOŞ-VICTOR ANGHEL Department of Theoretcal Physcs, Hora Hulube Natonal Insttute for Physcs and Nuclear Engneerng (IFIN-HH), 407 Atomstlor, Magurele-Bucharest 07725, Romana E-mal: Receved March 26, 2008 We show that fractonal excluson statstcs s manfested n general n nteractng systems and we dscuss the conjecture recently ntroduced (J. Phys. A: Math. Theor. 40, F03, 2007), accordng to whch f n a thermodynamc system the mutual excluson statstcs parameters are not zero, then they have to be proportonal to the dmenson of the Hlbert space on whch they act. By usng smple, ntutve arguments, but also concrete calculatons n nteractng systems models, we show that ths conjecture s not some abstract consequence of unphyscal modellng, but s a natural and for a long tme overlooked property of fractonal excluson statstcs. We show also that the fractonal excluson statstcs s the consequence of nteracton between the partcles of the system and t s due to the change from the descrpton of the system n terms of free-partcle energes, to the descrpton n terms of the quaspartcle energes. From ths result, the thermodynamc equvalence of systems of the same, constant densty of states, but any excluson statstcs follows mmedately. Key words: Quantum statstcs, ensemble equvalence, thermodynamc equvalence.. INTRODUCTION Fractonal excluson statstcs (FES) s a generalzaton of Bose and Ferm statstcs, ntroduced by Haldane n Ref. [], and wth ts thermodynamc propertes calculated manly by Isakov [2] and Wu [3]. The concept have been appled to a large number of systems of nteractng partcles [4 4] and descrbes quaspartcles that exst n fnte dmensonal Hlbert spaces n general, quaspartcles n a fnte regon of condensed matter; the Hlbert spaces are extensve, ncreasng proportonally to the sze of the condensed matter regon []. Quaspartcles that exst n these Hlbert spaces are called n general speces each subspace contans one speces. Let us denote the Hlbert spaces by H, wth =0,,, each of them contanng N partcles and G avalable states [5]. Then, by ncreasng the number of partcles of one speces, say N ncreases to N + δn, the number of avalable states n any of the Hlbert spaces Rom. Journ. Phys., Vol. 54, Nos. 3 4, P , Bucharest, 2009

2 282 Dragoş-Vctor Anghel 2 changes by δg j = α j δn. The proportonalty factors, α j, are called drect (when = j) and mutual (when j) excluson statstcs parameters. The smplest examples are the Bose and Ferm statstcs. For deal bosons, α j = 0 for any and j, whereas for deal fermons α = for any and α j = 0 for any and j, f. 2. PROPERTIES OF THE MUTUAL EXCLUSION STATISTICS PARAMETERS The FES (other than Bose and Ferm statstcs) s the result of nteracton between quaspartcles. The fnte dmensonal Hlbert spaces nvolved, H, are formed of quaspartcle wavefunctons correspondng to egenvalues contaned n fnte regons of the phase space. For example Isakov [2] and Iguch and Sutherland [6, 7] defne the Hlbert spaces by the wavevector egenvalues, k, whereas Murthy and Shankar [0], Sen and Bhadur [], Hansson, Lenaas, and Vefers [2] defne the Hlbert spaces by the quaspartcle energy egenvalues, ε. In all the cases, the egenvalues of the wavefunctons contaned n any of the Hlbert spaces belong to a fnte range (or a mult-dmensonal volume); therefore the quaspartclequaspartcle nteracton, whch changes the densty of states (DOS) n the phasespace, changes also the number of quaspartcle states n the range, leadng to FES [8] (see fgure ). Fg.. In (a), H and H 2 are two Hlbert spaces defned by the ranges of quaspartcle energes (the energy levels n H are not relevant, and therefore are not represented). In (b), extra partcles are nserted nto H, whch changes the densty of states n H 2 and therefore the number of allowed snglepartcle states. Ths s a manfestaton of fractonal excluson statstcs.

3 3 Interactng partcle systems 283 Let s take for example a system where the quaspartcle energes may be wrtten as [7, 0 2, 8 20] ε =ε + Vn + Vn, () j j j= 0 2 where ε are the energes of the nonnteractng partcles; The total energy of the system s E = n. = 0 ε We assume that the system n large enough and the energy levels dense enough to ntroduce the (quas)contnuous densty of states, σ(ε) and wrte equaton () as ε 0 ( ) n( ) ( ) ε=ε+ V ε, ε ε σ ε d ε, (2) where we replaced the subscrpts and j of V j by the correspondng energy levels, ε and ε', respectvely. In the varable ε, the densty of states s dfferent from σ(ε) and we shall denote t by σε ( ). To calculate σ ( ε ), we take a small nterval, δε, contanng σε δε ( ) states, and transform t nto the nterval δε, whch, obvously, wll contan the same number of states. Dvdng the number of states by the energy nterval and usng equaton (2), we fnd σε σ ε ( ε) =. ε V ( ε, ε ) + σ( ε ) n( ε ) + V( ε, ε ε) σ( ε) n( ε) 0 ε In fgure 2 we show an example. In some arbtrary, dmensonless unts, wth the populaton of the sngle partcle energy levels shown n the left plot, the DOS σ(ε) shown as the sold lne n the mddle plot, and constant nteracton potental, V( εε, ) V, we calculate σ ε ( ε) (pont lne n the mddle plot) and ε (pont lne n the rght plot) for the concrete calculatons of fgure 2 we choose V =, σ( ε ) = ε, and n(ε)=[exp(ε + ) ] }. ( ) (3) Fg. 2. The DOS on the quaspartcle energy axs, ( ) σ ε the pont lne n the mddle plot, and the quaspartcle energy, ε the pont lne n the rght plot for a bosonc sngle partcle energy levels populaton, n( ε ) = exp( ε+ ), a DOS σ( ε ) = ε and a constant nteracton potental, V,. ε ε ε ε (sold lne) for comparson. ( εε ) = In the rght plot we show both, ( ) (pont lne) and ( )

4 284 Dragoş-Vctor Anghel 4 To show how FES s manfested n the system, we splt the quaspartcle energy axs nto small ntervals, [ ε 0, ε ],...,[ ε, ε ],... (as shown n fgure 3), so that each nterval contans large enough numbers of partcles and avalable sngle partcle N ε, ε N ε, ε, states; we denote by ( ) the number of partcles n the nterval [ ] + ε + ε+ ( ) ( ) ( ) ( ) ( ) N N ε, ε = σ ε n ε = σ ε n ε, + ε ε + and, assumng that the energy ntervals are small enough so that we can replace n V, ε ε, ε we wrte the ntegral of equaton (2) ( ) M, V ε ε by ( ε ε ) for any ( ] M j j j ( j, ) (, + ) = 0 If we nsert δn partcles nto the nterval [ ε, ε ] ε =ε + V ε ε N ε ε. (4) + and we keep all the free partcle energy levels ε j( > ) unchanged, then we change the quaspartcle energy levels ε j( > ) by δε ( ) = V( ε j, ε j ) δn > (see the small, round, arrows on the ε axs n fgure 3). In ths way, n all the ntervals ε j, ε j+, wth j, the N j and G j reman ε0, ε,..., ε,,..., j εj on the ε axs, and the correspondng ntervals, [ ε 0, ε ],..., ε j, ε j,... on the ε axs. The ε, ε changes Fg. 3. The ntervals [ ] nserton of partcles n the nterval [ + ] the ntervals on the ε axs, from [, ] ε ε above, + f the ntervals on the ε axs are held fxed. unchanged. Therefore, f we maxmze the partton functon wth respect to the ε, ε, = 0,,..., we have to take nto account populaton of the energy ntervals [ ] +

5 5 Interactng partcle systems 285 the change of the quaspartcle energes due to the change of the populatons. We shall come back to ths method later. Another method to calculate the partton functon and ts maxmum, s to fx the ntervals [ ε 0, ε ],..., ε j, ε j,... along the ε axs. In ths case, the nserton of the δn partcles nto the nterval [ ε, ε + ] changes the values of ε j (for j > ) gven by equaton (4) nto ε j, whch should be calculated lke n [8]: j (, ), (, + ), (, + ) ε j =ε j + V ε j ε I + V ε j εk N ε k ε k + V ε j εk N ε k ε k. (5) k= 0 k= + Because σ(ε) s ndependent of the populaton, the change of ε j( > ) (see the δε s j on the left axs n fgure 3) leads to a change n the number of avalable sngle partcle states n the nterval ε j, ε j+ and therefore a change of the number of states n the nterval ε j, ε j+ and a change of σ ( ε ), accordng to equaton (3). Ths s the manfestaton of FES. The excluson statstcs parameters of the FES gas have been calculated n Ref. [8]. There we showed n the general case that these parameters obey the propertes conjectured n Ref. [5], namely that the mutual excluson statstcs parameters are proportonal to the dmenson of the Hlbert space on whch they act. Let us take as example a system of bosons of constant nteracton potental, (, ) V ε ε = V. Then, the drect and mutual excluson statstcs parameters are [8] and ( ), α εε =α εε = Vσ ε ε (6a) { ln σ( ε) } d α εε =θ( ε ε ) V σ ε( ε) δε αεε δg( ε), (6b) respectvely, where δε s the energy nterval n whch the mutual statstcs s manfested and δ G =σ ε ( ε ) δε s the number of energy levels contaned n t. Havng the excluson statstcs parameters (6), we can wrte the ntegral equaton (9) of Ref. [5] for the most probable partcle populaton as: V σ( ε ) + n( ε) dσ β µ ε( ) ln ε + = V ln + n( ε ) d ε. n ε ( ε ) d ε (7) Dfferentatng both sdes of (7) wth respect to ε, we obtan the equaton, εε ( ) Vn( ) ( ) ( ε) + n( ε) dn + ε σ ε d ε = β. n (8)

6 286 Dragoş-Vctor Anghel 6 If we use (2) nto (8), the dfferental equaton reduces to dn( ε) β = n( ε) + n( ε), (9) whch, provded that n(ε) s postve and converges to zero at ε admts only the soluton n ( ε ) = { exp β( ) } ε µ. (0a) Ths s, of course, the Bose populaton n the varable ε, and relaton (0a) s true for any σ(ε) and constant V. To determne µ, we plug (0a) back nto (7) and obtan µ =µ VN, (0b) where N = σ( ε) n( ε) s the total number of partcles n the system. From 0 n(ε), the populaton n( ε ) follows by a smple change of varable, gven by equaton (). Equatons (0) may seem surprsng, but let s have now another perspectve on the problem. Snce V s a constant, the total energy of the system reduces to 2 VN E = ε n +, () 2 whch s the energy of the gas n the mean-feld approxmaton. Ths case s easy to treat also from the perspectve of the free partcle energes, ε. If we turn back to the dvson of the energy axs ε nto the ntervals [ ε, ε + ], = 0,,..., of ε+ N = n( ε) σ( ε) partcles and G ε+ ( ) ε = σεstates, we wrte the partton ε functon and we maxmze t wth respect to the populatons n(ε), then we get n ε ( ) { exp ( ) } ε = β ε µ, (2a) whch s exactly equaton (0a), but wth a redefnton of the quaspartcle energy, ε =ε+ VN. (2b) The quaspartcle energes (2b) are dfferent from the ones defned n (). Moreover, ε ε s ndependent of, unlke ε ε, whch s Vn j 0 j j + Vn = 2 and depends on. Another dfference s that whle the total energy of the system can be wrtten as E = ε n, the summaton 2 ε n gves E + VN / 2. So, the defnton of quaspartcle energes () leads to the manfestaton of FES along the quaspartcle energy axs, ε. If V s constant, the FES partcle dstrbuton, n( ε ), corresponds to the Bose dstrbuton n the free-partcle energes, n( ε ) (equatons 0). If ( ) σ ε s also constant, then the mutual excluson

7 7 Interactng partcle systems 287 statstcs parameters, αεε (equaton 6b) vanshes and the drect excluson statstcs parameters become ndependent of ε : α εε = V σ; f Vσ =, then the nteractng Bose gas may be nterpreted as a Ferm gas. From here, the thermodynamc equvalence of systems of the same, constant densty of states and any excluson statstcs [2, 22, 9] follows drectly. 3. CONCLUSIONS Ths paper s a contnuaton of Ref. [8], where we showed that the fractonal excluson statstcs (FES) s manfested n general n systems of nteractng partcles and the mutual excluson statstcs parameters are proportonal to the dmenson of the Hlbert space on whch they act. Here we calculated the densty of quaspartcle states n general and we analyzed n partcular a system n whch the nter-partcle nteracton potental does not depend on the partcles quantum numbers (V j V). Usng the formalsm presented n Refs. [5, 8], we showed that n a gas of nteractng bosons, the resultng FES quaspartcle populaton, obtaned by the maxmzaton of the partton functon, s actually the orgnal Bose dstrbuton wrtten as a functon of the quaspartcle energy, ε, nstead of the freepartcle energy, ε. Therefore the FES reduces to a change of varable, from ε the free-partcle energy to ε the quaspartcle energy n the populaton of the sngle partcle energy levels, n. Form ths result, the thermodynamc equvalence of systems of the same, constant, densty of states but any excluson statstcs [2, 22, 9] follows mmedately. Acknowledgments. The work was partally supported by the NATO grant, EAP.RIG REFERENCES. F.D.M. Haldane, Phys. Rev. Lett., 67, 937 (99). 2. S.B. Isakov, Phys. Rev. Lett., 73, 250 (994). 3. Yong-Sh Wu, Phys. Rev. Lett., 73, 922 (994). 4. J.M.P. Carmelo, P. Horsch, A.A. Ovchnnkov, D.K. Campbell, A.H. Castro Neto, and N.M.R. Peres, Phys. Rev. Lett., 8, 489 (998). 5. A.D. de Vegy, S. Ouvry, Phys. Rev. Lett., 72, 600 (994). 6. R.K. Bhadur, S.M. Remann, S. Vefers, A.G. Choudhury, M.K. Srvastava, J. Phys. B, 33, 3895 (2000). 7. M.V.N. Murthy, R. Shankar, Phys. Rev. B, 60, 657 (999). 8. T.H. Hansson, J.M. Lenaas, S. Vefers, Nucl. Phys. B, 470, 29 (996). 9. S.B. Isakov, S. Vefers, Int. J. Mod. Phys. A, 2, 895 (997). 0. M.V.N. Murthy, R. Shankar, Phys. Rev. Lett., 73, 333 (994).. D. Sen, R.K. Bhadur, Phys. Rev. Lett., 74, 392 (995). 2. T.H. Hansson, J.M. Lenaas, S. Vefers, Phys. Rev. Lett., 86, 2930 (200).

8 288 Dragoş-Vctor Anghel 8 3. P. Bouwknegt, L. Chm, D. Rdout, Nucl. Phys. B, 572, 547 (2000). 4. S. Ouvry, arxv: v (2007). 5. D.V. Anghel, J. Phys. A: Math. Theor., 40, F03 (2007); arxv: K. Iguch, Phys. Rev. Lett., 80, 698 (998). 7. K. Iguch, B. Sutherland, Phys. Rev. Lett., 85, 278 (2000). 8. D.V. Anghel, arxv: (2007). 9. D.V. Anghel, J. Phys. A: Math. Gen., 35, 7255 (2002). 20. D.V. Anghel, Rom. Rep. Phys., 59, 235 (2007); cond-mat/ M. Howard Lee, Phys. Rev. E, 55, 58 (997). 22. M. Crescmanno, A.S. Landsberg, Phys. Rev. A, 63, 3560 (200).