Physica A 392 (2013) 4868–4873

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Physica A

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Stability and anyonic behavior of systems withM-statisticsMarcelo R. Ubriaco ∗

Laboratory of Theoretical Physics, Department of Physics, University of Puerto Rico, Río Piedras Campus, San Juan, PR 00931, USA

h i g h l i g h t s

• We calculate the thermodynamic curvature for systems withM-statistics.• We study the anyonic behavior and stability of the system in two and three dimensions.• We compare the stability of systems forM > 1 with the fermionic (M = 1), and bosonic (M → ∞) cases.

a r t i c l e i n f o

Article history:Received 17 April 2013Received in revised form 20 June 2013Available online 8 July 2013

Keywords:M-statisticsThermodynamic curvatureAnyonic behaviorStability

a b s t r a c t

Starting with the partition function Z for systems with M-statistics, as proposed in Chenet al. (1996) [1], we calculate from themetric gαη =

∂ ln Z∂βαβη the scalar curvature R in two and

three dimensions. Our results exhibit the details of the anyonic behavior as a function ofthe fugacity z and the identical particle maximum occupancy numberM . We also comparethe stability of systems for M > 1 with the fermionic (M = 1), and bosonic (M → ∞),cases.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Several years ago a new fractional statistics was proposed [1] in arbitrary dimensions which allows a finite number ofmulti-occupancy states of a single state. The formalism of new statistics is quite different than the para-Bose and para-Fermistatistics [2], and fractional exclusion statistics formalisms [3]. Since that a single quantum state can be occupied by n ≤ Midentical particles, this formalism opens the way to a new thermodynamics based on an extension of the Pauli exclusionprinciple. From a calculation of the second virial coefficient for low values of the fugacity z it was found in Ref. [1] that anideal gas becomes bosonic for M > 1. Also, a calculation of the heat capacity CV for high z values showed that the systemremains fermionic for M > 1. Certainly, these approximate calculations show that a thermodynamic system that followsM-statistics exhibits anyonic behavior for low z values, but miss the details that could tell us how this behavior dependson the fugacity z and the maximum occupancy M . In addition, one important question that one may address is related tothe stability of systems based on M-statistics as compared with fermionic (M = 1) and bosonic (M → ∞) cases. Theanswer to these questions can be given by performing a calculation of the thermodynamic curvature R, which in our case,is nothing else than the scalar curvature in the two-dimensional space defined by the parameters β and γ = −βµ. Theidea of using geometry to study some properties of thermodynamic systems [4–8] opened the way to the basic formalismof defining a metric in a two-dimensional parameter space and calculate the corresponding scalar curvature as a measureof the correlations strength of the system [9–18], with applications to classical and quantum gases [19,14,20,21], magneticsystems [22–25], non-extensive statistical mechanics [26–28], anyon gas [29,30], fractional statistics [31], deformed bosonand fermion systems [32], systems with fractal distribution functions [33] and quantum group invariance [34]. Some of the

∗ Tel.: +1 787 4610309.E-mail address: ubriaco@ltp.uprrp.edu.

0378-4371/$ – see front matter© 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physa.2013.06.062

M.R. Ubriaco / Physica A 392 (2013) 4868–4873 4869

basic results of this formalism is the relationship between the departure of the scalar curvature R from the zero value andthe stability of the system, and the fact that R vanishes for a classical gas, R > 0 (R < 0) for a boson (fermion) gas andbecomes singular at a critical point. Here, in order to study the stability and anyonic behavior of systems with M-statisticswe calculate the dependence of the scalar curvature on the fugacity z and the maximum occupation numberM . In Section 2we briefly describe the formalism of M-statistics. In Section 3 we calculate the scalar curvature for M-statistics in two andthree dimensions. In Section 4 we discuss and summarize our results.

2. QuantumM-statistics

In this section we briefly introduce the formalism of M-statistics, the details and the comparison of M-statistics withfractional exclusion statistics can be found in Ref. [1]. For a single species of particles the set of orthonormal eigenstates ofthe number operator N̂ is denoted by |j⟩ where j = 0, 1, 2, . . . ,M and

N̂|j⟩ = j|j⟩. (1)The annihilation and creation operators a and aĎ satisfy a set of relations

(aaĎ ± aĎa)|j⟩ = (|fj+1|2± |fj|2)|j⟩, (2)

wherea|j⟩ = fj|j − 1⟩, aĎ|j⟩ = f ∗

j+1|j + 1⟩. (3)

In particular, f0 = 0 and f1 = eiα . It is clear that aj = aĎj = 0 for all values of j such that j > M . For multi-species ofparticles the states are simplywritten as |ijk . . .⟩with the corresponding number operators satisfying commutation relations[N̂i, N̂j] = 0 and the a and aĎ operators satisfying two possible commutation rules

aiaj − e±i 2πM+1 ajai = 0,

aĎi aj − e∓i 2πM+1 aja

Ďi = 0 i < j. (4)

It is simple to see that the states are fermionic forM = 1, and become bosonic forM → ∞. The partition function takes thesimple form

Z =

p

Mj=0

z je−jβϵ(p). (5)

In the thermodynamic limit, we obtain in arbitrary dimensions D

ln Z = VA(D)

∞

0dppD−1 ln

1 − zM+1e−(M+1)βϵ(p)

1 − ze−βϵ(p), (6)

with ϵ(p) = p2/2m and the factor A(D) =1

2D−1πD/2Γ (D/2).

3. Scalar curvature

The metric gij for a thermodynamic system is defined [9] as the second order term after expanding the informationdistance I(ρ(β i), ρ(β i

+ dβ i)) = Trρ(ln ρ(β i) − ln ρ(β i+ dβ i)) between two statistical close states ρ(β i) and ρ(β i

+ dβ i)leading to

gij = −

∂2 ln ρ

∂β i∂β j

, (7)

where in our case the thermodynamic coordinates are β1= β and β2

= γ ≡ −βµ. For exponential distributions,

ρ =exp(−

i β

iF i)Z , the metric simply reduces to

gij =∂2 ln Z∂β i∂β j

. (8)

From Eqs. (5) and (8) it is simple to obtain that the metric tensor components are written in terms of the second momentsof the energy and particle number as follows

g11 = ⟨H2⟩ − ⟨H⟩

2,

g22 = ⟨N2⟩ − ⟨N⟩

2,

g12 = ⟨NH⟩ − ⟨N⟩⟨H⟩. (9)From the metric we obtain the scalar curvature R from the definition

R =2

det gR1212, (10)

4870 M.R. Ubriaco / Physica A 392 (2013) 4868–4873

Fig. 1. The scalar curvature R, in units of λ3V−1 , as a function of the fugacity z at D = 3 and constant β for the cases ofM = 1 (solid line),M = 5 (dashedline) andM = 25 (dotted line).

where det g = g11g22−g12g12 and due to the obvious identities like ∂gij∂β j ≡ gij,j = gjj,i the non-vanishing part of the curvature

tensor Rijkl is given in terms of the Christoffel symbols Γijk =12gij,k as follows

Rijkl = gmn ΓmilΓnjk − ΓmikΓnjl

. (11)

The calculation of R simply reduces to solve

R =1

2(det g)2

g11 g22 g12g11,1 g22,1 g21,1g11,2 g22,2 g21,2

where gij,k = ∂gij/∂βk. From Eq. (6) we find that the metric components can be written as

g11 =V

β2λDΓ (D/2)H D

2 +1

g12 =V

βλDΓ (D/2)H D

2

g22 =V

λDΓ (D/2)H D

2 −1, (12)

where the function Hη is the integral

Hη =

∞

0dxxη

f ′

0

f 20−

(M + 1)f ′

M

f 2M

, (13)

where the function fL = 1 − zL+1e−(L+1)x (L = 0, 1, . . . ,M) and f ′

L = ∂ fL/∂γ . The function Hη satisfies the simple relation

∂Hη

∂γ= −ηHη−1. (14)

We obtain for the scalar curvature R for arbitrary D

R =λDΓ

D2

2V

1 +

D2

H2

D2H D

2 −1 +D2 − 1

H D

2 +1H D2H D

2 −2 − DH D2 +1H

2D2 −1

(H D2 +1H D

2 −1 − H2D2)2

. (15)

Figs. 1 and 2 show the results of a numerical calculation of Eq. (15) for the scalar curvature in D = 3. Fig. 1 shows thedependence of R on the fugacity z = e−γ for the valuesM = 1, 5, 25. ForM = 5 the system is bosonic for very low values ofz and becomes fermionic for z > 0.72. The switch from bosonic to fermionic occurs at higher values of z asM increases. For

M.R. Ubriaco / Physica A 392 (2013) 4868–4873 4871

Fig. 2. The scalar curvature R at D = 3, in units of λ3V−1 , as a function ofM and values for the fugacity z = 0.5 (solid line), z = 0.9 (dashed line), z = 1.5(dotted line) and z = 15 (dashed–dotted line).

Fig. 3. The scalar curvature R, in units of λ2A−1 , as a function of the fugacity z at D = 2 and constant β for the cases ofM = 1 (solid line),M = 5 (dashedline) andM = 25 (dotted line).

large values of z andM , these systems aremore stable than the fermionicM = 1 system. There is a value of z like z ≈ 0.7 forM = 5 and z ≈ 1 forM = 25 where the scalar curvature vanishes and thusM-systemsmimic a classical gas behavior. Fig. 2shows the values of the scalar curvature R as a function of M ≥ 1 for the fugacity values z = 0.5, 0.9, 1.5, 15. For z < 1,M-systems become bosonic asM increases. For z > 1 the behavior is fermionic for all values ofM . For any z there is a valueof M > 1 after which the scalar curvature becomes constant. The instability increases at those values of z ≈ 1 and largervalues ofM , as expected. The behavior at D = 2, shown in Figs. 3 and 4, is quite similar to the one in three dimensions. Fig. 5displays the scalar curvature for 0 < z < 1 for the standard Bose–Einstein caseM → ∞,M = 25 andM = 50. The stabilityis practically identical up to a value of z where the Bose–Einstein system becomes more unstable approaching the onset ofBose–Einstein condensation and theM-systems are more stable and switching to fermionic (R < 0) at higher values of z.

4872 M.R. Ubriaco / Physica A 392 (2013) 4868–4873

Fig. 4. The scalar curvature R at D = 2, in units of λ2A−1 , as a function ofM and values for the fugacity z = 0.5 (solid line), z = 0.9 (dashed line), z = 1.5(dotted line) and z = 15 (dashed–dotted line).

Fig. 5. The scalar curvature R, in units of λ3V−1 , as a function of the fugacity z, for 0 < z < 1, at D = 3 and constant β for the cases a Bose–Einstein gas(M → ∞) (solid line),M = 25 (dashed line) andM = 50 (dotted line).

4. Conclusions

In this manuscript we have calculated the scalar curvature for systems following M-statistics in two and threedimensions. Since the scalar curvature R is positive (negative) for bosons (fermions) our results give us a detailed pictureabout the anyonic behavior than a calculation of the virial coefficients could provide. In addition, the values of the scalarcurvature tell us about the correlations and thus the stability of the system. A particular feature of M-systems is that theirbehavior is very similar in two and three dimensions. For those values 0 < z < 1 these systems are bosonic andmore stablethan the standard Bose–Einstein case, and become fermionic at higher values of z. The change from bosonic to fermionicoccurs at higher values of z as the maximum occupation number M increases. Therefore, M-statistics provides a differentframework, other than quantum group invariant systems [34], to study anyonic behavior if ever observed in either two orthree dimensions.

M.R. Ubriaco / Physica A 392 (2013) 4868–4873 4873

Acknowledgment

I thank Y.J. Ng for reading the manuscript and useful comments.

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