Fugacity versus chemical potential in nonadditive generalizations of the ideal Fermi-gas
FFugacity versus chemical potential innonadditive generalizations of the idealFermi-gas
Andrij Rovenchak, Bohdana Sobko
Department for Theoretical Physics,Ivan Franko National University of Lviv,12 Drahomanov St., Lviv, UA–79005, Ukraine
April 22, 2019
Abstract
We compare two approaches to the generalization of the ordinaryFermi-statistics based on the nonadditive Tsallis q -exponential usedin the Gibbs factor instead of the conventional exponential function.Both numerical and analytical calculations are made for the chemi-cal potential, fugacity, energy, and the specific heat of the ideal gasobeying such generalized types of statistics. In the approach based onthe Gibbs factor containing the chemical potential, high temperaturebehavior of the specific heat significantly deviates from the expectedclassical limit, while at low temperatures it resembles that of the or-dinary ideal Fermi-gas. On the contrary, when the fugacity entersas a multiplier at the Gibbs factor, the high-temperature limit repro-duces the classical ideal gas correctly. At low temperatures, however,some interesting results are observed, corresponding to non-zero spe-cific heat at the absolute zero temperature or a finite (non-zero) min-imal temperature. These results, though exotic from the first glance,might be applicable in effective modeling of physical phenomena invarious domains. Key words:
Fermi-statistics, Tsallis q -exponential, nonadditivestatistics, ideal Fermi-gas, minimal temperature a r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r Introduction
Nonextensive and nonadditive generalizations of entropy originated inthe information theory [1, 2] and were introduced in physical problemsby Tsallis [3]. The approaches based on Tsallis’s generalization areapplicable to problems, where, for instance, long-range interactions ornon-Markovian memory effects are essential [4]. These include bothphysical phenomena [5–7] and interdisciplinary applications [8–13].Various formulations of nonadditive generalizations are known forquantum Bose- and Fermi-distributions [14–21]. Note that the Bose-systems are studied to a larger extent, perhaps due to the fascinatingBose-condensation phenomenon [22, 23]. While the deformations ofthe Fermi-statistics are often studied alongside the Bose-statistics, insome works deformed fermions are a sole subject of analysis [15, 19,24].In the present paper, we use a phenomenological model previouslystudied for nonadditive generalizations of the fractional Polychronakosstatistics [25] and the ideal Bose-gas [26]. To obtain thermodynamicproperties as functions of temperature T , we apply the standard pro-cedure linking the number of particles N in a system with chemicalpotential µ or fugacity z = e µ/T : N = (cid:88) j n ( ε j , z, T ) = ∞ (cid:90) dε g ( ε ) n ( ε, z, T ) , (1)where n ( ε, z, T ) are mean occupation numbers, given in the conven-tional Fermi-statistics by n ( ε, z, T ) = 1 z − e ε/T + 1 . (2)The density of states g ( ε ) is introduced for convenience to substitutethe summation over all levels ε j with integration over energies ε . Tocover a vast diversity of problems, the density of states can be writtenas g ( ε ) = N A ε s − , (3)where, for instance, s = D/ D -dimensionalspace, s = D for D -dimensional harmonic oscillators oscillators, etc.Note that for convergence of the integral in Eq. (1) we must requirethat s >
1. The factor A is a constant independent of energy and is efined by such parameters as concentration of particles, mass of par-ticles, harmonic oscillator frequencies, etc., depending of the specificsystem under consideration.With the chemical potential or fugacity as functions of tempera-ture, we can calculate the total energy E = (cid:88) j ε j n ( ε j , z, T ) = ∞ (cid:90) dε εg ( ε ) n ( ε, z, T ) (4)and the isochoric heat capacity C V = (cid:18) ∂E∂T (cid:19) V . (5)The abovementioned functions, µ ( T ), z ( T ), E ( T ), and C V ( T )are in the focus of the present paper. In the following sections, weapply two different procedures to introduce nonadditive generaliza-tions of the Fermi-distribution (2), provide numerical results for thewhole temperature range and then analyze low- and high-temperatureregimes analytically. We analyze two approaches to the generalization of the Fermi-distributionusing the nonadditive Tsallis q -exponential [27] e xq = e x for q = 1 , [1 + (1 − q ) x ] / (1 − q ) for q (cid:54) = 1 and 1 + (1 − q ) x > , / (1 − q ) for q (cid:54) = 1 and 1 + (1 − q ) x ≤ . (6)As the first modification, we consider the following substitution of theGibbs factor e ( ε − µ ) /T in the Fermi-distribution: n ( ε, µ, T ) = 1 e ( ε − µ ) /Tq + 1 . (7)The second modification is very similar, namely n ( ε, z, T ) = 1 z − e ε/Tq + 1 . (8) .00.20.40.60.81.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 e n ( e , m , T ) Fermi, T = 0 Fermi, T = 0.1Fermi, T = 0.5 q = 1.2, T = 0.1 q = 1.2, T = 0.5 q = 0.9, T = 0.1 q = 0.9, T = 0.5 e n ( e , z , T ) Fermi, T = 0 Fermi, T = 0.1Fermi, T = 0.5 q = 1.2, T = 0.3 q = 1.2, T = 0.5 q = 0.9, T = 0.1 q = 0.9, T = 0.5 Figure 1: Occupation numbers n and n at different values of q and tem-perature T . The chemical potential and fugacity are fixed by Eq. (1).4 ince the factorization rule breaks for the q -exponentials, e x + yq (cid:54) = e xq e yq , one cannot establish a simple connection between the chemi-cal potential and fugacity similar to z = e µ/T for the ordinary case.Moreover, even if we can introduce a deformed fugacity as z q = e µ/Tq ,its inverse would not be related to the chemical potential in a simplemanner either, z − q (cid:54) = e − µ/Tq . To be precise, the following relationshold for the Tsallis q -exponentials [28]:( e xq e yq ) − q = ( e x + yq ) − q + (1 − q ) xy, (cid:2) e xq (cid:3) − = e − x − q . (9)In Figure 1 the functions n and n are shown for different valuesof q and temperature T . Here and further, for numerical calculationswe set in the density of states (3) constant A = 1 and power s = 3 / n involving the chemical potential,it is quite straightforward to demonstrate that the limit of T → q <
1, the expression [1 +(1 − q )( ε − µ ) /T ] is negative for ε < µ yielding zero when raised to thepositive power 1 / (1 − q ), according to Eq. (6). And vice versa, for q > n involving fugacity, is abit more complicated and we will postpone it for subsequent sections. The temperature dependences of the chemical potential µ and fugacity z calculated numerically are shown in Figs. 2 and 3. They are bysolving numerically the equation N = ∞ (cid:90) g ( ε ) n , ( ε, · , T ) dε, (10)where the dot stands for the chemical potential µ in the case of n and for the fugacity z in the case of n . T m m : q = 0.8 m : q = 0.9 m : q = 1.2Fermi Figure 2: Chemical potential as a function of temperature T at differentvalues of q for the first model. Tz z : q = 0.8 z : q = 0.9 z : q = 1.2Fermi Figure 3: Fugacity as a function of temperature T at different values of q forthe second model. 6 e then calculate energy E = ∞ (cid:90) ε g ( ε ) n , ( ε, · , T ) dε (11)and the isochoric heat capacity (5). The results for the isochoric spe-cific heat C V /N are shown in Fig. 4. TC V __N m : q = 0.9 m : q = 1.2Fermi z : q = 0.9 z : q = 1.2 Figure 4: Specific heat as a function of temperature T at different values of q .Models 1 and 2 demonstrate different behavior in low and high-temperaturelimits. The results of numerical calculations give us some hints facilitatinganalytical derivations presented in the next sections. In particular, thebehavior of µ and z for different values of the statistics parameter q in Figs. 2 and 3 suggests that there is a smooth transition betweenthe domains of q < q > q = 1.The most interesting, however, is the graph for the specific heat.One would expect that in the high-temperature limit the value of C V tends to a constant (equal to sN in the case of an ordinary idealgas). But the first model, where the modified Gibbs factor containsthe chemical potential, the heat capacity seems to increase infinitelyif q < q >
1, as shown for the nonadditive deal Bose-gas [26]. For the second model, with z , the classical limitis a constant depending on q .The low-temperature domain has its own peculiarities. Namely,the temperature behavior of first model nearly coincides with that ofthe ordinary Fermi-gas, while for the second model two unexpectedresults are observed. For q <
1, the value of the heat capacity doesnot approach zero at T →
0. For q >
1, the heat capacity becomeszero at a finite (non-zero) temperature. We confirm these observationsanalytically in subsequent sections.
In [26] it was shown how to obtain the high-temperature behavior ofthe ideal Bose-gas satisfying the nonadditive modification of the Gibbsfactor involving the chemical potential, with e ( ε − µ ) /T substituted by e ( ε − µ ) /Tq . The same approach can be used for the respective Fermi-gasmodel.From Fig. 2 we can see that, naturally, the ordinary Fermi-gasseparates cases q < q >
1. For fermions, the chemical potentialtends to the classical limit as T → ∞ , µT = − | µ | T → −∞ . (12)So, N = N A ∞ (cid:90) ε s − dεe ( ε − µ ) /T + 1 (cid:39) N AT s e µ/T ∞ (cid:90) dx x s − e − x . (13)We have thus the high-temperature limit µ (cid:39) − sT ln T. (14)Now we consider the deformed case for q <
1. The number ofparticles equals N = N A ∞ (cid:90) ε s − dεe ( ε − µ ) /Tq + 1 (cid:39) N AT s ∞ (cid:90) dx x s − (cid:104) e x + | µ | /Tq (cid:105) − (15) ince the unity in the denominator is small comparing to the q -exponential.In the same fashion, the energy yields E = N A ∞ (cid:90) ε s dεe ( ε − µ ) /Tq + 1 (cid:39) N AT s +1 ∞ (cid:90) dx x s (cid:104) e x + | µ | /Tq (cid:105) − . (16)After simple calculations [26] we arrive at the following high-temperaturebehavior for the chemical potential µ = − (cid:20) A (1 − q ) q − B (cid:18) s, − q − s (cid:19)(cid:21) q − s ( q − T s ( q − , (17)where B( x, y ) is Euler’s beta-function, energy EN = B (cid:16) s + 1 , − q − s − (cid:17) B (cid:16) s, − q − s (cid:17) | µ | , (18)and, respectively, the specific heat: C V N ∝ T (1 − q ) s s ( q − = T γ . (19)The case of q > q -exponential (6) implies a finite upper limit of integration, N = N AT s q − − | µ | T (cid:90) x s − dx (cid:104) − q ) (cid:16) x − | µ | T (cid:17)(cid:105) / (1 − q ) + 1 (20)and the condition | µ | T < q − | µ | T = 1 q − − δ, (21)where δ is a small number tending to zero at large temperatures, wecan apply the procedure similar to the one described above for the q < µ = 11 − q + (cid:20) A ( q − q − B (cid:18) s, q − (cid:19)(cid:21) q − s ( q − T s ( q − (22)yielding the same temperature dependence for the specific heat as in(19). Obviously, the ordinary exponential corresponding to q = 1ensures C V = const as T → ∞ .The respective temperature behavior at high temperatures is shownin Figs. 5 and 6. T m __T q = 0.9 q = 0.8 q = 1.2Fermi Figure 5: Chemical potential as a function of temperature T at differentvalues of q at high temperatures. Dotted lines show the asymptotic behavior. TC V __N q = 0.9 q = 0.8 q = 1.2Fermi Figure 6: Specific heat as a function of temperature T at different values of q at high temperatures. Dotted lines show the asymptotic behavior.10 High temperatures and classical limitfor the second model
For the model involving fugacity, the same analysis can be made asfor the nonadditive Bose gas [26]. Since z → N = N A ∞ (cid:90) ε s − dεz − e ε/Tq + 1 (cid:39) zN AT s ∞ (cid:90) dx x s − (cid:2) e xq (cid:3) − (23)and E = N A ∞ (cid:90) ε s dεz − e ε/Tq + 1 (cid:39) zN AT s +1 ∞ (cid:90) dx x s (cid:2) e xq (cid:3) − . (24)The leading term in the fugacity behavior is thus z = (cid:20) A − q ) s B (cid:18) s, − q − s (cid:19)(cid:21) − T − s (25)for q < z = (cid:20) A q − s B (cid:18) s, q − (cid:19)(cid:21) − T − s (26)for q > E = sq ( s + 1) − s N T. (27)and C V = (cid:18) ∂E∂T (cid:19) V = sq ( s + 1) − s N. (28)for both q < q >
1. As expected, the case of q = 1 correspondsto E = sN T .Adding more terms in the expansions for the number of particles(23) and energy (24), we can obtain corrections to the classical limitsfor fugacity, energy, and specific heat.For q <
1, from (23) we have N (cid:39) zN AT s − q ) s (cid:88) k =0 ( − k z k B (cid:18) s, k − q − s (cid:19) (29) nd E (cid:39) zN AT s +1 − q ) s +1 (cid:88) k =0 ( − k z k B (cid:18) s + 1 , k − q − s − (cid:19) . (30)The expression for the fugacity up to two terms is z = z + ∆ z = z + z B (cid:16) s, − q − s (cid:17) B (cid:16) s, − q − s (cid:17) , (31)where the correction ∆ z ∝ T − s , yielding energy EN = sq ( s + 1) − s T + AT s +1 (1 − q ) s +1 ∆ z B (cid:18) s + 1 , q − q − s (cid:19) = sq ( s + 1) − s T + const T − s , where const > . (32)The high-temperature correction to the classical specific heat limit(28) is thus negative (since s >
1) and proportional to T − s . Thecase of q > z ] with the same temperaturedependence obtained. T → for the second model The low-temperature limits of the first model, with the definition in-volving the chemical potential, are very similar to those of the ordinaryFermi-gas. In particular, it is straightforward to show that at T = 0the chemical potential achieves the value µ ≡ µ (cid:12)(cid:12)(cid:12) T =0 = (cid:16) sA (cid:17) /s (33)independent of q .The second model involving fugacity, on the other hand, revealsmuch more interesting properties at T →
0. They are studied in detailin the reminder of this section.The behavior of the fugacity z in the low-temperature limit is ob-tained as follows. The integral in the expression defining the fugacity AT s ∞ (cid:90) x s − dxz − e xq + 1 = 1 (34) an be split into two parts, ∞ (cid:90) x s − dxz − e xq + 1 = x (cid:90) x s − dxz − e xq + 1 + ∞ (cid:90) x x s − dxz − e xq + 1 , (35)where e x q = z or equivalently, x = ln q z with the q -logarithm definedas ln q x ≡ x − q − − q , yielding in particular ln x = ln x. (36)In view of different convergence radii, the two integrals are expandedinto the following series: x (cid:90) dx x s − (cid:104) z − e xq (cid:105) − + ∞ (cid:90) x dx x s − z ( e xq ) − (cid:104) z ( e xq ) − (cid:105) − = x (cid:90) dx x s − ∞ (cid:88) k =0 ( − k z k (cid:0) e xq (cid:1) k + ∞ (cid:90) x dx x s − ∞ (cid:88) k =1 ( − k +1 z k (cid:0) e xq (cid:1) − k . (37)In the limit of T →
0, the fugacity tends to infinity, and so does x .We thus are interested in the leading contributions from the respectiveintegrals, x (cid:90) dx x s − (cid:0) e xq (cid:1) k = (1 − q ) s + k − q k + (1 − q ) s x s + k − q + . . . , (38) ∞ (cid:90) x dx x s − (cid:0) e xq (cid:1) − k = (1 − q ) s + kq − k + ( q − s x s + kq − + . . . . (39)Taking into account that for large z the q -logarithm reduces to thepower function, x = ln q z (cid:39) z − q − q , (40)we finally obtain from (34): A ( T ln q z ) s (1 − q ) (cid:34) ∞ (cid:88) k =0 ( − k k + (1 − q ) s − ∞ (cid:88) k =1 ( − k k + ( q − s (cid:35) = 1 . (41) his expression can be rewritten using the so called Lerch tran-scendent Φ( a ; b ; c ) being a generalization of Riemann’s and Hurtwitzzeta-functions: Φ( a ; b ; c ) = ∞ (cid:88) k =0 a k ( b + c ) k , (42)namelylim T → T ln q z = (cid:110) A (1 − q ) (cid:104) Φ (cid:0) −
1; 1; (1 − q ) s (cid:1) + Φ (cid:0) − , , q − s (cid:1)(cid:105)(cid:111) − /s . (43)Note that in the limit of q → − (cid:0) −
1; 1; (1 − q ) s (cid:1) + Φ (cid:0) − , , q − s (cid:1) = 1(1 − q ) s + . . . , (44)where only the leading term is written, so at T = 0 µ = T ln z = ( s/A ) /s , (45)which coincides with expression (33) for the Fermi energy.Denoting φ ( q, s ) = (1 − q ) (cid:104) Φ (cid:0) −
1; 1; (1 − q ) s (cid:1) + Φ (cid:0) − , , q − s (cid:1)(cid:105) , (46)so that lim T → T ln q z = [ Aφ ( q, s )] − /s , (47)we can also write immediately the low-temperature limit for energy E , where the integral contains x s instead of x s − for N . Thus, E ≡ lim T → E = N A ( T ln q z ) s +1 φ ( q, s + 1) = N φ ( q, s + 1)[ φ ( q, s )] /s +1 . (48)After some lengthy derivations one can show that the subsequent termis linear in temperature, E (cid:12)(cid:12)(cid:12) T → = E + T α ( q, s ) + O ( T ) , (49)where α ( q, s ) is a complex expression satisfying α (1 , s ) = 0. Thedeviation of α ( q, s ) from zero is linked to the violation of the relation − x = [ e x ] − for the Tsallis q -exponentials, e − xq (cid:54) = [ e xq ] − , cf. theanalysis of the degenerate Fermi gas in [29, Chap. V].As a consequence of (49) we have an obvious result, C V (cid:12)(cid:12)(cid:12) T → = α ( q, s ) (cid:54) = 0 for q < . (50)Certainly, this violates the third law of thermodynamics. However,non-zero heat capacities at absolute zero are known for non-equilibriumsystems as well as in glasses [30] or some magnetic systems [31].In the limit of T →
0, we have for fugacity from (47) z = e T [ Aφ ( q,s )] − /s q , (51)so the occupation numbers after simple transformations become n ( ε, z, T ) = 1 (cid:16) ε [ Aφ ( q,s )] − /s (cid:17) / (1 − q ) + 1 (52)demonstrating thus a smooth step, unlike the ordinary Fermi-gas,which is obtained from the above expression in the limit of q → q >
1, fugacity becomes infinite as some temperature T yield-ing zero heat capacity as T . To obtain T , consider the relation N = N AT s ∞ (cid:90) x s − dxz − e xq + 1 = N AT s x (cid:90) x s − dxz − [1 + (1 − q ) x ] / (1 − q ) + 1 . (53)The finite upper limit of integration is obtained from the condition1 + (1 − q ) x = 0, see definition (6). So, for z → ∞ , T = ( q − (cid:16) sA (cid:17) /s . (54)Thus, in the case of q > T in the model. It becomes zero in the limit of q → T ln q z occurs naturally, which corresponds to µ = T ln z . We thuscan define the quantity µ q = T ln q z (55) s an equivalent of the chemical potential. On the other hand, from z − = e µ/T , substituting the ordinary exponential with the Tsallisone, we arrive at ˜ µ q = − T ln q z − . (56)In Figure 7, different definitions of the chemical potentials in the sec-ond model are shown in comparison with the chemical potential in thefirst model. As one can see, the first suggested definitions (55) hasthe behavior most similar to ordinary chemical potential. T m q = 0.9 q = 1.2Fermi q = 0.9, m = T ln zq = 0.9, m q = T ln q zq = 0.9, m ~ q = − T ln q z −1 q = 1.2, m q = T ln q z Figure 7: Various definitions of the chemical potential in the second modelas a function of temperature T compared to the chemical potential in thefirst model and in the ordinary Fermi-gas. We have analyzed two phenomenological approaches to the gener-alization of the Fermi-distribution using the nonadditive Tsallis q -exponential. The first model obtained by substitution of the Gibbsfactor e ( ε − µ ) /T → e ( ε − µ ) /Tq demonstrates the behavior at high temper-atures, which significantly deviates from the classical limit. Namely,the heat capacity is not constant but instead grows infinitely for q < q >
1. The low-temperature limits of this model, n the other hand, closely resemble those of the ordinary ideal Fermi-gas.The second model was obtained by substituting e ( ε − µ ) /T = z − e ε/T → z − e ε/Tq . Note that properties of the Tsallis q -exponentialdo not allow factorization e ( ε − µ ) /Tq (cid:54) = e − µ/Tq e ε/Tq . The differences withthe first model appear fundamental. In particular, at high tempera-tures the specific heat tends to a constant value. The most interestingresults, however, are obtained in the low-temperature domain.For q < T = 0. This property might be usefulin modeling magnetic systems [31] or non-ergodic systems like glasses[30]. Moreover, extrapolation of the C V curve to achieve zero valuewould mean negative absolute temperatures, which appeared recentlyin some cosmological models [32, 33] but are not limited to those[34, 35].A finite (non-zero) minimal temperature obtained for q > κ -exponential [39, 40], or some other types of statisticsdeformation [41, 42]. Acknowledgment
We are grateful to Dr. Volodymyr Pastukhov and Yuri Krynytskyi fordiscussions.This work was partly supported by Project FF-83F (No. 0119U002203)from the Ministry of Education and Science of Ukraine.
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