Micro-Canonical Ensemble Model of Particles Obeying Bose-Einstein and Fermi-Dirac Statistics

Indian Journal of Pure & Applied Physics Vol. 42, October 2004, pp. 749-757

Micro-canonical ensemble model of particles obeying Bose-Einstein and Fermi-Dirac statistics

Y K Ayodo1, K M Khanna2 & T W Sakwa1 1Department of Physical Sciences, Western University College, Box 190, Kakamega, Kenya 2Department of Physics, Moi University, Box 1125 Eldoret, Kenya Received 13 February 2004; accepted 12 April 2004

A micro-canonical ensemble for an assembly of bosons and fermions is considered in which the number of particles, internal energy and volume are kept constant. A statistical distribution model, which is fermion dominated and where bosons and fermions interact in pairs, is developed. The partition function is derived. Macroscopic thermodynamic quantities such as entropy, internal energy and specific heat are obtained in terms of the partition function. The model equations are applied to a mixture of liquid helium-3 and liquid helium-4 atoms.

[Keywords: Bose-Einstein Statistics, Fermi-Dirac statistics, Partition function, 3He- 4He mixture] IPC Code: C01B 23/00

1 Introduction was made by Gentile1. He proposed statistics in which A micro-canonical ensemble represents a collection up to N particles (N>>1) were allowed to occupy a of configurations of isolated systems that have single quantum state instead of just one particle for reached thermal equilibrium. A system is isolated Fermi case due to the Pauli exclusion principle, and from its environment if it does not exchange either infinitely many for the Bose case. However, Gentile’s particles or energy with its surroundings. The volume, approach was found to be too much of a internal energy and the number of particles of such a generalisation and contained the violation of the system are constant and are the same for all conventionally accepted Pauli principle. Furthermore, configurations that are part of the same micro- his model did not distinguish which particles were canonical ensemble. In this paper, a configuration of a fermions and which ones were bosons. However, mixture of bosons and fermions is studied and a Gentile’s work laid the emphasis and the foundation partition function is developed for the same. that the statistical mechanics of a mixture of bosons Thermodynamic quantities, such as internal energy, and fermions can be worked out. specific heat and entropy can be calculated from the The next attempt was that of Medvedev2. In his knowledge of the statistical distribution and the paper entitled ‘properties of particles obeying partition function. So far most of the studies deal ambiguous statistics’, Medvedev proposed a new either with a system of bosons or with a system of class of identical particles, which may exhibit both fermions. In nature, there do exist systems, which are Bose and Fermi statistics with respective probabilities 1 2 mixtures of bosons and fermions such as 1 H, 1 H and P′b and P′f. The model admits only primary Bose- Einstein and Fermi-Dirac statistics as existing. He 3 H, and the most interesting mixture is 4 He and 1 2 assumed that a particle is neither a pure boson nor a 3 2 He. It should be clearly understood that in the pure fermion. He let another particle, which interacts mixture, bosons obey Bose-Einstein statistics and with the first one, play the role of an external fermions obey Fermi-Dirac statistics. What observer. During the interaction it performs a distribution law or what will be the expression for the measurement at the first particle and identifies it as most probable distribution-in-energy in the mixture is either a boson or a fermion with respective the subject matter of study in this paper. probabilities P′b and P′f. According to the result of The first attempt to generalise quantum Bose and this measurement, it interacts with the first particle as Fermi statistics for a mixture of Bosons and Fermions if the last is a fermion or a boson, respectively. The 750 INDIAN J PURE & APPL PHYS, VOL. 42, OCTOBER 2004

first particle is the observer for the second particle and (i) Statistical count for an ensemble that is a mixture so the process is symmetric. Note that (P′b + P′f) is of the bosons and fermions assuming a pair not necessarily equal to one, and, if not, it means that interaction between the bosons and fermions. the second particle (observer) does not detect the first (ii) The most probable distribution in energy for a particle. The probability of this is (1−P′b−P′f). The mixture of the bosons and the fermions in the statistical uncertainty introduced here may be either ensemble. the intrinsic property of a particle itself or the (iii) The partition function for such an ensemble. experimental uncertainty of the measurement process. (iv) Using the partition function, the calculations were Another attempt in this direction is the so-called done for internal energy, specific heat and ‘statistical independence model’ of Landau and entropy. Lifshitz 3 in which two weakly interacting subsystems (bosons and fermions) are together regarded as one The above expressions are used to study the composite system, and the subsystems are assumed to thermodynamic properties of a mixture of liquid be quasi-closed. The statistical distribution or count helium-3 (fermions) and liquid helium-4 (bosons) for such a mixture is the product of the individual with different concentrations. For our model probabilities for two subsystems, one corresponding calculations a fermion concentration of 0.70 was used. to bosons and the other corresponding to fermions. The theoretical results obtained are compared with the With these assumptions, the statistical independence experimental observations on the properties of a model will hold only for an ideal gas assembly of mixture of liquid helium-3 and liquid helium-4. bosons and fermions. In reality, such an assembly does not exist and hence the statistical independence 2 Theory model cannot be used for real mixtures of bosons and Consider a micro-canonical assembly of N particles fermions like 3He and 4He mixtures. in which there are Nb bosons and Nf fermions such Chan et al.4 studied the ‘effect of disorder on that superfluid 3He-4He mixtures, and the ‘thermodyna- 3 4 mics of He- He mixtures in aerogel’. However, the N= Nb+ Nf … (1) studies related to more of an ideal system rather than a real system. Let ε1, ε2, ε3… εj… be the energy states of the In our earlier paper5 entitled “statistical mechanics assembly, and in the statistical equilibrium the and thermodynamics for a mixture of bosons and number of particles assigned to these energy levels be fermions”, the statistical distribution model for a n1, n2, n3, nj, respectively, such that the numbers nj mixture that was dominated by bosons was must satisfy the conditions requiring the conservation developed. The partition function that was derived of particles, N, and conservation of energy, E, i.e., worked well for a liquid helium-3 and liquid helium-4 mixture. In this paper, an assembly that is fermion ∞ ∑ nNj = … (2) dominated is studied. j=1 Therefore, the properties of a mixture of bosons and fermions assuming there exists a pair interaction and between the bosons and fermions; and that the concentrations of the bosons and fermions are ∞ different from each other are studied. Since the ∑ nEjjε = … (3) j=1 concentrations of the bosons and fermions will not be the same, and considering only pair interaction, in a such that, given state of equilibrium some fermions will be left unpaired. The value of the occupation number of nn= + n … (4) fermions in a given state will not exceed, rather, will j jb jf be much less, than the degeneracy of that state so that Pauli exclusion principle is not violated. With these where, basic assumptions, the expressions for the following were derived: njb = number of bosons in the energy level j … (5) AYODO et al.: MICRO-CANONICAL ENSEMBLE FOR BOSONS & FERMIONS 751

and njb ωω! ( jj) ( ) Pjbf = … (9) njf =number of fermions in the energy level j … (6) ()(nnnnjb!!!ω j−− jf )( jf jb )

Let ω j be the number of states in the j-th level, i.e, The statistical count, Cbf, for such a distribution

ω j is the degeneracy of the j-th level. Then the among all the levels (j = 1,2,3…) available to the assembly is the product of such expressions as given number of ways Pjb in which njb bosons can be assigned to ω states in the j-th level is given by, in Eq. (9) since every arrangement in a given energy j level can be considered independently of the other energy levels. Thus we can write, n P = ω jb … (7) jb( j ) ∞ CPbf=∏ jbf j=1 Similarly, the number of ways, Pjf, in which njf njb ∞ ⎧ ⎫ … (10) fermions can be assigned to the ω j states is given by, ⎪ ()()ωωjj! ⎪ = ∏⎨ ⎬ j=1 ⎪()(nnnnjb!!!ω j−− jf )( jf jb )⎪ ω j ! ⎩⎭ P = … (8) jf ω − n ! ()jjf Equation (10) will now be used to calculate the most probable distribution in energy for the ensemble. To satisfy Pauli exclusion principle, it is necessary to assume that ω j >> njf. 3 Most probable distribution in energy for the Once the particles are placed in the ω sub-levels ensemble j The objective is to calculate for what values of n in the j-th level, we shall further assume that bosons jb and njf, the statistical count Cbf, is maximum under the and fermions may interact in pairs. However, not all conditions of N and E being fixed. The distribution the fermions may form pairs with bosons since the numbers and the corresponding energies must satisfy bosons and the fermions will not be in equal the following relations: proportions in the mixture. This statement implies that njf > njb. ∞ It is the number of bosons njb that will determine ∑ nNjbb= … (11) the number of boson-fermion pairs. Hence, the j=1 number of boson-fermion pairs will be njb and the ∞ number of unpaired fermions will be (njf −njb). Since ∑ nEjbjε = b … (12) the permutations among the particles and the j=1 permutations among the pairs in the same energy level do not give a new complexion, in the statistical ∞ ∑ nNjff= … (13) distribution model proposed here, the following j=1 permutations must be excluded from the number of ways in which njb bosons, njf fermions and njb boson- ∞ fermion pairs are distributed in the j-th level: ∑ nEjfjε = f … (14) j=1

(i) Permutations among identical pairs should be where Nb and Nf are the total number of bosons and excluded by dividing by (njb)! fermions in the ensemble such that the total number of (ii) Permutations among identical unpaired fermions particles N is given by, should be excluded by dividing by (njf −njb)! N= N + N … (15) Hence the total number of ways, P in which n b f jbf , jb bosons, n fermions and n pairs of bosons and jf jb Similarly, E and E are the total internal energies of fermions can be distributed among the ω sub-levels b f j bosons and fermions such that the total energy of the in the j-th level is given by, ensemble E is given by, 752 INDIAN J PURE & APPL PHYS, VOL. 42, OCTOBER 2004

E= Eb + Ef … (16) then the corresponding coefficients will be zero. Thus we can write, Now to find the values of njb and njf for which Cbf is maximum, the procedure is to allow the variation of ∂ ln(C )−+ (α βε ) =0 … (21) C with respect to n and n , and put the result equal bf b j bf jb jf ∂n jb to zero.

∂ For Cbf to be maximum, ln(Cbf )−+ (α fβε j ) =0 … (22) ∂n jf ∞ ⎪⎪⎧⎫∂ ∂=ln(CCdnbf )∑⎨⎬ ln( bf ) jb Eqs [(21) and (22)] are true for all values of j. j=1 ⎪⎪∂njb ⎩⎭ … (17) Substituting for Cbf from Eq. (10) in Eq. (21) gives, ∞ ⎪⎪⎧⎫∂ +=⎨⎬ln(Cdnbf ) jf 0 ∑ ω jjfjb()nn− j=1 ⎪⎪∂njf ⎩⎭ =+exp(αbjβε ) … (23) n jb

The variations dnjb and dnjf are not independent since Similarly, substituting for C from Eq. (10) in the njb’s and njf’s must continue to satisfy the bf restrictions given in Eqs (11-14). Since N and E are Eq. (22) gives,

fixed, the variations in njb and njf must satisfy the following equations: ()ω jjf− n =+exp(α βε ) … (24) f j ()nnjf− jb ∞∞

∑∑dnjb+= dn jf 0 … (18) jj==11 Eqs [(23) and (24)] are solved for njb and njf to get:

2 and njb=−−−ωααβε jexp( b f 2 j ) ⎡ ωααβε2 exp(−− − 2 ) ⎤ ∞∞ ×−1 jbfj εεdn+= dn 0 … (19) ⎢ ⎥ ∑∑jjb jjf 1exp(+−−+αbjjβε ) ω exp( −−− α bf α 2 βε j ) jj==11 ⎣⎢ ⎦⎥ … (25) Hence, along with Eq. (17), Eqs [(18) and (19)] must ωααβε2 exp(−− − 2 ) n = jbfj also be satisfied. These equations can be combined by jf 1+−−+ exp(α βε ) ω exp( −−− α α 2 βε ) the method of Lagrange’s undetermined multipliers bjj bf j … (26) which are denoted by α and β. Thus, multiplying the first and the second terms in Eq. (18) by (−αb) and μ μ 1 where, α =− b , α =− f and β = (−αf), respectively, and Eq. (19) by (−β) and adding to b kT f kT kT Eq. (17), we get, … (27)

Equation (25) gives the most probable distribution in ∞ ⎪⎪⎧⎫∂ energy for bosons in the ensemble, and Eq. (26) gives ∑⎨⎬ln(Cdnbf )−+ (αβε b j ) jb j=1 ⎪⎪∂njb the most probable distribution in energy for fermions ⎩⎭ … (20) ∞ ⎪⎪⎧⎫∂ in the ensemble. In Eq. (27) μb is the chemical +−+=ln(Cdn ) (αβε ) 0 ∑⎨⎬bf f j jf potential for bosons and μf is the chemical potential j=1 ⎩⎭⎪⎪∂njf for fermions. We should note that njf is contained in the expression for njb. Now Eq. (20) demands that all the terms should be separately equal to zero, and the terms for which dnjb 4 Partition function for the ensemble and dnjf are not equal to zero, then, the coefficients of The general expression for the partition function Q dnjb and dnjf should be, respectively, equal to zero. If for an ensemble of bosons and fermions can be we assume that one of the dnjb’s and dnjf’s is non-zero, written as: AYODO et al.: MICRO-CANONICAL ENSEMBLE FOR BOSONS & FERMIONS 753

⎡⎤−−μ f μb 5 Results and Discussion QN= exp ⎢⎥ The general expressions for entropy S, the internal ⎣⎦kT 3,11 energy E and the specific heat Cv are given as ∞ ⎡⎤⎡⎤−−μ fbμ = nn+ exp … (28) ⎢⎥∑ ()jb jf ⎢⎥ 2 ⎛⎞∂lnQ ⎣⎦j=1 ⎣⎦kT ENkT= ⎜⎟ … (30) ⎝⎠∂T

Substituting for njb from Eq. (25) and njf from Eq. (26) ⎡ ⎛⎞Q ∂ ⎤ SkN=+⎢ln⎜⎟ T ln Q⎥ … (31) in Eq. (28), we get ⎣ ⎝⎠NT∂ ⎦

⎛⎞∂∂ln(QQ )⎛⎞2 ln( ) ∞ Cv 2 … (32) ⎛⎞−2ε =+2NkT⎜⎟ NkT ⎜⎟2 2 j ⎝⎠∂∂TT Q=∑ω j exp⎜⎟ V ⎝⎠V j=1 ⎝⎠kT Together with these equations, let us introduce the ⎡⎤⎛⎞μμbf+−2 ε j ⎢⎥1exp+ω j ⎜⎟ fermion concentration η given by kT ×+⎢⎥⎝⎠1 ⎢⎥με−+− μμ2 ε ⎛⎞bj ⎛ b f j ⎞ η=Nf /N … (33) ⎢⎥1exp++⎜⎟ω j exp ⎜ ⎟ ⎣⎦⎢⎥⎝⎠kT ⎝ kT ⎠ … (29) or

Equation (29) is the expression for the partition N=Nf /η … (34) function for a mixture of bosons and fermions in the ensemble, in which pair interaction between the two To perform the calculations, we substituted for N and types of particles is considered. It must be understood Q in the above thermodynamic relations from that the partition function Q has two significant terms: Eqs [(15), (26) and (29)]. The molar thermodynamic One term is quantities are of interest. The experiments done by Chan et al.4,7 mainly focused on the molar quantities of liquid helium-3 and liquid helium-4. Wilks and 8 ⎛⎞μbf+−με2 j Bett give the molar volume of liquid helium-3 as exp ⎜⎟ 3 −3 ⎝⎠kT 40.0 cm , and its molar density as 0.07 g cm ; and the 3 molar volume of liquid helium-4 as 28.0 cm and its molar density as 0.14 g cm−3. This, therefore, means This quantity contains 2εj which is a consequence that the molar mass of liquid helium-3 is 2.80 g and of the pair interaction between the bosons and that of liquid helium-4 is 3.92 g. Although the fermions. It can also be interpreted as the energy of a chemical potential should have temperature boson –fermion pair in the j-th energy level. The dependence, at low temperatures it assumes a nearly second term is, constant value given by the expression 8,

2 ⎛⎞μbj−ε 22 exp ⎜⎟ π h ⎛⎞3N 3 kT μ = ⎜⎟ … (35) ⎝⎠ 2mV⎝⎠π

The absence of μf, chemical potential for fermions, in where, m is the molar mass, V is the molar volume this quantity, is an indication that the distribution of and N is the number of particles in one mole, and this bosons is not affected by how fermions are is Avogadro’s number = 6.025 × 1023 particles mol−1. distributed. Substitution of the empirical data into Eq. (35) gives The number of bosons, the number of boson- the chemical potential μf, for fermions as, fermion pairs that shall be formed, and the number of fermions shall determine how many free or −27 μ f =×3.184 10 eV … (36) unattached fermions shall remain to be distributed among the available energy states. and for bosons as, 754 INDIAN J PURE & APPL PHYS, VOL. 42, OCTOBER 2004

μ =×6.215 10−28 eV … (37) Table 1⎯Essential parameters for liquid helium-3 and b helium-4

Since the transition temperature 8 of liquid helium-3 is Parameter Liquid helium-3 Liquid helium-4

3 very much lower than the transition temperature of Volume ( cm ) 40.00 28.00 liquid helium-4, it is interesting to know what Density (gcm−3) 0.07 0.14 happens in the vicinity of the transition temperature of Mass (g) 2.80 3.92 Chemical potential(ev) 3.184×10−27 6.215×10−28 liquid helium-4. Furthermore, the temperature ranges 4,5,7,8 used in experiments on such mixtures are so Table 2⎯Values of the partition function against temperature much higher than the transition temperature of liquid 22 helium-3, which is lower than 2.5 mK. However, T (K) Q (T)×10 particles given that both liquid helium-3 and liquid helium –4 2.01 4.137 are at the same temperature, they are considered to be 2.02 5.552 in thermal equilibrium. 2.03 7.430 2.04 9.914 2.05 13.190 5.1 Calculation of partition function Q 2.06 17.500 Using the essential parameters for liquid helium-3 2.07 23.160 and liquid helium-4 given in Table 1, we calculated 2.08 30.560 the partition function Q, using Eq. (29) for the 2.09 52.810 mixture in the temperature range 2.01 K to 2.30 K in 2.10 69.140 2.11 90.290 steps of 0.01 K. Table 2 gives the values of Q for the 2.12 117.600 mixture at different temperatures. Figure 1 depicts the 2.13 152.800 variation of Q with temperature T. Figure 1 shows 2.14 198.100 that there is an exponential rise in the value of the 2.15 256.200 partition function Q. The reason for this kind of 2.16 306.500 1 2.17 355.900 behaviour is that at very low temperatures there are 2.18 425.400 8 very few energy states that can be occupied by the 2.19 546.200 particles in the assembly. However, at higher 2.20 699.800 temperatures, the number of energy levels available 2.21 894.500 2.22 1141.000 for particle occupation could be large, and these could 2.23 1452.000 be called the excited states of the assembly of 2.24 1844.000 particles. 2.25 2336.000 2.26 2954.000 2.27 3728.000 5.2 Calculation of internal energy 2.28 4695.000 Equation (30) is used to calculate how the internal 2.29 5900.000 energy E varies with temperature using the parameters 2.3 7401.000 listed in Table 1. Table 3 gives the variation in the values of E with temperature in the range 2.01 K to approximately 2.150 kJ as the temperature approaches 2.30 K in steps of 0.01K. To convert the values of E 2.30 K. from electronvolt (eV) to Joules (J), E values are −19 multiplied by a factor of 1.6×10 . The graph of 5.3 Calculation of entropy S internal energy variation against temperature is given Equation (31) is used to calculate the variation of S in Fig. 2. with T. The values are given in Table 4 and the graph Figure 2 shows that there is a rise in the total showing the variation of S with T is plotted in Fig. 3. internal energy with temperature. This is not unusual The graph in Fig. 3 has the same shape as the one for since for any given thermodynamic system, the higher the variation of E with T. The values of entropy the temperature, the higher should be the internal increases with temperature just like the internal energy. The increase in internal energy tends to be energy. By definition, entropy is a measure of the exponential in the temperature range 2.01 K to 2.15 K molecular disorder of any given system. Naturally, but tends to assume a nearly constant value of there should be greater molecular disorder at higher AYODO et al.: MICRO-CANONICAL ENSEMBLE FOR BOSONS & FERMIONS 755

temperatures as opposed to lower temperatures. At particles in a system. In fact it is meaningless to talk higher temperatures molecules have higher vibrational about the entropy of a single particle. Therefore, the energies, creating more disorder and hence more fewer the particles the lower is the value of entropy entropy. These trends are clearly depicted in Fig. 3. and vice versa. The shape of the curve in Fig. 3 is in Entropy is also an extensive thermodynamic quantity. good agreement with the result obtained in Fig. 1, That means it is proportional to the number of where at lower temperatures there are fewer energy

Fig. 1⎯ Variation of partition function with temperature Fig. 2⎯ Variation of internal energy with temperature

Table 4⎯Values of entropy S at different temperatures Table 3⎯Values of internal energy E at different temperatures T (K) Entropy, S (J/K) T (K) Internal Energy, E (J) 2.01 29.854 2.01 59.638 2.02 39.400 2.02 79.084 2.03 51.708 2.03 104.278 2.04 67.432 2.04 136.618 2.05 87.306 2.05 177.682 2.06 112.105 2.06 229.156 2.07 142.585 2.07 292.700 2.08 179.385 2.08 369.742 2.09 222.900 2.09 461.209 2.10 273.140 2.10 567.216 2.11 329.606 2.11 686.782 2.12 391.219 2.12 817.675 2.13 456.354 2.13 956.460 2.14 522.985 2.14 1099.000 2.15 588.927 2.15 1240.000 2.16 652.121 2.16 1376.000 2.17 710.867 2.17 1502.000 2.18 763.984 2.18 1616.000 2.19 810.844 2.19 1717.000 2.20 851.314 2.20 1805.000 2.21 885.646 2.21 1879.000 2.22 914.345 2.22 1941.000 2.23 938.053 2.23 1992.000 2.24 957.457 2.24 2034.000 2.25 973.223 2.25 2068.000 2.26 985.965 2.26 2096.000 2.27 996.222 2.27 2136.000 2.28 1004.000 2.28 2150.000 2.29 1011.000 2.29 2161.000 2.30 1016.000

756 INDIAN J PURE & APPL PHYS, VOL. 42, OCTOBER 2004

Table 5⎯Values of specific heat (×10−15) at different temperatures

T (K) Specific heat, C (J/mol. K)

2.01 −5.383 2.02 −94.560 2.03 −258.300 2.04 −265.000 2.05 −191.200 2.06 −371.700 2.07 −92.040 2.08 −85.240 2.09 −59.300 2.10 21.010 2.11 1496.000 2.12 232.800 Fig. 3⎯ Variation of entropy with temperature 2.13 338.300 2.14 593.500 states for the distribution of particles. This means 2.15 8798.000 2.16 2537.000 entropy is expected to be low at lower temperatures. 2.17 7023.000 2.18 2460.000 2.19 2362.000 5.4 Calculation of specific heat C v 2.20 −3580.000 Equation (32) is used to calculate how the specific 3 2.21 −1742.000 heat at constant volume for a mixture of liquid He 2.22 −2021.000 4 and liquid He varies with temperature T. Since the 2.23 −2516.000 molar quantities of the constituents of the mixture are 2.24 −1648.000 the ones that were of interest, the values of the 2.25 −7491.000 specific heat were multiplied by a factor of 1.6×10−19 2.26 −3437.000 2.27 −3714.000 and divided by 297.619 to convert the units of −1 −1 −1 −1 2.28 −1276.000 specific heat from eV kg K to J mol K . This is 2.29 −5627.000 because from Table 1, 6.72g is the mass of 2 moles of 2.30 −1587.000 the mixture and thus 1 kg of the mixture is roughly 3 equal to 297.619 moles. Table 5 gives the values of Cv liquid He concentrations. However, our theoretical at different temperatures. model assumes a bulk mixture, meaning without The shape of the specific heat curve exhibits a lot aerogel, of the two liquids. Furthermore, our of fluctuations. Specific heat values are too low in the calculations do not include the flow properties of the temperature range from 2.01 K to 2.09 K. One two liquids, for instance, superfluidity in liquid 4He is particular feature that was found to be more supposed to disappear above a certain critical interesting is that the highest peak of the curve velocity. The normal-superfluid phase transition in occurred at 2.15 K. From here we see a significant pure liquid 4He is a second order phase transition, change of phase at 2.15 K. This happens to be very whereas the phase change in the mixture of liquid 3He near the λ-transition temperature8,9 for liquid 4He, into liquid 4He are characterised with a lot of which is 2.166 K. Below this temperature liquid 4He fluctuations with no discontinuity. This may be due to becomes a superfluid. the fact that in our case the atoms of the two liquids However, experimental observations by Chan have not been considered to be entirely independent et al.4,7,9 showed shifts in the transition temperature at but exchange energy through pair interaction. which peaks in the value of the specific heat occurred. This can be accounted for due to the fact that, 6 Conclusions experimentally, a highly porous material called Different authors1-5 studied the statistical aerogel was used to control the flow of liquid 3He into thermodynamics for a mixture of bosons and fermions liquid 4He and changes in the thermodynamic by putting forth different models. In these models quantities of the mixture were observed for different particles were considered independent or weakly AYODO et al.: MICRO-CANONICAL ENSEMBLE FOR BOSONS & FERMIONS 757

(iii) Specific heat Cv —The shape of the specific heat Cv and that of the internal energy E is the same in Ref. 5, whereas we find that the specific heat Cv has maxima and minima, and the maximum value of Cv is around 2.15 K.The shape of the Cv curve is different from the shape of the curve for the internal energy E. It should be acceptable that the specific heat Cv for a fermion dominated system will be different from a boson dominated system. It is the Pauli exclusion principle that restricts the flow of fermions from one level to another as the temperature changes, whereas no such restriction exists for a boson dominated system. Thus, a fermion-dominated system may

Fig. 4⎯ Variation of specific heat (×10−15) with temperature refuse to absorb heat resulting in a negative specific heat. interacting. In our model, the bosons and fermions are supposed to be interacting via a pair interaction, and The actual transition temperature of the mixture is the whole assembly is supposed to be in thermal at 2.15 K, below which the whole mixture goes into equilibrium. Furthermore, there are more fermions the superfluid state. The magnitudes of the than bosons in this ensemble. Comparing the thermodynamic quantities increases as the value of η calculations presented in Tables 1-5; and the graphs increases and this is evident from Eqs [(30) to (34)]. presented in Figs (1-4), with the corresponding results The phase transition is one that is not smooth but is in Ref. 5, in which bosons were more than fermions, characterised by fluctuations. the following marked differences in the shapes of the curves can be observed: References 1 Gentile G, Phys Rev Lett, 17 (1952) 493. (i) Partition function Q—The partition function in 2 Medvedev M V, Phys Rev Lett, 78 (1996) 4147. Ref. 5 becomes roughly constant after T≅2.2 K, 3 Landau & Lifshitz, Statistical physics, Vol. 1,Third edition (Pergamon Press, New York) (1981). whereas, in the present calculation, Q varies 4 Chan M H, Blum K I & Murphy S Q, Phy Rev Lett, 61 (1992) exponentially after T≅2.2 K. This means that at 1950. higher temperatures the occupation of excited 5 Khanna K M & Ayodo Y K, Indian J Pure & Appl Phys, 41 states increases and this is the basic character of (2003) 280. fermion-dominated systems. 6 Baierhein R, Thermal Physics, First edition (Cambridge (ii) Internal Energy E—The internal energy in Ref. 5 university Press), 1999. becomes maximum around T ≅ 2.14 K and then 7 Chan M H, Mulders N & Reppy J, Physics Today, 71 (1996) 31. decreases as the temperature increases. In the 8 Wilks J & Bett D S, An Introduction to liquid helium, Second present calculations, the value of the internal edition (Clarendon Press, Oxford \) (1994). energy smoothly increases as the temperature is 9 Greenberg O W, Phys Rev Lett, 43 (1995) 411. increased, and then becomes constant after 10 Khanna K M & Mehrotra S N, Physica, 81A (1975) 311. T≅2.3 K. This means that a fermion-dominated 11 Khanna K M, Statistical Mechanics and Many-Body Problems system behaves like an electron gas. (Today and Tomorrow Publishers, New Delhi), (1986).