Indian Journal o Pure & Applied Physics Vol. 4, October 004, pp. 749-757 Micro-canonical ensemble model o particles obeying Bose-Einstein and Fermi-Dirac statistics Y K Ayodo, K M Khanna & T W Sakwa Department o Physical Sciences, Western University College, Box 90, Kakamega, Kenya Department o Physics, Moi University, Box 5 Eldoret, Kenya Received 3 February 004; accepted April 004 A micro-canonical ensemble or an assembly o bosons and ermions is considered in which the number o particles, internal energy and volume are kept constant. A statistical distribution model, which is ermion dominated and where bosons and ermions interact in pairs, is developed. The partition unction is derived. Macroscopic thermodynamic quantities such as entropy, internal energy and speciic heat are obtained in terms o the partition unction. The model equations are applied to a mixture o liquid helium-3 and liquid helium-4 atoms. [Keywords: Bose-Einstein Statistics, Fermi-Dirac statistics, Partition unction, 3 He- 4 He mixture] IPC Code: C0B 3/00 Introduction A micro-canonical ensemble represents a collection o conigurations o isolated systems that have reached thermal equilibrium. A system is isolated rom its environment i it does not exchange either particles or energy with its surroundings. The volume, internal energy and the number o particles o such a system are constant and are the same or all conigurations that are part o the same microcanonical ensemble. In this paper, a coniguration o a mixture o bosons and ermions is studied and a partition unction is developed or the same. Thermodynamic quantities, such as internal energy, speciic heat and entropy can be calculated rom the knowledge o the statistical distribution and the partition unction. So ar most o the studies deal either with a system o bosons or with a system o ermions. In nature, there do exist systems, which are mixtures o bosons and ermions such as H, H and 3 H, and the most interesting mixture is 4 He and 3 He. It should be clearly understood that in the mixture, bosons obey Bose-Einstein statistics and ermions obey Fermi-Dirac statistics. What distribution law or what will be the expression or the most probable distribution-in-energy in the mixture is the subect matter o study in this paper. The irst attempt to generalise quantum Bose and Fermi statistics or a mixture o Bosons and Fermions was made by Gentile. He proposed statistics in which up to N particles (N>> were allowed to occupy a single quantum state instead o ust one particle or Fermi case due to the Pauli exclusion principle, and ininitely many or the Bose case. However, Gentile s approach was ound to be too much o a generalisation and contained the violation o the conventionally accepted Pauli principle. Furthermore, his model did not distinguish which particles were ermions and which ones were bosons. However, Gentile s work laid the emphasis and the oundation that the statistical mechanics o a mixture o bosons and ermions can be worked out. The next attempt was that o Medvedev. In his paper entitled properties o particles obeying ambiguous statistics, Medvedev proposed a new class o identical particles, which may exhibit both Bose and Fermi statistics with respective probabilities P b and P. The model admits only primary Bose- Einstein and Fermi-Dirac statistics as existing. He assumed that a particle is neither a pure boson nor a pure ermion. He let another particle, which interacts with the irst one, play the role o an external observer. During the interaction it perorms a measurement at the irst particle and identiies it as either a boson or a ermion with respective probabilities P b and P. According to the result o this measurement, it interacts with the irst particle as i the last is a ermion or a boson, respectively. The
750 INDIAN J PURE & APPL PHYS, VOL. 4, OCTOBER 004 irst particle is the observer or the second particle and so the process is symmetric. Note that (P b + P is not necessarily equal to one, and, i not, it means that the second particle (observer does not detect the irst particle. The probability o this is ( P b P. The statistical uncertainty introduced here may be either the intrinsic property o a particle itsel or the experimental uncertainty o the measurement process. Another attempt in this direction is the so-called statistical independence model o Landau and Lishitz 3 in which two weakly interacting subsystems (bosons and ermions are together regarded as one composite system, and the subsystems are assumed to be quasi-closed. The statistical distribution or count or such a mixture is the product o the individual probabilities or two subsystems, one corresponding to bosons and the other corresponding to ermions. With these assumptions, the statistical independence model will hold only or an ideal gas assembly o bosons and ermions. In reality, such an assembly does not exist and hence the statistical independence model cannot be used or real mixtures o bosons and ermions like 3 He and 4 He mixtures. Chan et al. 4 studied the eect o disorder on superluid 3 He- 4 He mixtures, and the thermodynamics o 3 He- 4 He mixtures in aerogel. However, the studies related to more o an ideal system rather than a real system. In our earlier paper 5 entitled statistical mechanics and thermodynamics or a mixture o bosons and ermions, the statistical distribution model or a mixture that was dominated by bosons was developed. The partition unction that was derived worked well or a liquid helium-3 and liquid helium-4 mixture. In this paper, an assembly that is ermion dominated is studied. Thereore, the properties o a mixture o bosons and ermions assuming there exists a pair interaction between the bosons and ermions; and that the concentrations o the bosons and ermions are dierent rom each other are studied. Since the concentrations o the bosons and ermions will not be the same, and considering only pair interaction, in a given state o equilibrium some ermions will be let unpaired. The value o the occupation number o ermions in a given state will not exceed, rather, will be much less, than the degeneracy o that state so that Pauli exclusion principle is not violated. With these basic assumptions, the expressions or the ollowing were derived: (i Statistical count or an ensemble that is a mixture o the bosons and ermions assuming a pair interaction between the bosons and ermions. (ii The most probable distribution in energy or a mixture o the bosons and the ermions in the ensemble. (iii The partition unction or such an ensemble. (iv Using the partition unction, the calculations were done or internal energy, speciic heat and entropy. The above expressions are used to study the thermodynamic properties o a mixture o liquid helium-3 (ermions and liquid helium-4 (bosons with dierent concentrations. For our model calculations a ermion concentration o 0.70 was used. The theoretical results obtained are compared with the experimental observations on the properties o a mixture o liquid helium-3 and liquid helium-4. Theory Consider a micro-canonical assembly o N particles in which there are N b bosons and N ermions such that N= N b + N ( Let ε, ε, ε 3 ε be the energy states o the assembly, and in the statistical equilibrium the number o particles assigned to these energy levels be n, n, n 3, n, respectively, such that the numbers n must satisy the conditions requiring the conservation o particles, N, and conservation o energy, E, i.e., n = N ( = and = n ε = E (3 such that, n = nb + n (4 where, n b = number o bosons in the energy level (5
AYODO et al.: MICRO-CANONICAL ENSEMBLE FOR BOSONS & FERMIONS 75 and n =number o ermions in the energy level (6 P b = n ( b ω ( ω ( nb ( ω n ( n nb (9 Let ω be the number o states in the -th level, i.e, ω is the degeneracy o the -th level. Then the number o ways P b in which n b bosons can be assigned to ω states in the -th level is given by, P b n ( ω b = (7 Similarly, the number o ways, P, in which n ermions can be assigned to the ω states is given by, P = ω ( ω n (8 To satisy Pauli exclusion principle, it is necessary to assume that ω >> n. Once the particles are placed in the ω sub-levels in the -th level, we shall urther assume that bosons and ermions may interact in pairs. However, not all the ermions may orm pairs with bosons since the bosons and the ermions will not be in equal proportions in the mixture. This statement implies that n > n b. It is the number o bosons n b that will determine the number o boson-ermion pairs. Hence, the number o boson-ermion pairs will be n b and the number o unpaired ermions will be (n n b. Since the permutations among the particles and the permutations among the pairs in the same energy level do not give a new complexion, in the statistical distribution model proposed here, the ollowing permutations must be excluded rom the number o ways in which n b bosons, n ermions and n b bosonermion pairs are distributed in the -th level: (i Permutations among identical pairs should be excluded by dividing by (n b (ii Permutations among identical unpaired ermions should be excluded by dividing by (n n b Hence the total number o ways, P b, in which n b bosons, n ermions and n b pairs o bosons and ermions can be distributed among the ω sub-levels in the -th level is given by, The statistical count, C b, or such a distribution among all the levels ( =,,3 available to the assembly is the product o such expressions as given in Eq. (9 since every arrangement in a given energy level can be considered independently o the other energy levels. Thus we can write, C b = Pb = nb ( ω ( ω ( n ( ω n ( n n = = b b (0 Equation (0 will now be used to calculate the most probable distribution in energy or the ensemble. 3 Most probable distribution in energy or the ensemble The obective is to calculate or what values o n b and n, the statistical count C b, is maximum under the conditions o N and E being ixed. The distribution numbers and the corresponding energies must satisy the ollowing relations: n = N ( = = b b n ε = E ( = b b n = N (3 = n ε = E (4 where N b and N are the total number o bosons and ermions in the ensemble such that the total number o particles N is given by, N= N b + N (5 Similarly, E b and E are the total internal energies o bosons and ermions such that the total energy o the ensemble E is given by,
75 INDIAN J PURE & APPL PHYS, VOL. 4, OCTOBER 004 E= E b + E (6 Now to ind the values o n b and n or which C b is maximum, the procedure is to allow the variation o C b with respect to n b and n, and put the result equal to zero. For C b to be maximum, ln( Cb = ln( Cb dnb = nb + ln( Cb dn = 0 = n (7 The variations dn b and dn are not independent since the n b s and n s must continue to satisy the restrictions given in Eqs (-4. Since N and E are ixed, the variations in n b and n must satisy the ollowing equations: dn + dn = 0 (8 b = = and b = = ε dn + ε dn = 0 (9 Hence, along with Eq. (7, Eqs [(8 and (9] must also be satisied. These equations can be combined by the method o Lagrange s undetermined multipliers which are denoted by α and β. Thus, multiplying the irst and the second terms in Eq. (8 by ( α b and ( α, respectively, and Eq. (9 by ( β and adding to Eq. (7, we get, = ln( Cb ( αb + βε dnb nb + ln( Cb ( α + βε dn = 0 = n (0 Now Eq. (0 demands that all the terms should be separately equal to zero, and the terms or which dn b and dn are not equal to zero, then, the coeicients o dn b and dn should be, respectively, equal to zero. I we assume that one o the dn b s and dn s is non-zero, then the corresponding coeicients will be zero. Thus we can write, n n b ln( C ( α + βε =0 ( b b ln( C ( α + βε =0 ( b Eqs [( and (] are true or all values o. Substituting or C b rom Eq. (0 in Eq. ( gives, ω ( n nb = exp( αb + βε (3 n b Similarly, substituting or C b rom Eq. (0 in Eq. ( gives, ( ω n = exp( α + βε ( n n b (4 Eqs [(3 and (4] are solved or n b and n to get: nb = ω exp( αb α βε ω exp( αb α βε + exp( αb βε + ω exp( αb α βε (5 ω exp( αb α βε n = + exp( α b βε + ω exp( α b α βε (6 μ μ b where, α b =, α = and β = kt kt kt (7 Equation (5 gives the most probable distribution in energy or bosons in the ensemble, and Eq. (6 gives the most probable distribution in energy or ermions in the ensemble. In Eq. (7 μ b is the chemical potential or bosons and μ is the chemical potential or ermions. We should note that n is contained in the expression or n b. 4 Partition unction or the ensemble The general expression or the partition unction Q or an ensemble o bosons and ermions can be written as:
AYODO et al.: MICRO-CANONICAL ENSEMBLE FOR BOSONS & FERMIONS 753 μ μb Q= Nexp kt μ μb = ( nb + n exp (8 kt = 5 Results and Discussion The general expressions or entropy S, the internal energy E and the speciic heat C v are given as 3, lnq E= NkT T (30 Substituting or n b rom Eq. (5 and n rom Eq. (6 in Eq. (8, we get Q S = kn ln + T ln Q N T (3 ε Q= ω exp = kt μb+ μ ε + ω exp kt + μb ε μb+ μ ε + exp + ω exp kt kt (9 Equation (9 is the expression or the partition unction or a mixture o bosons and ermions in the ensemble, in which pair interaction between the two types o particles is considered. It must be understood that the partition unction Q has two signiicant terms: One term is μb+ μ ε exp kt This quantity contains ε which is a consequence o the pair interaction between the bosons and ermions. It can also be interpreted as the energy o a boson ermion pair in the -th energy level. The second term is, ln( Q ln( Q = NkT + NkT T V T C v V (3 Together with these equations, let us introduce the ermion concentration η given by η=n /N (33 or N=N /η (34 To perorm the calculations, we substituted or N and Q in the above thermodynamic relations rom Eqs [(5, (6 and (9]. The molar thermodynamic quantities are o interest. The experiments done by Chan et al. 4,7 mainly ocused on the molar quantities o liquid helium-3 and liquid helium-4. Wilks and Bett 8 give the molar volume o liquid helium-3 as 40.0 cm 3, and its molar density as 0.07 g cm 3 ; and the molar volume o liquid helium-4 as 8.0 cm 3 and its molar density as 0.4 g cm 3. This, thereore, means that the molar mass o liquid helium-3 is.80 g and that o liquid helium-4 is 3.9 g. Although the chemical potential should have temperature dependence, at low temperatures it assumes a nearly constant value given by the expression 8, μb ε exp kt π h 3N μ = m πv 3 (35 The absence o μ, chemical potential or ermions, in this quantity, is an indication that the distribution o bosons is not aected by how ermions are distributed. The number o bosons, the number o bosonermion pairs that shall be ormed, and the number o ermions shall determine how many ree or unattached ermions shall remain to be distributed among the available energy states. where, m is the molar mass, V is the molar volume and N is the number o particles in one mole, and this is Avogadro s number = 6.05 0 3 particles mol. Substitution o the empirical data into Eq. (35 gives the chemical potential μ, or ermions as, 7 μ 3.84 0 ev = (36 and or bosons as,
754 INDIAN J PURE & APPL PHYS, VOL. 4, OCTOBER 004 8 μ b 6.5 0 ev = (37 Since the transition temperature 8 o liquid helium-3 is very much lower than the transition temperature o liquid helium-4, it is interesting to know what happens in the vicinity o the transition temperature o liquid helium-4. Furthermore, the temperature ranges used in experiments 4,5,7,8 on such mixtures are so much higher than the transition temperature o liquid helium-3, which is lower than.5 mk. However, given that both liquid helium-3 and liquid helium 4 are at the same temperature, they are considered to be in thermal equilibrium. 5. Calculation o partition unction Q Using the essential parameters or liquid helium-3 and liquid helium-4 given in Table, we calculated the partition unction Q, using Eq. (9 or the mixture in the temperature range.0 K to.30 K in steps o 0.0 K. Table gives the values o Q or the mixture at dierent temperatures. Figure depicts the variation o Q with temperature T. Figure shows that there is an exponential rise in the value o the partition unction Q. The reason or this kind o behaviour is that at very low temperatures there are very ew energy states 8 that can be occupied by the particles in the assembly. However, at higher temperatures, the number o energy levels available or particle occupation could be large, and these could be called the excited states o the assembly o particles. 5. Calculation o internal energy Equation (30 is used to calculate how the internal energy E varies with temperature using the parameters listed in Table. Table 3 gives the variation in the values o E with temperature in the range.0 K to.30 K in steps o 0.0K. To convert the values o E rom electronvolt (ev to Joules (J, E values are multiplied by a actor o.6 0 9. The graph o internal energy variation against temperature is given in Fig.. Figure shows that there is a rise in the total internal energy with temperature. This is not unusual since or any given thermodynamic system, the higher the temperature, the higher should be the internal energy. The increase in internal energy tends to be exponential in the temperature range.0 K to.5 K but tends to assume a nearly constant value o Table Essential parameters or liquid helium-3 and helium-4 Parameter Liquid helium-3 Liquid helium-4 Volume ( cm 3 40.00 8.00 Density (gcm 3 0.07 0.4 Mass (g.80 3.9 Chemical potential(ev 3.84 0 7 6.5 0 8 Table Values o the partition unction against temperature T (K Q (T 0 particles.0 4.37.0 5.55.03 7.430.04 9.94.05 3.90.06 7.500.07 3.60.08 30.560.09 5.80.0 69.40. 90.90. 7.600.3 5.800.4 98.00.5 56.00.6 306.500.7 355.900.8 45.400.9 546.00.0 699.800. 894.500. 4.000.3 45.000.4 844.000.5 336.000.6 954.000.7 378.000.8 4695.000.9 5900.000.3 740.000 approximately.50 kj as the temperature approaches.30 K. 5.3 Calculation o entropy S Equation (3 is used to calculate the variation o S with T. The values are given in Table 4 and the graph showing the variation o S with T is plotted in Fig. 3. The graph in Fig. 3 has the same shape as the one or the variation o E with T. The values o entropy increases with temperature ust like the internal energy. By deinition, entropy is a measure o the molecular disorder o any given system. Naturally, 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 higher temperatures molecules have higher vibrational energies, creating more disorder and hence more entropy. These trends are clearly depicted in Fig. 3. Entropy is also an extensive thermodynamic quantity. That means it is proportional to the number o particles in a system. In act it is meaningless to talk about the entropy o a single particle. Thereore, the ewer the particles the lower is the value o entropy and vice versa. The shape o the curve in Fig. 3 is in good agreement with the result obtained in Fig., where at lower temperatures there are ewer energy Fig. Variation o partition unction with temperature Table 3 Values o internal energy E at dierent temperatures T (K Internal Energy, E (J.0 59.638.0 79.084.03 04.78.04 36.68.05 77.68.06 9.56.07 9.700.08 369.74.09 46.09.0 567.6. 686.78. 87.675.3 956.460.4 099.000.5 40.000.6 376.000.7 50.000.8 66.000.9 77.000.0 805.000. 879.000. 94.000.3 99.000.4 034.000.5 068.000.6 096.000.7 36.000.8 50.000.9 6.000 Fig. Variation o internal energy with temperature Table 4 Values o entropy S at dierent temperatures T (K Entropy, S (J/K.0 9.854.0 39.400.03 5.708.04 67.43.05 87.306.06.05.07 4.585.08 79.385.09.900.0 73.40. 39.606. 39.9.3 456.354.4 5.985.5 588.97.6 65..7 70.867.8 763.984.9 80.844.0 85.34. 885.646. 94.345.3 938.053.4 957.457.5 973.3.6 985.965.7 996..8 004.000.9 0.000.30 06.000
756 INDIAN J PURE & APPL PHYS, VOL. 4, OCTOBER 004 Table 5 Values o speciic heat ( 0 5 at dierent temperatures T (K Speciic heat, C (J/mol. K Fig. 3 Variation o entropy with temperature states or the distribution o particles. This means entropy is expected to be low at lower temperatures. 5.4 Calculation o speciic heat C v Equation (3 is used to calculate how the speciic heat at constant volume or a mixture o liquid 3 He and liquid 4 He varies with temperature T. Since the molar quantities o the constituents o the mixture are the ones that were o interest, the values o the speciic heat were multiplied by a actor o.6 0 9 and divided by 97.69 to convert the units o speciic heat rom ev kg K to J mol K. This is because rom Table, 6.7g is the mass o moles o the mixture and thus kg o the mixture is roughly equal to 97.69 moles. Table 5 gives the values o C v at dierent temperatures. The shape o the speciic heat curve exhibits a lot o luctuations. Speciic heat values are too low in the temperature range rom.0 K to.09 K. One particular eature that was ound to be more interesting is that the highest peak o the curve occurred at.5 K. From here we see a signiicant change o phase at.5 K. This happens to be very near the λ-transition temperature 8,9 or liquid 4 He, which is.66 K. Below this temperature liquid 4 He becomes a superluid. However, experimental observations by Chan et al. 4,7,9 showed shits in the transition temperature at which peaks in the value o the speciic heat occurred. This can be accounted or due to the act that, experimentally, a highly porous material called aerogel was used to control the low o liquid 3 He into liquid 4 He and changes in the thermodynamic quantities o the mixture were observed or dierent.0 5.383.0 94.560.03 58.300.04 65.000.05 9.00.06 37.700.07 9.040.08 85.40.09 59.300.0.00. 496.000. 3.800.3 338.300.4 593.500.5 8798.000.6 537.000.7 703.000.8 460.000.9 36.000.0 3580.000. 74.000. 0.000.3 56.000.4 648.000.5 749.000.6 3437.000.7 374.000.8 76.000.9 567.000.30 587.000 liquid 3 He concentrations. However, our theoretical model assumes a bulk mixture, meaning without aerogel, o the two liquids. Furthermore, our calculations do not include the low properties o the two liquids, or instance, superluidity in liquid 4 He is supposed to disappear above a certain critical velocity. The normal-superluid phase transition in pure liquid 4 He is a second order phase transition, whereas the phase change in the mixture o liquid 3 He into liquid 4 He are characterised with a lot o luctuations with no discontinuity. This may be due to the act that in our case the atoms o the two liquids have not been considered to be entirely independent but exchange energy through pair interaction. 6 Conclusions Dierent authors -5 studied the statistical thermodynamics or a mixture o bosons and ermions by putting orth dierent models. In these models particles were considered independent or weakly
AYODO et al.: MICRO-CANONICAL ENSEMBLE FOR BOSONS & FERMIONS 757 Fig. 4 Variation o speciic heat ( 0 5 with temperature interacting. In our model, the bosons and ermions are supposed to be interacting via a pair interaction, and the whole assembly is supposed to be in thermal equilibrium. Furthermore, there are more ermions than bosons in this ensemble. Comparing the calculations presented in Tables -5; and the graphs presented in Figs (-4, with the corresponding results in Re. 5, in which bosons were more than ermions, the ollowing marked dierences in the shapes o the curves can be observed: (i Partition unction Q The partition unction in Re. 5 becomes roughly constant ater T. K, whereas, in the present calculation, Q varies exponentially ater T. K. This means that at higher temperatures the occupation o excited states increases and this is the basic character o ermion-dominated systems. (ii Internal Energy E The internal energy in Re. 5 becomes maximum around T.4 K and then decreases as the temperature increases. In the present calculations, the value o the internal energy smoothly increases as the temperature is increased, and then becomes constant ater T.3 K. This means that a ermion-dominated system behaves like an electron gas. (iii Speciic heat C v The shape o the speciic heat C v and that o the internal energy E is the same in Re. 5, whereas we ind that the speciic heat C v has maxima and minima, and the maximum value o C v is around.5 K.The shape o the C v curve is dierent rom the shape o the curve or the internal energy E. It should be acceptable that the speciic heat C v or a ermion dominated system will be dierent rom a boson dominated system. It is the Pauli exclusion principle that restricts the low o ermions rom one level to another as the temperature changes, whereas no such restriction exists or a boson dominated system. Thus, a ermion-dominated system may reuse to absorb heat resulting in a negative speciic heat. The actual transition temperature o the mixture is at.5 K, below which the whole mixture goes into the superluid state. The magnitudes o the thermodynamic quantities increases as the value o η increases and this is evident rom Eqs [(30 to (34]. The phase transition is one that is not smooth but is characterised by luctuations. Reerences Gentile G, Phys Rev Lett, 7 (95 493. Medvedev M V, Phys Rev Lett, 78 (996 447. 3 Landau & Lishitz, Statistical physics, Vol.,Third edition (Pergamon Press, New York (98. 4 Chan M H, Blum K I & Murphy S Q, Phy Rev Lett, 6 (99 950. 5 Khanna K M & Ayodo Y K, Indian J Pure & Appl Phys, 4 (003 80. 6 Baierhein R, Thermal Physics, First edition (Cambridge university Press, 999. 7 Chan M H, Mulders N & Reppy J, Physics Today, 7 (996 3. 8 Wilks J & Bett D S, An Introduction to liquid helium, Second edition (Clarendon Press, Oxord \ (994. 9 Greenberg O W, Phys Rev Lett, 43 (995 4. 0 Khanna K M & Mehrotra S N, Physica, 8A (975 3. Khanna K M, Statistical Mechanics and Many-Body Problems (Today and Tomorrow Publishers, New Delhi, (986.