Advanced Thermodynamics: Lecture 19 Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Bose–Einstein Statistics The thermodynamic probability W k of a macrostate of an assembly depends on the particular statistics obeyed by the assembly. In Bose–Einstein (B–E) statistics, the particles are considered indistinguishable, and there is no restriction on the number of particles that occupy a particular energy state. The energy states themselves are however distinguishable. Considering a possible energy level j . The distribution of the particles in different states which are denoted by numbers in () can be represented as below. For convenience sake each particle is labeled with an alphabet though they are indistinguishable. [(1) ab ][(2) c ][(3)][(4) def ] . . . (1) Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
There are g j states, therefore there are g j ways in which the sequences can begin and in each of these sequences the remaining ( g j + N j − 1) numbers and letters can be arranged in any order. The number of different sequences in which N distinguishable objects can be arranged is N !. N ( N − 1)( N − 2) . . . 1 = N ! (2) For example 3 letters a , b , c can be arranged in following ways, abc , cab , bac , bac , cab , cab There are six possible sequences, which 3!. Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
The factorial for a large number x is given by Stirling’s approximation, ln x ! = x ln x − x Hence, for x = 70 ln 70! = 70 ln 70 − 70 = 245 log 10 70! = 106 70! = 10 106 Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
The number of different possible sequences of the ( g j + N j − 1) numbers and letters is therefore ( g j + N j − 1)! and the total number of possible sequences of g j numbers and N j letters is g j [( g j + N j − 1)!] (3) Even though each possible sequence represents a possible distribution of particles, many will represent the same distribution. For example sequences given by equations ( 4) and (1) are one and the same. [(3)][(1) ab ][(4) def ][(2) c ] . . . (4) There are g j groups in the sequence, one for each state, so the number of different sequences of groups is g j ! and we must divide Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
equation (3) by g j ! to avoid counting the same distribution more than once. Also, the particles are indistinguishable. Thus a different sequence of letters similar to [(1) ca ][(2) e ][(3)][(4) bdf ] . . . (5) also represents the same distribution as equation (1). The N j letters can be arranged in a sequence in N j different ways, so equation (3) must be also be divided by N j !. Hence the number of different distributions for the j th level is ω j = g j [( g j + N j − 1)!] (6) g j ! N j ! which can be written as Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
ω j = ( g j + N j − 1)! (7) ( g j − 1)! N ! since g j ! = g j ( g j − 1)! (8) Suppose an energy level j includes 3 states ( g j = 3) and 2 particles ( N j = 2). The number of possible distributions are ω j = (3 + 2 − 1)! = 6 (3 − 1)!2! Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
If a level is non–degenerate, that is, if there is only one state in the level and g j = 1, and there is only one possible way in which particles in the level can be arranged, hence ω j = 1. Then equation (3) becomes. ω j = N j ! 0! N j ! = 1 Therefore as a convention for the right answer, we must set 0! = 1. Also if a level j is unoccupied and N j = 0 and ω j = 1 for that level, ω j = ( g j − 1)! ( g j − 1)!0! = 1 Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
For each of the possible distributions in any level, we may have any one of the possible distributions in each of the other levels, so the total number of possible distributions, or the thermodynamic probability W B . E of a macrostate in the B–E statistics is the product over all levels of the values ω j for each level, given by ( g j + N j − 1)! � � W B . E = W k = ω j = (9) ( g j − 1)! N j ! i j For example, an assembly includes two level p and q , with g p = 3 , N p = 2, g q = 2 and N q = 1. The thermodynamics probability of the macrostate N p = 2, and N q = 1 is W B . E = 4! 2!2! × 2! 1!1! = 6 × 2 = 12 (10) Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Although all microstates of an isolated, closed system are equally probable, the only possible microstates are those in which the total number of particles equals the number N in the system and in which the total energy of the particles equals the energy U of the system. For example, let us consider a system of 6 particles with all the permitted energy levels equally spaced, i.e ǫ 0 = 0 , ǫ 1 = ǫ, ǫ 2 = 2 ǫ . . . . Let us assume that each energy level has 3 states, g j = 3 and the total energy U of the system is equals 6 ǫ . Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
N j k= 1 2 3 4 5 6 7 8 9 10 11 N=6 ε j /ε= 6 1 0.041 U=6ε 5 1 0.008Ω=1532 g j =3 4 1 1 0.205 3 2 1 1 0.41 2 1 1 3 2 1 0.83 1 1 2 1 3 2 4 6 1.6 0 5 4 4 3 4 3 2 3 2 1 2.83 W k = 63 135 135 180 90 270 180 100 216 135 28 Eleven possible states of an assembly of 6 particles obeying B–E statistics. Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Thus for macrostate k = 1, since g j = 3 is all levels. The thermodynamic probability is W 1 = (3 + 1 − 1)! × (3 + 5 − 1)! = 3 × 21 = 63 (11) 2!1! 2!5! The single particle in level 6 could be in anyone of 3 states and the 5 particles in level 0 can be distributed in 21 different ways. Therefore there are 63 possible arrangements. The total number of possible microstates or the thermodynamic probability of the system is � Ω = W k = 1532 (12) k Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
The average occupation number of level 2 is computed as N 2 = 1 N 2 k W k = 1272 ¯ � 1532 = 0 . 83 (13) Ω k The most probable macrostate is the one with the largest number of microstates (270), is the sixth state. Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Fermi–Dirac Statistics Developed by Enrico Fermi and Paul Dirac. Applies to indistinguishable particles that obey Pauli’s exclusion principle, according to which there can be no more than one particle in each permitted energy state. Thus the number of particles N j at an energy level j cannot exceed the degeneracy g j of that level. A possible arrangement for F–D statistics might be: [(1) a ][(2) b ][(3)][(4) c ] . . . (14) The possible number of unique sequences are g j ! ω j = (15) ( g j − N j )! N j ! Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
The thermodynamic probability in F–D statistics is g j ! � � W F . D = ω j = (16) ( g j − N j )! N j ! j j Consider again a system of 6 particles with all the permitted energy levels equally spaced, i.e ǫ 0 = 0 , ǫ 1 = ǫ, ǫ 2 = 2 ǫ . . . . And let us assume that each energy level has 3 states, g j = 3 and the total energy U of the system is equals 6 ǫ . Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
N j k= 1 2 3 4 5 N=6 ε j /ε= 4 1 0.123 U=6ε 3 1 1 0.494 Ω=73 g j =3 2 1 3 2 1.15 1 2 1 3 2 1.73 0 3 3 2 3 2 2.51 W k = 9 27 9 1 27 Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Five possible states of an assembly of 6 particles obeying F–D statistics. Thus in macrostate 1, 3! 3! 3! W 1 = (3 − 3)!3! = 3 × 3 × 1 = 9 (3 − 1)!1! × (3 − 2)!2! × The total number of possible macrostates is � Ω = W k = 73 k Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Maxwell–Boltzmann Statistics In Maxwell–Boltzmann (M–B) statistics, the particles are considered distinguishable and like B–E statistics there is no restriction on the placement on the particles in various energy states. Consider a j level with a degeneracy of g j and N j particles. The number of ways the particles can be arranged in g j states is, ω j = g N j (17) j The total number of distributions between all levels, with a specified set of particles in each level is, g N j � � ω j = (18) j j j Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Though, this will not be the thermodynamic probability W k since the an interchange of particle between levels will also give rise to a different macrostate (by virtue of particles being distinguishable). The total number of ways in which N particles are distributed among levels with N 1 particles in level 1, N 2 particles in level 2 and so on, is given by. N ! N ! N 1 ! N 2 ! . . . = (19) � j N j ! The thermodynamic probability of M − − B statistics is the product of the above 2 equations and is given by g N j N ! g N j � � j W M . B = = N ! (20) j � j N j ! N j ! j j Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Consider again a system of 6 particles with all the permitted energy levels equally spaced, i.e ǫ 0 = 0 , ǫ 1 = ǫ, ǫ 2 = 2 ǫ . . . . The arrangement is similar to that of B–E statistics but has far more number of states since particles are now distinguishable. Thus for macro state k = 1, in which levels zero and six are occupied. W 1 = 6!3 5 3 1 1! = 18 × 3 5 (21) 5! The total number of possible macrostates is given by W k = 1386 × 3 5 = 3 . 37 × 10 5 � Ω = (22) k Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
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