NOTES FOR STATISTICAL MECHANICS

  BOSE-EINSTIEN DISTRIBUTION

Suppose we have a number of energy levels, labeled by index \displaystyle i, each level having energy \displaystyle \varepsilon_i and containing a total of \displaystyle n_i particles. Suppose each level contains \displaystyle g_i distinct sublevels, all of which have the same energy, and which are distinguishable. For example, two particles may have different momenta, in which case they are distinguishable from each other, yet they can still have the same energy. The value of \displaystyle g_i associated with level \displaystyle i is called the "degeneracy" of that energy level. Any number of bosons can occupy the same sublevel.
Let \displaystyle w(n,g) be the number of ways of distributing \displaystyle n particles among the \displaystyle g sublevels of an energy level. There is only one way of distributing \displaystyle n particles with one sublevel, therefore \displaystyle w(n,1)=1. It is easy to see that there are \displaystyle (n+1) ways of distributing \displaystyle n particles in two sublevels which we will write as:
w(n,2)=\frac{(n+1)!}{n!1!}.
With a little thought (see Notes below) it can be seen that the number of ways of distributing \displaystyle n particles in three sublevels is
w(n,3) = w(n,2) + w(n-1,2) + \cdots + w(1,2) + w(0,2)
so that
w(n,3)=\sum_{k=0}^n w(n-k,2) = \sum_{k=0}^n\frac{(n-k+1)!}{(n-k)!1!}=\frac{(n+2)!}{n!2!}
where we have used the following theorem involving binomial coefficients:
\sum_{k=0}^n\frac{(k+a)!}{k!a!}=\frac{(n+a+1)!}{n!(a+1)!}.
Continuing this process, we can see that \displaystyle w(n,g) is just a binomial coefficient (See Notes below)
w(n,g)=\frac{(n+g-1)!}{n!(g-1)!}.
For example, the population numbers for two particles in three sublevels are 200, 110, 101, 020, 011, or 002 for a total of six which equals 4!/(2!2!). The number of ways that a set of occupation numbers \displaystyle n_i can be realized is the product of the ways that each individual energy level can be populated:
W = \prod_i w(n_i,g_i) = \prod_i \frac{(n_i+g_i-1)!}{n_i!(g_i-1)!} \approx\prod_i \frac{(n_i+g_i)!}{n_i!(g_i-1)!}
where the approximation assumes that n_i \gg 1.
Following the same procedure used in deriving the Maxwell–Boltzmann statistics, we wish to find the set of \displaystyle n_i for which W is maximised, subject to the constraint that there be a fixed total number of particles, and a fixed total energy. The maxima of \displaystyle W and \displaystyle \ln(W) occur at the value of \displaystyle N_i and, since it is easier to accomplish mathematically, we will maximise the latter function instead. We constrain our solution using Lagrange multipliers forming the function:
f(n_i)=\ln(W)+\alpha(N-\sum n_i)+\beta(E-\sum n_i \varepsilon_i)
Using the n_i \gg 1 approximation and using Stirling's approximation for the factorials \left(x!\approx x^x\,e^{-x}\,\sqrt{2\pi x}\right) gives
f(n_i)=\sum_i (n_i + g_i) \ln(n_i + g_i) - n_i \ln(n_i) +\alpha\left(N-\sum n_i\right)+\beta\left(E-\sum n_i \varepsilon_i\right)+K.
Where K is the sum of a number of terms which are not functions of the ni. Taking the derivative with respect to \displaystyle n_i, and setting the result to zero and solving for \displaystyle n_i, yields the Bose–Einstein population numbers:
n_i = \frac{g_i}{e^{\alpha+\beta \varepsilon_i}-1}.
By a process similar to that outlined in the Maxwell-Boltzmann statistics article, it can be seen that:
d\ln W=\alpha\,dN+\beta\,dE
which, using Boltzmann's famous relationship S=k\,\ln W becomes a statement of the second law of thermodynamics at constant volume, and it follows that \beta = \frac{1}{kT} and \alpha = - \frac{\mu}{kT} where S is the entropy, μ is the chemical potential, k is Boltzmann's constant and T is the temperature, so that finally:
n_i = \frac{g_i}{e^{(\varepsilon_i-\mu)/kT}-1}.
Note that the above formula is sometimes written:
n_i = \frac{g_i}{e^{\varepsilon_i/kT}/z-1},

A derivation of the Maxwell–Boltzmann distribution

Suppose we have a container with a huge number of very small identical particles. Although the particles are identical, we still identify them by drawing numbers on them in the way lottery balls are being labelled with numbers and even colors.
All of those tiny particles are moving inside that container in all directions with great speed. Because the particles are speeding around, they do possess some energy. The Maxwell–Boltzmann distribution is a mathematical function that speaks about how many particles in the container have a certain energy.
It can be so that many particles have the same amount of energy εi. The number of particles with the same energy εi is Ni. The number of particles possessing another energy εj is Nj. In physical speech this statement is lavishly inflated into something complicated which states that those many particles Ni with the same energy amount εi, all occupy a so called "energy level" i . The concept of energy level is used to graphically/mathematically describe and analyse the properties of particles and events experienced by them. Physicists take into consideration the ways particles arrange themself and thus there is more than one way of occupying an energy level and that's the reason why the particles were tagged like lottery ball, to know the intentions of each one of them.
To begin with, let's ignore the degeneracy problem: assume that there is only one single way to put Ni particles into energy level i . What follows next is a bit of combinatorial thinking which has little to do in accurately describing the reservoir of particles.
The number of different ways of performing an ordered selection of one single object from N objects is obviously N. The number of different ways of selecting two objects from N objects, in a particular order, is thus N(N − 1) and that of selecting n objects in a particular order is seen to be N!/(N − n)!. The number of ways of selecting 2 objects from N objects without regard to order is N(N − 1) divided by the number of ways 2 objects can be ordered, which is 2!. It can be seen that the number of ways of selecting n objects from N objects without regard to order is the binomial coefficient: N!/(n!(N − n)!). If we now have a set of boxes labelled a, b, c, d, e, ..., k, then the number of ways of selecting Na objects from a total of N objects and placing them in box a, then selecting Nb objects from the remaining N − Na objects and placing them in box b, then selecting Nc objects from the remaining N − Na − Nb objects and placing them in box c, and continuing until no object is left outside is
\begin{align} W & = \frac{N!}{N_a!(N-N_a)!} \times \frac{(N-N_a)!}{N_b!(N-N_a-N_b)!} ~ \times \frac{(N-N_a-N_b)!}{N_c!(N-N_a-N_b-N_c)!} \times \ldots \times \frac{(N-\ldots-N_l)!}{N_k!(N-\ldots-N_l-N_k)!} = \\ \\ & = \frac{N!}{N_a!N_b!N_c!\ldots N_k!(N-\ldots-N_l-N_k)!} \end{align}
and because not even a single object is to be left outside the boxes, implies that the sum made of the terms Na, Nb, Nc, Nd, Ne, ..., Nk must equal N, thus the term (N - Na - Nb - Nc - ... - Nl - Nk)! in the relation above evaluates to 0! which makes possible to write down that relation as
\begin{align} W & = N!\prod_{i=a,b,c,...}^k \frac{1}{N_i!} \end{align}
Now going back to the degeneracy problem which characterize the reservoir of particles. If the i-th box has a "degeneracy" of gi, that is, it has gi "sub-boxes", such that any way of filling the i-th box where the number in the sub-boxes is changed is a distinct way of filling the box, then the number of ways of filling the i-th box must be increased by the number of ways of distributing the Ni objects in the gi "sub-boxes". The number of ways of placing Ni distinguishable objects in gi "sub-boxes" is g_i^{N_i}. Thus the number of ways W that a total of N particles can be classified into energy levels according to their energies, while each level i having gi distinct states such that the i-th level accommodates Ni particles is:
W=N!\prod \frac{g_i^{N_i}}{N_i!}
This is the form for W first derived by Boltzmann. Boltzmann's fundamental equation S=k\,\ln W relates the thermodynamic entropy S to the number of microstates W, where k is the Boltzmann constant. It was pointed out by Gibbs however, that the above expression for W does not yield an extensive entropy, and is therefore faulty. This problem is known as the Gibbs paradox The problem is that the particles considered by the above equation are not indistinguishable. In other words, for two particles (A and B) in two energy sublevels the population represented by [A,B] is considered distinct from the population [B,A] while for indistinguishable particles, they are not. If we carry out the argument for indistinguishable particles, we are led to the Bose-Einstein expression for W:
W=\prod_i \frac{(N_i+g_i-1)!}{N_i!(g_i-1)!}
Both the Maxwell-Boltzmann distribution and the Bose-Einstein distribution are only valid for temperatures well above absolute zero, implying that g_i\gg 1. The Maxwell-Boltzmann distribution also requires low density, implying that g_i\gg N_i. Under these conditions, we may use Stirling's approximation for the factorial:
N! \approx N^N e^{-N},
to write:
W\approx\prod_i \frac{(N_i+g_i)^{N_i+g_i}}{N_i^{N_i}g_i^{g_i}}\approx\prod_i \frac{g_i^{N_i}(1+N_i/g_i)^{g_i}}{N_i^{N_i}}
Using the fact that (1+N_i/g_i)^{g_i}\approx e^{N_i} for g_i\gg N_i we can again use Stirlings approximation to write:
W\approx\prod_i \frac{g_i^{N_i}}{N_i!}
This is essentially a division by N! of Boltzmann's original expression for W, and this correction is referred to as correct Boltzmann counting.
We wish to find the Ni for which the function W is maximized, while considering the constraint that there is a fixed number of particles \left(N=\textstyle\sum N_i\right) and a fixed energy \left(E=\textstyle\sum N_i \varepsilon_i\right) in the container. The maxima of W and ln(W) are achieved by the same values of Ni and, since it is easier to accomplish mathematically, we will maximize the latter function instead. We constrain our solution using Lagrange multipliers forming the function:
f(N_1,N_2,\ldots,N_n)=\ln(W)+\alpha(N-\sum N_i)+\beta(E-\sum N_i \varepsilon_i)
\ln W=\ln\left[\prod\limits_{i=1}^{n}\frac{g_i^{N_i}}{N_i!}\right] \approx \sum\limits_{i=1}^n\left(N_i\ln g_i-N_i\ln N_i + N_i\right)
Finally
f(N_1,N_2,\ldots,N_n)=\alpha N +\beta E + \sum\limits_{i=1}^n\left(N_i\ln g_i-N_i\ln N_i + N_i-(\alpha+\beta\varepsilon_i) N_i\right)
In order to maximize the expression above we apply Fermat's theorem (stationary points), according to which local extrema, if exist, must be at critical points (partial derivatives vanish):
\frac{\partial f}{\partial N_i}=\ln g_i-\ln N_i -(\alpha+\beta\varepsilon_i) = 0
By solving the equations above (i=1\ldots n) we arrive to an expression for Ni:
N_i = \frac{g_i}{e^{\alpha+\beta \varepsilon_i}}
Substituting this expression for Ni into the equation for ln W and assuming that N\gg 1 yields:
\ln W = \alpha N+\beta E\,
or, differentiating and rearranging:
dE=\frac{1}{\beta}d\ln W-\frac{\alpha}{\beta}dN
Boltzmann realized that this is just an expression of the second law of thermodynamics. Identifying dE as the internal energy, the second law of thermodynamics states that for variation only in entropy (S) and particle number (N):
dE=T\,dS+\mu\,dN
where T is the temperature and μ is the chemical potential. Boltzmann's famous equation S=k\,\ln W is the realization that the entropy is proportional to ln W with the constant of proportionality being Boltzmann's constant. It follows immediately that β = 1 / kT and α = − μ / kT so that the populations may now be written:
N_i = \frac{g_i}{e^{(\varepsilon_i-\mu)/kT}}
Note that the above formula is sometimes written:
N_i = \frac{g_i}{e^{\varepsilon_i/kT}/z}
where z = exp(μ / kT) is the absolute activity.
Alternatively, we may use the fact that
\sum_i N_i=N\,
to obtain the population numbers as
N_i = N\frac{g_i e^{-\varepsilon_i/kT}}{Z}
where Z is the partition function defined by:
Z = \sum_i g_i e^{-\varepsilon_i/kT}

Derivation of Fermi-Dirac Statistics

Consider a system of particles with allowed energy levels tex2html_wrap_inline1025 . Let tex2html_wrap_inline1027 be the number of allowed states at energy tex2html_wrap_inline1029 , and let tex2html_wrap_inline1031 be the actual number of particles at energy tex2html_wrap_inline1029 . The values tex2html_wrap_inline1029 and tex2html_wrap_inline1027 are fixed, and the values tex2html_wrap_inline1031 are random according to the particular arrangement of electrons. Two important physical parameters are the total number of particles, tex2html_wrap_inline1041 , and the total energy of the system, tex2html_wrap_inline1043 .
We impose the following hypotheses:

  hypothesis54


  hypothesis58


  hypothesis64


definition68


theorem71

Given a specified tex2html_wrap_inline1053 , then the conditional probability that a state at energy level tex2html_wrap_inline1029 is occupied is tex2html_wrap_inline1057 since exactly tex2html_wrap_inline1031 out of tex2html_wrap_inline1027 states are filled; this is a consequence of hypotheses 1 and 3. By hypothesis 2, the probability tex2html_wrap_inline1063 that the electron distribution is specified by tex2html_wrap_inline1065 is given by the ratio of tex2html_wrap_inline1067 , the number of such arrangements, to tex2html_wrap_inline1069 , the total number of possible arrangements. By the total probability theorem:
  equation83
Thus, tex2html_wrap_inline1071 is obtained as a weighted average of the occupancy distribution conditioned on the various allowed values of tex2html_wrap_inline1065 .

  hypothesis89

Hypothesis 4 is valid as long as the number of energy levels, states and particles is sufficiently large. This hypothesis permits us to approximate the weighted average formula (1.5) for the (unconditional) occupancy distribution very well with the term corresponding to the most likely arrangement: tex2html_wrap_inline1083 . Thus, we need only determine the most likely arrangement and study its properties in order to accurately characterize the system in general.
Our task now is to compute tex2html_wrap_inline1067 , and to find tex2html_wrap_inline1087 which maximizes it. The number of ways to place tex2html_wrap_inline1031 indistinguishable particles in tex2html_wrap_inline1027 states (hypothesis 3), with no more than one particle in a single state (hypothesis 1), is:
  equation97
and therefore:
  equation105
To simplify the algebra to follow, instead of maximizing tex2html_wrap_inline1067 , we maximize tex2html_wrap_inline1095 ; since the logarithmic function tex2html_wrap_inline1097 is strictly monotonic increasing, the maxima of W and tex2html_wrap_inline1101 occur at the same point tex2html_wrap_inline1087 . Thus:
  equation110

We now invoke Stirling's approximation formula: for large n, tex2html_wrap_inline1107 , where the approximation is valid in the sense that the ratio of both sides approaches 1 as tex2html_wrap_inline1111 . Thus,
  equation117
We apply this approximation:
  equation121
We now maximize (1.10) subject to the constraints that tex2html_wrap_inline1007 and tex2html_wrap_inline1009 are fixed, and that tex2html_wrap_inline1117 . We impose tex2html_wrap_inline1119 and tex2html_wrap_inline1121 using the methods of Lagrange multipliers: find tex2html_wrap_inline1065 and tex2html_wrap_inline1125 such that the function tex2html_wrap_inline1127 :
  equation129
is maximized. We first locate critical points where tex2html_wrap_inline1129 for all i are tex2html_wrap_inline1133 , tex2html_wrap_inline1135 . The last two simply return the constraints imposed by tex2html_wrap_inline1007 and tex2html_wrap_inline1009 . To compute tex2html_wrap_inline1141 , first note that tex2html_wrap_inline1143 . Thus:
  equation137
Note that tex2html_wrap_inline1141 involves only tex2html_wrap_inline1031 , and not other tex2html_wrap_inline1149 's; the significance of the introduction of the Lagrange multipliers is that the parameters tex2html_wrap_inline1151 and tex2html_wrap_inline1153 that appear in (1.12) are the same for all indices i; specifically, it is important to emphasize that they do not depend on the energy level tex2html_wrap_inline1029 .
Letting tex2html_wrap_inline1159 denote the value of tex2html_wrap_inline1031 at which tex2html_wrap_inline1129 , we obtain:
  equation146
and thus:
  equation161
for some constants tex2html_wrap_inline1125 which do not depend on tex2html_wrap_inline1029 .
As a final remark, we often write tex2html_wrap_inline1169 instead of tex2html_wrap_inline1071 , thus suppressing the discrete nature of the allowed energy levels. This corresponds to either approximating the range of discrete levels with a continuum, considering the small energy difference between the allowed levels, or to extending the results of this discrete analysis to the case where a continuum of energy levels in indeed allowed.

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