Having calculated the three overall weight functions for the Maxwell-Boltzmann, Bose-Einstein,
and Fermi-Dirac distribution we now have to calculate the distribution functions themselves. For this we have to answer
the question: What is the most probable configuration?
This question we have to answer taking into
account possible restrictions like conserving the overall energy and/or particle number. Both restrictions represent extensive
parameters (i.e. they scale with the size of the system), which is not true for the weight functions \(W\)
(here the \(\prod_i\) shows up and not \(\sum_i\)). This we can cure by not maximizing \(W\) but \(\ln(W)\). Further justification will be given in the subsequent sections; here it is enough
to state that the entropy \(S := k \ln(W)\) becomes an extensive parameter, i.e. for two independent
occupation weights \(W_1\) and \(W_2\), having a combined weight \(W = W_1 W_2\),
we find the overall entropy
|
| \begin{equation*} S = S_1 + S_2 = k \ln\left(W_1\right) + k \ln\left(W_2\right) = k \ln\left(W_1 W_2\right) = k \ln\left(W\right) \label{S1__S2}\quad. \end{equation*} | (5.11) |
Our mathematical task is thus
Maximize \(k\,\ln\left(W\right)\), i.e. \(k\; d \,\ln\left(W\right) = 0 = \sum_i \left(\frac{\partial \ln\left(W\right)}{\partial n_i}\right) dn_i\)
Under restriction 1: Number of particles \(N = \sum_i n_i\) is constant, i.e. \(\sum_i dn_i =0\)
Under restriction 2: Overall energy \(\epsilon = \sum_i n_i \epsilon_i\) is constant, i.e. \(\sum_i \epsilon_i dn_i =0\)
To incorporate the restrictions into the optimization problem we will introduce Lagrange parameters \(\beta\) and \(\gamma\), leading to the final mathematical problem
In order to find an explicit solution we now have to specify \(W\), which in our example will be the Maxwell-Boltzmann weight of Eq. (5.6). Using the Stirling formula \(\ln(x!) \approx x \ln(x) - x\) we get
|
| \begin{equation*} \ln\left(n_i/g_i\right) + \gamma + \beta \epsilon_i = 0 \quad \mbox{for all $i$,} \end{equation*} | (5.15) |
leading to
|
| \begin{equation*} n_i= g_i e^{ - \gamma - \beta \epsilon_i} \quad . \label{MB_distr} \end{equation*} | (5.16) |
The physical meaning of the two Lagrange parameters can now easily be extracted by including Eq. (5.16) into Eq. (5.14). We get
|
| \begin{equation*} \frac{\partial S}{\partial N} = - \frac{\mu}{T} \quad \mbox{and} \quad \frac{\partial S}{\partial \epsilon} = \frac{1}{T} \quad . \end{equation*} | (5.18) |
Comparison with Eq. (5.17) gives
|
| \begin{equation*} \gamma = - \frac{\mu}{kT} \quad \mbox{and} \quad \beta = \frac{1}{kT} \quad , \end{equation*} | (5.19) |
leading to the well known Maxwell-Boltzmann distribution function
|
| \begin{equation*} n_i= g_i e^{-\frac{\epsilon_i-\mu}{kT}} \quad . \label{MB_distr_2} \end{equation*} | (5.20) |
The two Lagrange parameter can now be determined by fulfilling the restrictions. From restriction 1 we get
|
| \begin{equation*} N = \sum_i n_i = \sum_i g_i e^{ - \gamma - \beta \epsilon_i} = e^{ - \gamma} \sum_i g_i e^{- \beta \epsilon_i} = e^{ - \gamma} Z \quad . \end{equation*} | (5.21) |
So by introducing the partition function
|
| \begin{equation*} Z = \sum_i g_i e^{- \beta \epsilon_i} \label{eq_Z_MB} \end{equation*} | (5.22) |
we get
|
| \begin{equation*} n_i= \frac{N}{Z} g_i e^{- \beta \epsilon_i} \quad . \label{eq_ni_Z_MB} \end{equation*} | (5.23) |
Note: That \(\gamma\) (and thus \(\mu\)) does not show up in
the final results is a common feature of the Maxwell-Boltzmann distribution function and thus of classical particles. Consequently
the canonical ensemble (cf. section 5.4) is typically used to describe systems of classical particles.
From restriction 2 we get
|
| \begin{equation*} \epsilon = \frac{N}{Z} \sum_i g_i e^{- \beta \epsilon_i} \epsilon_i \end{equation*} | (5.24) |
which in an alternative form can be written as
|
| \begin{equation*} \epsilon = - \frac{N}{Z} \frac{dZ}{d \beta} = - N \frac{d}{d \beta} \ln\left(Z\right) \end{equation*} | (5.25) |
So having calculated the partition function \(Z\), the entropy of the system can be calculated. The fundamental meaning of \(\ln\left(Z\right)\) will be discussed in larger detail in the remaining sections. There entropy will be discussed from a more general point of view and makes it unnecessary to solve the maximization problem for the entropy for the Bose-Einstein and Fermi-Dirac distributions separately. Here we will just state the final results:
|
| \begin{equation*} \mbox{Bose-Einstein distribution:} \quad n_i= \frac{g_i}{ e^{\frac{\epsilon_i-\mu}{kT}} - 1} \quad . \label{BE_distr} \end{equation*} | (5.26) |
|
| \begin{equation*} \mbox{Fermi-Dirac distribution:} \quad n_i= \frac{g_i}{ e^{\frac{\epsilon_i-\mu}{kT}} + 1} \quad . \label{FD_distr} \end{equation*} | (5.27) |
|
|
|
|
© J. Carstensen (TD Kin II)