The Boltzmann distribution, formulated in 1868 by Ludwig Boltzmann, describes the probability distribution 
of objects (particles or oscillation modes) in a system over various energy states, 
.
It is mathematically expressed as:
where
is the Boltzmann constant.
is the number of objects in the energy state .
is the total number of objects in the system.
The derivation of the Boltzmann distribution equation involves the following steps:
-
- Derivation of the total differential of
.
- Application of the Lagrange method of undetermined multipliers on the total differential of .
- Simplification of solution using Stirling’s approximation.
- Evaluation of
.
Step 1
Consider a system with molecules randomly occupying different energy states, . At any time, the configuration of the system can be represented by 
with 
molecules in energy state 
, 
molecules in energy state 
, and so on. The total number of molecules is therefore:
where is the number of molecules in the energy state .
The number of ways, 
, to achieve an instantaneous configuration of is given by the combinatorial mathematics of
or in the natural logarithmic form:
Question
Eq2 implies that the molecules are distinguishable. Why?
Answer
Boltzmann statistics is rooted in classical physics, where particles of the same type are considered distinguishable because they can be differentiated by their physical states, such as position and velocity. In contrast, particles of the same type in quantum mechanics are considered indistinguishable, which leads to different statistical distributions like Fermi-Dirac statistics for fermions and Bose-Einstein statistics for bosons.
As the number of molecules in each energy state varies with time, the configuration of the system changes and so does the number of ways of achieving the new configurations. We can therefore express the LHS of eq3 in its total differential form of
_{all\;other\;n\neq n_i}dn_i})
, or for simplicity:
dn_i\;\;\;\;\;\;\;\;4)
Let’s further define our system as a closed system with total energy, 
, given by:
Eq5 restricts the number of configurations of the system. For example, the configurations of 
and 
cannot coexist as they have different total energies. If we assume, under the conditions imposed by eq1 and eq5, that all possible configurations of the system have the same probability of occurring, the configuration with the maximum number of ways of achieving will most likely be the one the system adopts.
Step 2
The most probable configuration is found by evaluating the maximum point for the function in eq4, i.e.
dn_i=0\;\;\;\;\;\;\;\;6)
To solve eq 6, we employ the Lagrange method of undetermined multipliers. We begin by rearranging eq1 to 
, where 
is dependent on the rest of the variables, which are all independent. Eq1 can also be written in the form of a new function, :
Likewise, by rearranging eq5 to })
we have another dependent variable, 
and another function:
This results in eq6 having two dependent variables. The total differential of 
and 
are
dn_i=0\;\;\;\;\;\;\;\;9)
and
dn_i=0\;\;\;\;\;\;\;\;10)
respectively.
Since 
, we can multiply eq9 and eq10 by the factors 
and 
respectively and add them to eq6 to give:
+\alpha\biggr\(\frac{\partial g}{\partial n_i}\biggr\)-\beta\varepsilon_i\biggr\(\frac{\partial h}{\partial n_i}\biggr\)\biggr\]dn_i=0\;\;\;\;\;\;\;\;11)
The factors, and , are called Lagrange multipliers. Two of the variables, e.g. and , are dependent variables, while the rest are independent variables. If there is some value of and some value of that render the 
-th and -th terms of eq11 zero, we have
+\alpha\biggr\(\frac{\partial g}{\partial n_j}\biggr\)-\beta\varepsilon_j\biggr\(\frac{\partial h}{\partial n_j}\biggr\)=0\;\;\;\;\;\;\;\;12)
+\alpha\biggr\(\frac{\partial g}{\partial n_k}\biggr\)-\beta\varepsilon_k\biggr\(\frac{\partial h}{\partial n_k}\biggr\)=0\;\;\;\;\;\;\;\;13)
Consequently, we are left with all independent variables terms. 
in eq11 can now vary arbitrarily, which implies that all the remaining coefficients equal to zero. Substituting eq9 and eq10 in eq11,
dn_i+\alpha dg-\beta dh=0\;\;\;\;\;\;\;\;14)
Noting that and are constants, eq7 and eq8 become 
and 
respectively. Substituting 
and 
in eq14,
+\alpha-\beta\varepsilon_i\biggr\] dn_i=0\;\;\;\;\;\;\;\;15)
Since all coefficients are now equal to zero,
+\alpha-\beta\varepsilon_i=0\;\;\;\;\;\;\;\;16)
Step 3
To simplify eq16, we take the natural logarithm on both sides of eq3 to give })
. Since \cdots]=lnN+ln(N-1)+\cdots=\sum_{K=1}^NlnK})
,
\;\;\;\;\;\;\;\;17)
For large , we have 
. Integrating by parts, 
. Hence, 
, which is known as Stirling’s approximation. Eq17 becomes 
. Since, 
,
Substituting eq18 in eq16,
where we have changed the summation index from 
to in eq19 to discriminate the summation variable from the differentiation variable.
Since 
, we have 
. By implicit differentiation, 
and 
. Furthermore, 
. Therefore, eq19 becomes,
Substituting eq20 in eq1, we have 
, which when substituted in eq20 gives
Step 4
An easy way to determine the value of is to use the equation for the distribution of molecules of an ideal gas in a cylinder:
where
is number of molecules at a height 
, which implies that is number of molecules in energy state 
.
is the number of molecules at a height , where 
. It follows that is number of molecules in energy state 
.
is the mass of a molecule.
is the acceleration due to gravity.
is the difference in height between and .
Since 
represents the difference in energy between states and , we can rewrite eq22 as:
From eq21, the fractions of molecules in energy states and are 
and 
respectively. Dividing 
by 
,
}\;\;\;\;\;\;\;\;24)
Comparing eq23 and 24, 
. Therefore, eq21 becomes
which is the Boltzmann distribution.
The Boltzmann distribution is used to derive mathematical expressions of many scientific concepts, including the statistical entropy, the Maxwell-Boltzmann distribution, and the Planck radiation law.