The Maxwell-Boltzmann distribution is a type of probability distribution named after **James Clark Maxwell** and **Ludwig Boltzmann**. It is an integral part of statistical mechanics.

The distribution was first proposed by the Scottish physicist Maxwell in 1859 to describe the **distribution of velocities among the molecules in a gas**.

Later, in 1871, the German physicist Boltzmann generalised Maxwell’s calculations to describe the **distribution of energies in a molecule**.

Index

## Derivation of the Distribution

Consider a system of particles, where the particles are identical but are indistinguishable. For the total **number of particles** being **n**, the **fixed volume** of the system being **V** and the **system’s fixed energy **being **U**, we have to identify the fraction of particles present at a given energy level.

The **energy levels of the system** are fixed. Let the energy levels of the system be **ε _{0}, ε_{2}, ……, ε_{r}**.

The **number of particles in each energy level** can be represented by **n _{0}, n_{1}, ……., n_{r}**.

To move ahead, we need to define a **microstate**.

In physics, a microstate is defined as the **arrangement of each molecule in the system at a given instant**. For each macroscopic property of the system (macrostate), there exist many microstates that make up the macrostate.

The number of ways to attain a given microstate is given by the formula,

**ω= n! / n _{0}!n_{1}!…n_{r}!**

We need to know the values of **n _{i}** that can maximise the value of

**ω**. To maximise this value, we obtain the value of ln(ω).

According to **Sterling’s approximation**,

Thus:

Now, we take the derivative of the above relation to obtain:

Therefore, the sum of all the changes in the system must add up to zero. From this, we can infer that:

The total energy of the system must be constant. Thus:

where **U is a constant**. Thus, on differentiating:

Using Lagrange’s method of undefined multipliers, we obtain the following relation:

For the above relation to be true, the following must hold:

Simplifying the above relation, we get:

The above equation describes the number of particles in the** i ^{th} level**. Taking the sum of the equation over i, the only parameter that is a variable is

**ε**. We obtain:

_{i}Here, we define the partition function P such that:

Substituting the partition function in the above equation, we get the following:

Thus, the number of particles in the **i ^{th} level** becomes:

The value of β in the above equation is **β = 1/(k _{B}T)**, where

**k**is the

_{B}**Boltzmann constant**and

**T**is the

**temperature**of the system. Substituting the value of in the equation, we obtain:

The above relation describes the number of particles in the most probable microstates and is known as the **Maxwell Boltzmann Statistics Expression**.

Take the potential energy to be zero. Thus the only contribution to the total energy U comes from the kinetic energy E. The relationship between kinetic energy and momentum is given by:

where **p ^{2}** is the square of the momentum vector

**p = [p**. Thus, the

_{x}, p_{y}, p_{z}]**Maxwell Boltzmann Statistics Expression can be rewritten as**:

Here, m is the molecular mass of the gas, and T is the thermodynamic temperature (in Kelvin). The distribution of **n _{i}:n** is proportional to the

**probability density function f**for finding a molecule with the given momentum components. Thus:

_{p}The normalising constant can be determined through understanding that the probability of a molecule having *some* momentum must be the** Maxwell Boltzmann Statistics** relation. Integrating the above relation over all p_{x}, p_{y} and p_{z} yields a factor of:

Thus, the normalised function becomes:

From this equation, we can find the energy distribution by substituting the relation between E and p, and then integrating, to obtain:

And the velocity vector distribution can be derived by identifying that the **velocity probability density** **f _{v}** is related to the momentum probability distribution by:

Using **p = mv**, we get:

This is the Maxwell Boltzmann Velocity Distribution function. For the distribution for speed, we obtain speed from the velocity components by:

and the volume elements in the spherical coordinates is given by:

Integration over the **solid angle ** gives an additional factor of **4π**. Thus, the speed distribution is obtained as:

## Typical Speeds

From the properties of **Maxwell Distribution**, we obtain three speed functions:

- The
**mean speed <v>,**or**v**,_{avg} - The
**most probable speed**(mode)**v**,_{p} - The
**root mean squared speed**〈**√v**〉^{2}

### The Most Probable Speed (v_{p})

This is the speed most likely to be possessed by any molecule of the same mass m in the system. It corresponds to the peak (also called mode) of the distribution function.

### The Average Speed (V_{avg})

The average speed is the average of all the speeds of all the molecules. It is the expected value of the speed distribution.

### Root Mean Square Velocity (v_{rms})

The root mean square speed v_{rms} is the square root of the mean square speed, corresponding to the speed of a particle with median kinetic energy.

Thus, we can see that: