Principle of Equipartition of Energy

The energy ε of a particle is a function of all the coordinates of the cell in phase space in which it is located. Let z represent any arbitrary coordinate and ε_z, the energy associated with that coordinate.

(5.185)

To get the distribution in z-coordinate, we will integrate the equation over all the coordinates except z.

(5.186)

where C is a constant

(5.187)

Let  = mean energy of a single particle associated with coordinate z

(5.188)

If ε_z is a quadratic function of z, i.e.ε_z= az², where a is a constant,

(5.189)

Let

Numerator

Denominator

(5.190)

If the energy associated with a particular coordinate is a quadratic function of that coordinate, the average energy of a particle is .This is the principal of equipartition of energy.