Statistical entropy is a concept in thermodynamics that quantifies the number of different ways a system can be arranged while still maintaining its macroscopic properties.

The total energy of a system, , is given by eq5. Its total differential is:

Consider the transfer of heat to a closed system of particles. If the volume of the system is constant, then no work is done on the system and the energy level of each state is unchanged . However, the population of particles in each state varies according to temperature. Hence,

If we interpret as the internal energy of the system, , then according to the first law of thermodynamics of a constant volume system,

For a reversible transfer of heat, the second law of thermodynamics states that . Combining , eq26 and eq27,

Substituting eq16 in eq28,

Since the total number of particles for a closed system is constant, and

Substituting eq4 in eq29, we have or the integrated form: