The multiplicity function for a two state paramagnet, W(n,N), is the number of spin states such that n of the N spins point in the z-direction. This function is given by the combinatoric function C(N,n). That is:




It is primarily used in introductory statistical mechanics

Statistical mechanics

Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

 and thermodynamics

Thermodynamics

Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

 textbooks to explain the microscopic definition of entropy to students. If the spins are non-interacting, then the multiplicity function counts the number of states which have the same energy in an external magnetic field. By definition, the entropy

Entropy

Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

 S is then given by the natural logarithm

Natural logarithm

The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

 of this number:




Where k is the Boltzmann constant