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
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
and
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
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 is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
S is then given by the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
of this number:
[Schroeder, Daniel V.. An Introduction to Thermal Dynamics. San Francisco: Addison Wesley Longman 2002.]
Where k is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
References
Thermodynamics
{{thermodynamics-stub