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. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
and
thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
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, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
S is then given by the
natural logarithm
The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
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 thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
References
Thermodynamics
{{thermodynamics-stub