The symbol W in the Boltzmann formula engraved into the Boltzmann tombstone in the 1930s. [2]
In statistical thermodynamics, multiplicity W, from the German Wahrscheinlichkeit (war-shine-leash-kite), meaning probability, is the number of quantum states associated with one particular macroscopic thermodynamic state, defined typically by two state parameters, such as volume and energy, of a Boltzman-type system, containing a larger number of particles N, with non-correlated velocities. [1]

In simpler terms, W is the number of different ways P particles can be distributed in a system composed of N different, but connected, compartments. [5]

To note, there is not a good English translation equivalent of the German word Wahrscheinlichkeit, wherein the direct translation is "probability", which does not seem to connote its original meaning as was was first developed in the form of an H theorem, in 1872, by Austrian physicist Ludwig Boltzmann; hence the W goes by variants including "multiplicity", "complexions" P, as well as "permutations" Ω, or "thermodynamic probability, depending on the discussion. The symbol W, as used in the engraved formula, seems to have introduced by Max Planck in 1901. [4]

Formulations
There are a number of similar but different formulas (and accompanying explanations) given for this "multiplicity", or probability, as it was called by Boltzmann, each of which seems to differ details, depending on the components of the system, e.g. atoms, molecules, bosons, etc. [1]

In 1969, American chemical engineer Linus Pauling gave the following expression for W:

$W = \frac{n^N}{\prod_{j=1}^\infty N_j !} \,$

where n refers to a molecular quantum state and where is $\prod \,$ product function, signifying the result of the product of Nj factorial, where N is a number of identical molecules, and Nj is the number of molecules in a set of n molecular quantum states with energy close to Ej. [1]

In 2003, Danish chemical-physicist John Avery gave the following expression for W: [6]

$W = \frac{N!}{n_1 ! n_2 ! n_3 ! \ldots n_i ! \ldots } \,$

where N is a number of identical weakly-interacting systems and ni is the number of the systems that occupy a particular state. [6]

In 1999, American physicist Ralph Baierlein gives the following definition of entropy:

$S \equiv k \ln (multiplicity) \!$

and defines the change in entropy ΔS = Sf – Si in a process, such as between the entropy of ice as compared to water, as being a function of the difference between the two multiplicities of the two states:

$S = k \ln (multiplicity_f) - k \ln (multiplicity_i) \!$
or
$S = k \ln \frac{multiplicity_f}{multiplicity_i} \!$

and then defines the multiplicity of a macrostate to be the number of microstates that corresponds to the microstate. He uses a four ball (labeled A, B, C, D) two bowl (L, R) example, and tabulates the possible unique arrangements, to conclude that the multiplicity, or the different ways in which one can apportion the four balls to the two bowls, is 16 (1 for all balls in L bowl; 4 for three balls in L bowl; 6 for two balls in each bowl; 4 for three balls in R bowl, and 1 for four balls in R bowl). Hence, a four particle, two-container system is said to have a multiplicity of sixteen. [7] It is difficult, however, to see how the number sixteen is arrived at using one of the above expressions for W?

Two-container/eight-particle example
To go through a simple example, given by Belgian thermodynamicist Ilya Prigogine, suppose we have eight particles, N = 8, contained in a two-compartment system, such as two spherical gas bulbs connected via an open stopcock. The problem, then, is to find the probability of the various possible distributions of particles between the two compartments. There is, for instance, only way of placing the eight particles in a single half. If we assume the particles to be distinguishable, there are eight different ways of putting one particle in one half and seven in the other. Equal distribution of the eight distinguishable particles between the two halves can be done in:

$\frac{8!}{4!4!} = 70 \,$

different ways. Prigogine, however, stops here and does not go on to calculate the total number of complexions or multiplicity for this system [5]

Orbital description
In molecular orbital theory, multiplicity is a quantity used in atomic spectra to describe the energy levels of man-electron atoms characterized by Russell-Saunders coupling given by 2S+1, where S is the total electron spin quantum number. The multiplicity of an energy level is indicated by a left superscript to the value of L, where L is the resultant electron orbital momentum of the individual electron orbital angular momenta l. [3]

References
1. Pauling, Linus. (1969). Chemistry (ch. 10.4: Entropy: the Probable State of an Isolated System, pgs. 350-54). Dover.
2. Muller, Ingo. (2007). A History of Thermodynamics - the Doctrine of Energy and Entropy (pg. 102). New York: Springer.
3. Daintith, John. (2004). Oxford Dictionary of Science. Oxford University Press.
4. Planck, Max. (1901). "On the Law of Distribution of Energy in the Normal Spectrum". Annalen der Physik, vol. 4, p. 553 ff.
5. Prigogine, Ilya. (1984). Order Out of Chaos – Man’s New Dialogue with Nature (pg. 123). New York: Bantam Books.
6. Avery, John (2003). Information Theory and Evolution (pg. 77). New Jersey: World Scientific.
7. Baierlein, Ralph. (1999). Thermal Physics, (pg. 27). Cambridge University Press.