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Multiplicity (statistical mechanics)

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In statistical mechanics, multiplicity (also called statistical weight) refers to the number of microstates corresponding to a particular macrostate of a thermodynamic system.[1] Commonly denoted , it is related to the configuration entropy of an isolated system[2] via Boltzmann's entropy formula where is the entropy and is the Boltzmann constant.

Example: the two-state paramagnet

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A simplified model of the two-state paramagnet provides an example of the process of calculating the multiplicity of particular macrostate.[1] This model consists of a system of N microscopic dipoles μ which may either be aligned or anti-aligned with an externally applied magnetic field B. Let represent the number of dipoles that are aligned with the external field and represent the number of anti-aligned dipoles. The energy of a single aligned dipole is while the energy of an anti-aligned dipole is thus the overall energy of the system is

The goal is to determine the multiplicity as a function of U; from there, the entropy and other thermodynamic properties of the system can be determined. However, it is useful as an intermediate step to calculate multiplicity as a function of and This approach shows that the number of available macrostates is N + 1. For example, in a very small system with N = 2 dipoles, there are three macrostates, corresponding to Since the and macrostates require both dipoles to be either anti-aligned or aligned, respectively, the multiplicity of either of these states is 1. However, in the either dipole can be chosen for the aligned dipole, so the multiplicity is 2. In the general case, the multiplicity of a state, or the number of microstates, with aligned dipoles follows from combinatorics, resulting in where the second step follows from the fact that

Since the energy U can be related to and as follows:

Thus the final expression for multiplicity as a function of internal energy is

This can be used to calculate entropy in accordance with Boltzmann's entropy formula; from there one can calculate other useful properties such as temperature and heat capacity.

References

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  1. ^ a b Schroeder, Daniel V. (1999). An Introduction to Thermal Physics (First ed.). Pearson. ISBN 9780201380279.
  2. ^ Atkins, Peter; Julio de Paula (2002). Physical Chemistry (7th ed.). Oxford University Press.