In
statistical mechanics, multiplicity (also called statistical weight) refers to the number of
microstates
A microstate or ministate is a sovereign state having a very small population or very small land area, usually both. However, the meanings of "state" and "very small" are not well-defined in international law.Warrington, E. (1994). "Lilliputs ...
corresponding to a particular
macrostate
In statistical mechanics, a microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations. In contrast, the macrostate of a system refe ...
of a
thermodynamic system
A thermodynamic system is a body of matter and/or radiation, confined in space by walls, with defined permeabilities, which separate it from its surroundings. The surroundings may include other thermodynamic systems, or physical systems that are ...
.
Commonly denoted
, it is related to the
configuration entropy
In statistical mechanics, configuration entropy is the portion of a system's entropy that is related to discrete representative positions of its constituent particles. For example, it may refer to the number of ways that atoms or molecules pack to ...
of an isolated system
via
Boltzmann's entropy formula
In statistical mechanics, Boltzmann's equation (also known as the Boltzmann–Planck equation) is a probability equation relating the entropy S, also written as S_\mathrm, of an ideal gas to the multiplicity (commonly denoted as \Omega or W), the ...
where
is 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 ...
and
is
Boltzmann's 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 ...
.
Example: the two-state paramagnet
A simplified model of the two-state
paramagnet provides an example of the process of calculating the multiplicity of particular macrostate.
[ This model consists of a system of microscopic dipoles which may either be aligned or anti-aligned with an externally applied magnetic field . 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 ; 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 . For example, in a very small system with 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 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
Statistical mechanics
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