Abstract
The formalism described in Chap. 1 can be regarded as a macroscopic perspective on a system with a microscopic structure consisting of a collection of atoms or molecules of one or more types.
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Notes
- 1.
In cases where we wish to use the same letter for a thermodynamic variable and its classical statistical mechanical counterpart we make the distinction by adding a ‘hat’ to the statistical mechanical variable. In particular this relationship applies to the enthalpy \(H_\eta \) and the Hamiltonian \(\widehat{H}_\eta \) (Eq. 2.1.7).
- 2.
For a comment concerning the usage of the subscript \(\eta \), specifying the number of independent fields (and hence the number of independent extensive variables, or densities) see the footnote on page 7. For the sake of brevity it is convenient in most cases to drop reference to the independent fields and extensive variables as arguments, retaining only \(T\).
- 3.
In this and all subsequent formulae, the summation is over all microstates \({\mathbf {\sigma }}\) compatible with the given values of the independent extensive variables.
- 4.
The different distributions, for different values of \(\eta \) are usually called ensembles. This term is associated with the (relative-frequency flavoured) picture of a collection (or ensemble) of systems whose phase points have the density (2.1.2) in the space of microstates (Gibbs 1902).
- 5.
Although we do need to be careful about the consequences of the difference of sign in the two terms on the right of (2.3.10).
- 6.
Here it is clearer if we explicitly display representative members
and 
of the \(n_\mathrm {e}\) internal couplings (all independent) and the \(\eta \) independent external couplings. - 7.
In later work, when the context makes it unnecessary we shall omit the qualification ‘dimensionless’.
- 8.
Of course, all the extensive quantities in the system, including the partition function and Hamiltonian are functions of \(N\). However, it, or the lattice signifier \({\mathcal {N}}\), will be displayed explicitly only when it seems useful to do so. In the case of densities like \(\phi \) it is also sometimes useful to indicate the underlying lattice by including the infinite lattice signifier \({\mathcal {L}}\).
- 9.
It is obvious that the Hamiltonian of a pair-interaction model can be represented (in a decomposable way) as a sum over lattice faces and this is often useful in cases where the Hamiltonian also includes an indecomposable face interaction.
- 10.
Or indeed on an arbitrary planar lattice (Baxter 1978) or a three-dimensional lattice.
- 11.
Although, of course, an internal coupling
is simply replaced by a \(\zeta _j\), whereas the coupling corresponding to an external coupling 
is a \(-\zeta _j\).
and 
of the 
is simply replaced by a 
is a Author information
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Lavis, D.A. (2015). Statistical Mechanics. In: Equilibrium Statistical Mechanics of Lattice Models. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9430-5_2
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DOI: https://doi.org/10.1007/978-94-017-9430-5_2
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