Abstract
Thermodynamic variables fall into two classes, intensive and extensive and intensive variables again are of two types, fields such as the temperature\(^1\) \(\widetilde{T}\), the pressure \(\widetilde{P}\), the magnetic field \(\widetilde{\mathcal {H}}\) and the chemical potential \(\mu \), and densities which are defined below.
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Notes
- 1.
The reason for attaching the tilde symbol to these variables (apart from \(\mu \) and \(M\)) is that it is convenient to define all fields so that they have the dimensions of energy and extensive variables so that they are dimensionless; \(\mu \) and \(M\) are already of these types.
- 2.
To distinguish the use of the equals sign to signify a definition, as distinct from a derived equality, we use the symbol ‘\(:=\)’ to represent the former.
- 3.
In this context it is interesting to note the work of Underwood, Sutton and de Podesta at the NPL (de Podesta 2013) which aims to set a new standardised temperature scale by measuring, through a determination of the speed of sound in a monotonic gas at low densities, the kinetic energy per particle and thus \(k_{{{\text {B}}}}\widetilde{T}\) for the gas.
- 4.
In this formula the contributions to the summation are assumed to be such that each field \(\xi _i\) causes an increase in the corresponding extensive variable \(Q_i\). As we shall see in Sect. 1.4 this is the case for a magnetic system, when \(\xi _i=\mathcal {H}\) and \(Q_i=\mathcal {M}\). However, for a fluid system the pressure \(\xi _i=P\) leads to a decrease in volume \(Q_i=V\). Thus the contribution to (1.1.3) will in this case be \(-P\mathrm{{d}}V\).
- 5.
Because of the various choices possible for the independent variables in any thermodynamic system, it is often necessary to specify, using subscripts, the variable or variables kept constant during a partial differentiation. In general this is not necessary for free energies which are defined to be functions of a specific set of variables.
- 6.
For enthalpy and free energies and later in the book for Hamiltonians and partition functions the subscript \(\eta \) (or its particular value) is used to indicate the number of external fields (excluding the temperature). In later chapters where there is no possibility of ambiguity this subscript is omitted.
- 7.
When we consider lattice systems with a dipole on each lattice site \(M=N\) the number of lattice sites.
- 8.
That the field must correspond to minus the pressure has already been noted in Sect. 1.1.
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Lavis, D.A. (2015). Thermodynamics. In: Equilibrium Statistical Mechanics of Lattice Models. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9430-5_1
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DOI: https://doi.org/10.1007/978-94-017-9430-5_1
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