Series Methods

Abstract

As we saw in Sect. 1.3, the complete solution of a statistical mechanical model is known once an explicit expression has been obtained for the partition function. In this chapter we shall discuss methods which involve the derivation of expansions of partition functions and free energies, or their derivatives, in series of powers of temperature-dependent variables. These series are called low temperature, or high temperature, respectively according to whether the expansion variable tends to zero as T → 0 or T → ∞. If an expression for the coefficient of the general term were known then a series would constitute an exact solution. However, the phrase ‘series methods’ refers primarily to the term-by-term calculation of as many coefficients as possible and the deduction of results for the model from the behaviour of a limited number of coefficient values. Low- and high-temperature series expansions provide approximations to the thermodynamic functions of a model in their respective ranges of temperature. They are, however, more often used to study critical properties, specifically to obtain critical points and exponents. Suppose that

$$ f(\zeta ) = \sum\limits_{n = 0}^\infty {{a_n}{\zeta ^n}} $$

((7.1))

is some thermodynamic function of the model of interest, with ζ a function of temperature appropriate to high-temperature or low-temperature series.