The Critical Point: A Historical Introduction to the Modern Theory of Critical Phenomena

The Lee-Yang theory of equilibrium and nonequilibrium phase transitions

“…Clearly this scenario is impossible if the number of zeros M is finite, except at the isolated points z n where the free energy exhibits a logarithmic singularity. Since such a point cannot lie on the positive real z axis, there is no scope for a phase transition in a finite spin system, such as the simple example (1). On the other hand, if the partition function zeros accumulate towards a point z 0 on the real axis as we increase the number of spins N to infinity there is the possibility of a phase transition.…”

“…Finally, one can show that near the intersection point between the curve of zeros in the complex-α plane and the positive real α axis, the zeros of (24) sit on the curve x = 1 2 − (y 2 + 1 4 − ξ c ) 1/2 where x and y are the real and imaginary parts of α. For the case β < 1 2 , ξ c < 1 4 and the transition is first order. One finds that the curve of zeros is smooth at the transition point α = β, and that the density of zeros is (1 − 2β)/[2πβ(1 − β)] which is nonzero.…”

“…The details of the matrix product method are not necessary for the following-suffice to say that one ends up calculating a normalisation proportional to (23) through a product of matrices, often of infinite dimension. In this way one obtains some explicit formulas for the normalisation (22) which we shall use below 1 .…”

“…More specifically, the free energy is given by the logarithm of a partition function, a quantity that normalises the steady-state probability distribution of microscopic configurations. Initially it was not universally accepted that this approach could faithfully describe phase transitions, in particular the first-order solidfluid transition [1]. In order to show that the statistical mechanical approach can reproduce the correct discontinuities in the free energy at a first-order transition, Lee and Yang [2] introduced a description of phase transitions concerning zeros of the partition function when generalised to the complex plane of an intensive thermodynamic quantity.…”

Rigorous results, cross-model justification, and the transfer of empirical warrant: the case of many-body models in physics

“…To paraphrase Domb (1996), individual models may thus be tied into a general methodological framework, whereby they become part of a more general fabric of mathematical and numerical techniques, rather than being considered in isolation. This sense of 'cohesion', however, while no doubt suggestive, may be considered too weak to sanction any specific appeal to individual rigorous results as a source of justification.…”