Statistical systems near a classical critical point have been intensively studied from both theoretical and experimental points of view. In particular, correlation functions are of relevance in comparing theoretical models with the experimental data of real systems. In order to compute physical quantities near a critical point, one needs to know the model at the critical (conformal) point. In this line, recent progress in the knowledge of conformal field theories, through the conformal bootstrap, gives the hope of getting some interesting results also outside of the critical point. In this paper, we will review and clarify how, starting from the knowledge of the critical correlators, one can calculate in a safe way their behavior outside the critical point. The approach illustrated requires the model to be just scale invariant at the critical point. We will clarify the method by applying it to different kind of perturbations of the 2D Ising model.
This paper compares theory and experiment for behavior very near critical points. The primary experimental results are the "critical indices" which describe singularities in various thermodynamic derivatives and correlation functions. These indices are tabulated and compared with theory. The basic theoretical ideas are introduced via the molecular field approach, which brings in the concept of an order parameter and suggests that there are close relations among different phase transition problems. Although this theory is qualitatively correct it is quantitatively wrong, it predicts the wrong values of the critical indices. Another theoretical approach, the "scaling law" concept, which predicts relations among these indices, is described. The experimental evidence for and against the scaling laws is assessed. It is suggested that the scaling laws provide a promising approach to understanding phenomena near the critical point, but that they are by no means proved or disproved by the existing experimental data.
The fact noted by Schofield that the specific heat at constant volume, albeit singular, is less singular than the specific heat at constant pressure implies an asymptotic relation between the two-, three-, and four-body distribution functions. The experimental background of this fact is discussed and a fluctuation formula which expresses
CV in terms of two-, three-, and four-body correlation functions is derived. A heuristic explanation of the asymptotic properties of the distribution functions is given based on the fact that local fluctuations proportional to the critical eigenvector are overwhelmingly the most probable near the critical point. It is shown that if
CV is indeed infinite there exists a second critical eigenvector linearly independent of the first. Some consequences of the existence of two critical eigenvectors are discussed, and a form for the short-range behavior of the distribution functions near the critical point is conjectured.