The ice model

Introduction

The six-vertex model, which is the object of this chapter, is a special case of the eight-vertex model on a two-dimensional square lattice introduced by Fan and Wu (1970) in order to summarize a class of exactly solvable models in classical statistical mechanics. The thermodynamics of the six-vertex model is by now known in its full generality (Yang and Yang, 1966a–d; Lieb, 1967a) which is not the case for the eight-vertex model, but only the self-conjugate one (Baxter, 1971a).

The general model can be considered as a two-dimensional idealization of a crystalline system in which pairs of adjacent atoms or radicals on the network are linked by ‘hydrogen bonds’. Of ionic type, this link between two neighbouring electronegative atoms is realized by a proton H+ which is located closer to one of the atoms than to the other. On each link of the network there thus exist two equilibrium positions for H+ (Pauling, 1960). If the coordination number at each site equals 4, as is the case for the oxygen atom in hexagonal ice, we indeed have eight possible proton configurations on a given site, which give rise to the eight vertices.

In contrast, for the physical systems for which this model could be viewed as a valid idealization – ice H2O, the ferroelectric PO4H2K, the antiferroelectric PO4H2NH4 – there exist at most two H+ next to each site.

It is my pleasure to welcome this translation of the original French version of my book La fonction d'onde de Bethe into English.

The theory of exactly solvable models, perhaps more than any other subfield of many-body physics, has the distinct advantage of providing solid, reliable and longlasting knowledge. This latter characteristic perhaps explains why my original text, which is by now over three decades old, is still used by members of the scientific community, despite too many mistakes and neglects on my part. This is why the present work of J.-S. Caux is more than a translation, for by his revision he has drawn ‘new from old’ thanks to his style and rigour.

Despite all the developments which the field has known in the intervening period, and which are of course not treated or mentioned here, the fact probably remains that much of what is presented has not been too deprecated in the years since the original version appeared. I am convinced that this translation will bring to a much larger readership an accurate image of the status of the knowledge on these fascinating models at the moment of publication of the original. The initiative and merit belong to the translator, to whom I express my gratitude.

Abstract

Complexity science is the study of systems with many interdependent components. One of the main concepts is “emergence”: the whole may be greater than the sum of the parts. The objective of this chapter is to put emergence on a firm mathematical foundation in the context of dynamics of large networks. Both stochastic and deterministic dynamics are treated. To minimise technicalities, attention is restricted to dynamics in discrete time, in particular to probabilistic cellular automata and coupled map lattices. The key notion is space-time phases: probability distributions for state as a function of space and time that can arise in systems that have been running for a long time. What emerges from a complex dynamic system is one or more space-time phases. The amount of emergence in a space-time phase is its distance from the set of product distributions over space, using an appropriate metric. A system exhibits strong emergence if it has more than one space-time phase. Strong emergence is the really interesting case.

This chapter is based on MSc or PhD courses given at Warwick in 2006/7, Paris in April 2007, Warwick in Spring 2009 and Autumn 2009, and Brussels in Autumn 2010. It was written up during study leave in 2010/11 at the Université Libre de Bruxelles, to whom I am grateful for hospitality, and finalised in 2012.

The chapter provides an introduction to the theory of space-time phases, via some key examples of complex dynamic system.

I am most grateful to Dayal Strub for transcribing the notes into LaTeX and for preparing the figures.

Abstract

Economic behavior and market evolution present notoriously difficult complex systems, where physical interacting particles become purpose-pursuing interacting agents, thus providing a kind of a bridge between physics and social sciences.

We systematically develop the mathematical content of the basic theory of financial economics that can be presented rigorously using elementary probability and calculus, that is, the notions of discrete and absolutely continuous random variables, their expectation, notions of independence and of the law of large numbers, basic integration – differentiation, ordinary differential equations and (only occasionally) the method of Lagrange multipliers. We do not assume any knowledge of finance, apart from an elementary understanding of the idea of compound interest, which can be of two types: (i) simple compounding with rate r and a fixed period of time means your capital in this period is multiplied by (1 + r); (ii) continuous compounding with rate r means your capital in a period of time of length t is multiplied by ert.

This chapter is based on several lecture courses for statistics and mathematics students at the University of Warwick and on invited mini-courses presented by the author at various other places. Sections 6.2 and 6.3 are developed from the author's booklet [9]. The chapter is written in a rather concise (but comprehensive) style in attempt to pin down as clear as possible the mathematical relations that govern the laws of financial economics. Numerous heavy volumes are devoted to the detail discussion of the economic content of these mathematical relations, see e.g. [5], [6], [8], [15], [17].