Excursions in the Land of Statistical Physics

The following sections are included:

• Preface
• Contents

Michael Fisher spent the first 16 years of his academic life in the Physics Department of King’s College, London, starting as an undergraduate and ending as a full professor. A survey is undertaken of his activities and achievements during the various periods of this phase of his career.

The droplet or cluster theory of condensation is reviewed critically and extended. It is shown to imply that the condensation point is marked by a singularity of the thermodynamic potential as conjectured by Mayer. The singularity turns out to be an essential singularity at which all derivatives of the thermodynamic variables remain finite. The theory also yields an understanding of the uniqueness of the critical point (in contrast to an extended critical region or Derby-hat type of behaviour) and leads to relations between the various critical point singularities.

A one-dimensional model is described with a Hamiltonian containing short-range many-body potentials, The exact solution of the model is sketched and shown to exhibit condensation and critical phenomena for suitable (fixed) potentials. The analysis confirms the conclusions of the cluster theory and thereby lends support to the validity of its underlying assumptions.

Ladies and Gentlemen: I want to start by dedicating this lecture, this talk, which is, in fact, going to be a rather informal discussion, to Lars Onsager, whom I consider the greatest theoretical chemist of this century and who certainly has made the greatest contributions to the area that I am going to be talking about: namely, the states of matter and our theoretical understanding of them. Onsager got the Nobel Prize in chemistry, as most of you will know, in 1968 and died three years ago, on 5 October 1976, still at the height of his powers. He was a major inspiration to my own scientific career…

New results concerning the statistics of, in particular, p random walkers on a line whose paths do not cross are reported, extended, and interpreted. A general mechanism yielding phase transitions in one-dimensional or linear systems is recalled and applied to various wetting and melting phenomena in (d = 2)-dimensional systems, including fluid films and p × 1 commensurate adsorbed phases, in which interfaces and domain walls can be modelled by noncrossing walks. The heuristic concept of an effective force between a walk and a rigidwall, and hence between interfaces and walls and between interfaces, is expounded and applied to wetting in an external field, to the behavior of the twopoint correlations of a two-dimensional Ising model below T_c and in a field, and to the character of commensurate–incommensurate transitions for d = 2 (recapturing recent results by various workers). Applications of random walk ideas to three-dimensional problems are illustrated in connection with melting in a lipid membrane model.

Herman Feshbach, the organizer of this Symposium in honor of Niels Bohr, asked me, in his original invitation, for a review of the present state of condensed matter physics, with emphasis on major unsolved problems and comments on any overlap with Bohr’s ideas regarding the fundamentals of quantum mechanics. That is surely a difficult assignment and, indeed, goes well beyond what is attempted here; nevertheless, I will take the liberty of raising one issue of a philosophical or metaphysical flavor.

Gibbs introduced the fundamental thermodynamic potentials, U(S, V) and G(p, T), and their representation as convex surfaces. The developments and implications of these ideas, in particular as they arise in phase diagrams, are examined. These include: the nonexistence of “metastable extensions” of free energies, violations of the phase rule on critical lines, current concepts for characterizing phases going beyond broken symmetries, especially the response to “twisted” boundary conditions and the relevance to spin glasses, and the importance of dimensionality. Unanticipated singularities in phase diagrams at critical endpoints and multicritical points are remarked and the theory of anomalous first-order transitions at which, e.g., the pressure jumps discontinuously when the volume is changed isothermally, is reviewed. Issues in the thermodynamics and phase transitions of random systems are mentioned.

The essence of simulating criticality in asymmetric fluid models is to discover effective, unbiased finite-size scaling methods that (i) recognize that both the critical temperature, T_c, and the critical density, ρ_c, are unknown and (ii) that can resolve ‘nearby’ critical universality classes. To this end precise, focused simulations for a range of ‘box’ sizes, L^d (in d dimensions), are imperative…

Individual molecular motors, or motor proteins, are enzymatic molecules that convert chemical energy, typically obtained from the hydrolysis of ATP (adenosine triphosphate), into mechanical work and motion. Processive motor proteins, such as kinesin, dynein, and certain myosins, step unidirectionally along linear tracks, specifically microtubules and actin filaments, and play a crucial role in cellular transport processes, organization, and function. In this review some theoretical aspects of motor-protein dynamics are presented in the light of current experimental methods that enable the measurement of the biochemical and biomechanical properties on a single-molecule basis. After a brief discussion of continuum ratchet concepts, we focus on discrete kinetic and stochastic models that yield predictions for the mean velocity, V(F, [ATP],…), and other observables as a function of an imposed load force F, the ATP concentration, and other variables. The combination of appropriate theory with single-molecule observations should help uncover the mechanisms underlying motor-protein function.

The tasks posed for renormalization group theory (RGT) within statistical physics by critical phenomena theory in the 1960’s are set out briefly in contradistinction to quantum field theory (QFT), which was the origin for Ken Wilson’s concerns. Kadanoff’s 1966 block spin scaling picture and its difficulties are presented;Wilson’s early vision of flows is described from the author’s perspective. How Wilson’s subsequent breakthrough ideas, published in 1971, led to the epsilon expansion and the resulting clarity is related. Concluding sections complete the general picture of flows in a space of Hamiltonians, universality and scaling. The article represents a 40% condensation (but with added items) of an earlier account: Rev. Mod. Phys. 70, 653–681 (1998).

Starting from his early years selected contributions of C. N. Yang to statistical mechanics are high-lighted and the physics to which they led is briefly reviewed.