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Physics Physique Fizika

The theory of condensation and the critical point

Michael Ellis Fisher

Published Oct 1, 1967• 812 citations
Publisher

Abstract

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.

DOI
10.1103/physicsphysiquefizika.3.255

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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.
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The fermionic version of the two-dimensional Ising model in a uniform magnetic field is studied. Its approximate but rather accurate solution was obtained recently in a variational procedure based on the BCS ansatz for the ground state. Here, this solution is continued analytically into the complex H-plane in the vicinity of the origin. In the ferromagnetic phase an essential singularity in the free-energy is observed at the point H=0 of the type predicted by the droplet condensation theory. Simple phenomenological estimates for the radius of the critical droplet and for the rate of decay of metastable states are confirmed by microscopic calculations.
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 difficulty near the critical point encountered by the Landau theory or the classical theory of the second order phase transition is removed with the least modification of the original framework. The essential point of our idea is to note that the temperature region in which the Landau theory valids depends sensitively on the way of defining the local order parameter. It is shown that the relations among the singularities of various thermodynamic quantities predicted by this modified form of the Landau theory are in good agreement with those obtained by nonclassical theories such as the static scaling theory.
The analysis of selected nonlinear problems in the renormalization group is found to show striking contrasts between the usual local linearized fixed-point analysis and the properties of global solutions of nonlinear equations derived from an approximation of the Wegner-Houghton differential formulation. The competition between various fixed points that is incorporated in general global solutions can upset the asymptotically valid critical behavior deduced from the local analysis. In general, the critical-point exponents of such a solution will not satisfy equalities, but rather the corresponding inequalities. However, these nonscaling solutions have extraneous singularities that are not related to the thermodynamic singularities of the system. If singularities of this type are excluded, then the global solution has the same critical-point exponents as the local solution derived by linearizing around the stablest fixed point. It is shown that in this case the critical surface in the Hamiltonian space is closely related to the surface of order-2 critical points in a thermodynamic field space. The boundaries of this surface are correspondingly related to the critical points of higher order in this thermodynamic space. The nonlinear global solution predicts multiple power scaling behavior from a single scaling equation deduced from the renormalization group. Previously such behavior was obtained by postulating the simultaneous validity of two of more "linear" scaling hypotheses.
By an exact calculation on the planar Ising model we show that for all subcritical temperatures, the bubble model predicts the correct average distribution of matter contributing to the pair-spin correlation function in the two-particle sector. This considerably substantiates the bubble model which is generalized to higher dimensions with an external field h and then used to show that the free energy has an essential singularity in h of Andreev-Fisher form at all points along the phase boundary. We argue that such a singularity would also occur in the Baxter model.