Abstract
The paper aims at characterising and documenting a fundamental change in how phase transitions were modelled microscopically in the period 1937–1970. At first, physicists took what will be called a naturalistic approach to phase transitions such as the condensation of gases and the Curie point of ferromagnets. Here the purpose was to explain the phenomenon in question, i.e., to show that a model exhibits the same features as the phenomenon. The scope of this approach was broad, as the goal was to account for several aspects of the phenomenon. The employed model should be very realistic and close to the foundational theory, be it classical or quantum mechanics. In the 1960s, the physicists used an alternative approach that they termed a caricature approach. This approach not only required explanation in the above sense but also understanding of the physical phenomenon, i.e. insights into why the phenomenon behaves as it does. The scope was limited to certain aspects of the phenomenon, such as the behaviour near the critical point. The caricature approach used a hierarchy of models, ranging from realistic ones over more simplified models to models that were mere caricatures of the system in question. Hence, the two approaches represent very different orientations when it comes to the purpose and scope, the organisation of the resulting theories, and what models are acceptable.
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Notes
In 1950, Shigetoshi Katsura and Hisaaki Fujita echoed this description of the program [Katsura and Fujita [18]].
Mayer et al. had to make some inessential limitations as well, which can be removed at the cost of a more complicated examinations: a) The molecules are identical; b) the number of molecules N is so large that terms of the order \(\mathrm {N}^{\mathrm {-1/3}}\) can be ignored compared to unity; and c) the only degrees of freedom of the molecules are the translational ones.)
Published the following year as [Fisher [8]].
The model was introduced under the heading ‘simple models’ along with the Heisenberg model for magnetism and he wrote that “the classical continuum gas model is probably the most realistic.” [Fisher [8], p. 27].
Temperley wrote for instance that the interaction of another model “is a crude approximation to a Lennard-Jones type of interaction” [Temperley [32], p. 234]
C.N. Yang and T. D. Lee proposed a model in 1952, that was close to Mayer’s model [Yang and Lee [35]]. Benjamin Widom [Widom [34]] published a paper in 1957, where he examined a model like Mayer’s (though he added further approximations). Another example is Temperley’s influential book of 1956 on phase transitions. Here he reviewed “five possible liquid models,” [Temperley [32], p. 52.] and concluded that the one of the most promising lines of approach seem to Mayer-type theory.
The paper seems to have been written some years earlier.
It should be noted, however, that the details of their endeavors differed in that Domb and Miedema, in contrast to Temperley, did not (and perhaps did not want to) decide between more complex theories dealing with iterant nature of the electrons.
The title of his dissertation of 1967 at Harvard, “Critical Phenomena in Heisenberg Models of Magnetism,” reveals his preoccupation with the Heisenberg models, which are general spin models with nearest-neighbour interaction. These models were the subject of a series of papers that he wrote from the middle of the 1960s to the early 1970s. Their overall theme was to determine, in the words of one of the titles, the “[d]ependence of critical properties of Heisenberg magnets upon spin and lattice”. In the very early 1970s he turned the issue raised in the above title from a question that was interesting but of limited scope, into a question of central importance for the theory of critical phenomena.
These factors, which concerned the properties of the interactions, were (i) the uniformity of the direction of the interaction between nearest-neighbours (ii) the range of interaction (what happens if further than nearest neighbours are taken into account?); and (iii) the symmetry of the interaction between two spins (what happens if there is an asymmetry, for instance in the z-component of the interaction?).
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Niss, M. How to model phase transitions? The changing approaches 1937–1970. EPJ H 46, 25 (2021). https://doi.org/10.1140/epjh/s13129-021-00024-7
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DOI: https://doi.org/10.1140/epjh/s13129-021-00024-7