Abstract
The present paper focuses on a particular class of models intended to describe and explain the physical behaviour of systems that consist of a large number of interacting particles. Such many-body models are characterized by a specific Hamiltonian (energy operator) and are frequently employed in condensed matter physics in order to account for such phenomena as magnetism, superconductivity, and other phase transitions. Because of the dual role of many-body models as models of physical systems (with specific physical phenomena as their explananda) as well as mathematical structures, they form an important sub-class of scientific models, from which one can expect to draw general conclusions about the function and functioning of models in science, as well as to gain specific insight into the challenge of modelling complex systems of correlated particles in condensed matter physics. In particular, it is argued that many-body models contribute novel elements to the process of inquiry and open up new avenues of cross-model confirmation and model-based understanding. In contradistinction to phenomenological models, which have received comparatively more philosophical attention, many-body models typically gain their strength not from ‘empirical fit’ per se, but from their being the result of a constructive application of mature formalisms, which frees them from the grip of both ‘fundamental theory’ and an overly narrow conception of ‘empirical success’.
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Notes
- 1.
There may, of course, be independent reasons why one might represent, say, a specific trajectory by a certain set of deterministic equations, or by a (non-mathematical) pictorial representation. However, in such cases, as well as in contexts where the stochasticity of the causal process is irrelevant, one would not be dealing with a model of Brownian motion, in the proposed narrower sense of ‘mathematical model’.
- 2.
Systematic ‘tweaking’, as Martin Krieger observes, “has turned out to be a remarkably effective procedure” (Krieger 1981, p. 428). By varying contributions to the model, e.g., by adding disturbances, one can identify patterns in the response of the model, including regions of stability.
- 3.
Quoted in (Hughes 1999, p. 104).
- 4.
- 5.
The zero-temperature case \((T = 0)\) is of special significance in a variety of many-body models. However, in order to keep the presentation accessible, \(T \ne 0\) will be assumed throughout the following discussion.
- 6.
As Martin Niss notes, during the first decade of the study of the Lenz--Ising model “[c]omparisons to experimental results were almost absent”, and moreover, its initial “development was not driven by discrepancies between the model and experiments” (Niss 2005, pp. 311–312).
- 7.
Needless to say, a considerable number of background assumptions are necessary in order to identify which unit is indeed the smallest one that still captures the basic mechanisms that determine the behaviour of the extended system.
- 8.
Operators should be read from right to left, so if an operator product like \(\hat{a}_{i,\sigma }^{\dag } \hat{a}_{j,\sigma } \left|\Psi \right\rangle\) acts on a quantum state, the operator \(\hat{a}_{j,\sigma }\) directly in front of \(\left|\Psi \right\rangle\) acts first, followed by \(\hat{a}_{i,\sigma }^{\dag } .\) Because operators do not always commute, the order of operation is important.
- 9.
For a discussion of the formalism of creation and annihilation operators as a ‘mature mathematical formalism’, see (Gelfert 2011, pp. 281—282).
- 10.
The notions of ‘theorem’ and ‘rigorous result’ are frequently used interchangeably in scientific texts, especially in theoretical works such as (Griffiths 1972).
- 11.
This is noted in passing, though not elaborated on, by R.I.G. Hughes in his case study of one of the first computer simulations of the Ising model (Hughes 1999, p. 123): “In this way the verisimilitude of the simulation could be checked by comparing the performance of the machine against the exactly known behaviour of the Ising model.”
- 12.
This case of cross-model support between many-body models that were originally motivated by very different concerns is discussed in detail in (Gelfert 2009).
- 13.
As Cartwright argues, it is for this reason that warrant to believe in predictions must be established case by case on the basis of models. She criticizes the ‘vending-machine view’, in which “[t]he question of transfer of warrant from the evidence to the predictions is a short one since it collapses to the question of transfer of warrant from the evidence to the theory”. This, Cartwright writes, “is not true to the kind of effort hat we know it takes in physics to get from theories to models that predict what reliably happens”; hence, “[w]e are in need of a much more textured, and I am afraid much more laborious view” regarding the claims and predictions of science (Cartwright 1999, p. 185).
- 14.
A model whose predictions of the order parameter are systematically wrong (e.g., consistently too low) but which gets the qualitative behaviour right (e.g., the structure of the phase diagram), may be preferable to a model that is more accurate for most situations, but is vastly (qualitatively) mistaken for a small number of cases. Likewise, a model that displays certain symmetry requirements or obeys certain other rigorous relations may be preferable to a more accurate model (with respect to the physical observables in question) that lacks these properties.
- 15.
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Gelfert, A. (2015). Between Rigor and Reality: Many-Body Models in Condensed Matter Physics. In: Falkenburg, B., Morrison, M. (eds) Why More Is Different. The Frontiers Collection. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43911-1_11
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