Abstract
We sketch a new framework for the analysis of disordered systems, in particular mean field spin glasses, which is variational in nature and within the formalism of classical thermodynamics. For concreteness, only the Sherrington–Kirkpatrick model is considered here. For this we show how the Parisi solution (replica symmetric, or when replica symmetry is broken) emerges, in large but finite volumes, from a high temperature expansion to second order of the Gibbs potential with respect to order parameters encoding the law of the effective fields. In contrast with classical systems where convexity in the order parameters is the default situation, the functionals employed here are, at infinite temperature, concave: this feature is eventually due to the Gaussian nature of the interaction and implies, in particular, that the canonical Boltzmann-Gibbs variational principles must be reversed. The considerations suggest that thermodynamical phase transitions are intimately related to the divergence of the infinite expansions.
It is our pleasure to dedicate this work to Anton Bovier on the occasion of his 60th birthday.
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Notes
- 1.
It is known that the solution is unique for small \(\beta \), see e.g. [27, Prop. 1.3.8].
- 2.
We shall mention in passing that these considerations might also have some consequences on the delicate, and to these days still debated validity of the AT-line [5]: as the plots show, there are \((\beta , h)\)-regimes below the AT-line where low temperature behavior is already “hiding” behind the RS-solution. One is thus presumably better off by considering the K-RSB formulation [20] (potentially with \(K=\infty \)), which can still collapse to the RS-value. Indeed, it is known [3] that the full Parisi variational principle is convex in Parisi’s functional order parameter, the latter belonging to the convex set of increasing functions on the unit interval. Since the local minimum of a globally convex functional is also the global minimum, to (dis)prove the validity of the AT-line amounts to (dis)proving that the RS-solution is a local minimum.
- 3.
The (probably deep) reason eludes us: quite simply, regarding the \(\varvec{m}'s\) as fixed parameters leads seamlessly, as we are going to see, to the Parisi 1RSB solution; on the other hand, performing a Legendre transformation also on these parameters yields unwieldy/unrecognizable variational principles, see also the discussion in the last Section.
- 4.
These are presumably also critical points of the 1RSB Gibbs potential, but we haven’t checked that.
- 5.
Thanks to the self-similarity of the building bricks of the Parisi theory an analogous statement is expected for the K-RSB solution as well. The formulas become however so cumbersome that we restrain here from discussing the generic case.
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Acknowledgements
It is our pleasure to thank Hermann Dinges for help and advice on convex geometry. NK also wishes to express his enormous gratitude to Timm Plefka for much needed guidance through the conceptual subtleties of thermodynamics, and for sharing his deep insights on mean field spin glasses. Inspiring discussions with Yan V. Fyodorov are also gratefully acknowledged. Last but not least, our thanks to Pierluigi Contucci, Francesco Guerra, Emanuele Mingione, and Dmitry Panchenko for valuable inputs, criticism, and feedback.
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Kersting, G., Kistler, N., Schertzer, A., Schmidt, M.A. (2019). From Parisi to Boltzmann. In: Gayrard, V., Arguin, LP., Kistler, N., Kourkova, I. (eds) Statistical Mechanics of Classical and Disordered Systems . StaMeClaDys 2018. Springer Proceedings in Mathematics & Statistics, vol 293. Springer, Cham. https://doi.org/10.1007/978-3-030-29077-1_8
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