Abstract
The principle of maximum entropy has recently been applied to several problems of condensed matter theory. In this paper we discuss some technical aspects of the maxent approach to these problems, and show some general properties of the applications of the method. In particular, we show that maxent can be thought of as a convenient way to close hierarchies, and to extrapolate perturbation series for quantities of physical interest. An illustration of this viewpoint is provided by an examination of the dynamics of a quantum mechanical spin system. We discuss a general maxent method for the extrapolation of power series, and apply the method both to problems of condensed matter (a virial equation of state and spin resonance problems), and to a classic example of a difficult series to handle: the anharmonic quantum oscillator with octic perturbation. We show that the inclusion of information beside Taylor coefficients is critical to obtaining a satisfactory extrapolation for the divergent perturbation series. A general maxent criterion is proposed for optimal series extrapolation.
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References
Abragam, A.: 1961, Principles of Nuclear Magnetism, Oxford, Clarendon.
Akhiezer, N.I.: 1965, The Classical Moment Problem, Oliver and Boyd, Edinburgh, 74.
Baker, G., G. Gutierrez, and M. de Llano: 1984, Ann. Phys. 153, 283.
Bender, C.M. and S. Orszag: 1978, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York.
Bender, C.M., L.R.. Mead, and N. Papanicolaou: 1987, J. Math. Phys. 28, 1016.
Bretthorst, G.L.: 1987, (unpublished).
Collins, R. and A. Wragg: 1977, J. Phys. A 10, 1441.
Courant, R. and D. Hilbert: 1953, Methods of Mathematical Physics, Vol. 1, Interscience, New York.
Drabold, D.A., A.E. Carlsson, and P.A. Fedders: 1989, in Maximum Entropy and Bayesian Methods, J. Skilling (ed.), Kluwer, Dordrecht, Holland.
Fedders, P.A. and A.E. Carlsson: 1985, Phys. Rev. B 32, 229.
Grandy, W.T.: 1988, Foundations of Statistical Mechanics, I, Reidel, Dordrecht, Holland, Chapter 7.
Gull, S.: 1989, in Maximum Entropy and Bayesian Methods, J. Skilling (ed.), Kluwer, Dordrecht, Holland.
Hioe, F.T., D. MacMillen, and E.W. Montroll: 1976, J. Math. Phys. 17, 1320.
Jaynes, E.T.: 1957, Phys. Rev. 106, 620.
Jaynes, E.T.: 1968, IEEE Transactions on System Science and Cybernetics SSC-4, 227.
Jaynes, E.T.: 1978, in The Maximum Entropy Formalism, R.D. Levine and M. Tribus (eds.), MIT Press, Cambridge. The articles of E.T. Jaynes cited here may also be found in: Jaynes, E.T.: 1983, Papers on Probability, Statistics and Statistical Physics, Reidel, Dordrecht, Holland.
Kubo, R., M. Toda, and N. Hashitsume: 1985, Statistical Mechanics II: Nonequilibrium Statistical Mechanics, Springer, Berlin.
Mead, L.R. and N. Papanicolaou: 1984, J. Math. Phys. 25, 2404.
Ree, F.H. and W.G. Hoover: 1967, J. Chemical Phys. 46, 4181.
Skilling, J.: 1989, in Maximum Entropy and Bayesian Methods, J. Skilling (ed.), Kluwer, Dordrecht, Holland.
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© 1991 Springer Science+Business Media Dordrecht
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Drabold, D., Jones, G. (1991). Maximum Entropy in Condensed Matter Theory. In: Grandy, W.T., Schick, L.H. (eds) Maximum Entropy and Bayesian Methods. Fundamental Theories of Physics, vol 43. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3460-6_8
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DOI: https://doi.org/10.1007/978-94-011-3460-6_8
Publisher Name: Springer, Dordrecht
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