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Nonlinearity: Historical and Technological View

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Nonlinearity in Condensed Matter

Part of the book series: Springer Series in Solid-State Sciences ((SSSOL,volume 69))

Abstract

The title of this conference causes us to ask about the meaning of Condensed Matter Physics and of nonlinearity. The ambiguities in Condensed Matter Physics are trivial, and need no belaboring. Every highly developed branch of science has borderline subjects which can be excluded, or included, according to taste. The ambiguity in nonlinear is deeper, and reflects the mode of description. We tend to consider classical motion in a constant, linear, or quadratic potential as linear and refer to anharmonicity when we go beyond that. But the Hamilton-Jacobi equation, even for a free particle, is already nonlinear. On the other hand the Schrödinger equation is always linear, no matter how complex the potential, and as a limiting case can always describe the classical Hamilton-Jacobi equation. The classical Liouville equation is also linear and represents the universal trick for making things look linear, in the case of deterministic problems. We populate the space, in which the solutions evolve, with a density, and all the real complexities are shifted to the description of the motion of a point in this solution space. The kind of linearity obtained via the Liouville equation does not require deterministic behavior, the master equation is also linear, and treats stochastic problems. The low field Hall current, in crossed electric and magnetic fields, represents a linear conductance, if we consider the magnetic field to be a parameter. If, however, we consider electric and magnetic fields as simultaneous components of a total electromagnetic field, then it is clearly nonlinear. Chaos is taken as a symptom of nonlinearity. We can, however, ask about the deviations from a given chaotic solution and return to linearity [1]. It is possible that there is some kind of rational path through this perplexity, and that there is a broadly applicable definition of nonlinearity. I do not know about it, and interpret nonlinear in a more phenomenological sense. Nonlinearity has to do with thresholds, with multi-stability, with hysteresis, with phenomena which change qualitatively as we change excitation.

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  1. IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY, 10598, USA

    R. Landauer

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  1. Center of Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, 87545, NM, USA

    Alan R. Bishop , David K. Campbell , Pradeep Kumar  & Steven E. Trullinger , ,  & 

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Landauer, R. (1987). Nonlinearity: Historical and Technological View. In: Bishop, A.R., Campbell, D.K., Kumar, P., Trullinger, S.E. (eds) Nonlinearity in Condensed Matter. Springer Series in Solid-State Sciences, vol 69. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-83033-4_1

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