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Spectral Properties of Zero Temperature Dynamics in a Model of a Compacting Granular Column

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Abstract

The compacting of a column of grains has been studied using a one-dimensional Ising model with long range directed interactions in which down and up spins represent orientations of the grain having or not having an associated void. When the column is not shaken (zero “temperature”) the motion becomes highly constrained and under most circumstances we find that the generator of the stochastic dynamics assumes an unusual form: many eigenvalues become degenerate, but the associated multi-dimensional invariant spaces have but a single eigenvector. There is no spectral expansion and a Jordan form must be used. Many properties of the dynamics are established here analytically; some are not. General issues associated with the Jordan form are also taken up.

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Authors and Affiliations

  1. Physics Department, Clarkson University, Potsdam, NY, 13699-5820, USA

    L. S. Schulman

  2. Institut de Physique Théorique, IPhT, CEA Saclay and URA 2306, CNRS, 91191, Gif-sur-Yvette cedex, France

    J. M. Luck

  3. Theory Department, S.N. Bose National Centre, Block JD Sector III, Salt Lake, Calcutta, 700098, India

    Anita Mehta

Authors
  1. L. S. Schulman
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  2. J. M. Luck
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  3. Anita Mehta
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Correspondence to L. S. Schulman.

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Schulman, L.S., Luck, J.M. & Mehta, A. Spectral Properties of Zero Temperature Dynamics in a Model of a Compacting Granular Column. J Stat Phys 146, 924–954 (2012). https://doi.org/10.1007/s10955-012-0429-6

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  • DOI: https://doi.org/10.1007/s10955-012-0429-6

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