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Amoebas of Complex Hypersurfaces in Statistical Thermodynamics

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Abstract

The amoeba of a complex hypersurface is its image under the logarithmic projection. A number of properties of algebraic hypersurface amoebas are carried over to the case of transcendental hypersurfaces. We demonstrate the potential that amoebas can bring into statistical physics by considering the problem of energy distribution in a quantum thermodynamic ensemble. The spectrum \(\{\varepsilon_k\}\subset \mathbb{Z}^n\) of the ensemble is assumed to be multidimensional; this leads us to the notions of multidimensional temperature and a vector of differential thermodynamic forms. Strictly speaking, in the paper we develop the multidimensional Darwin–Fowler method and give the description of the domain of admissible average values of energy for which the thermodynamic limit exists.

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Authors and Affiliations

  1. Department of Mathematics, Stockholm University, Stockholm, Sweden

    Mikael Passare

  2. Institute of Core Undergraduate Programmes, Siberian Federal University, Krasnoyarsk, Russia

    Dmitry Pochekutov

  3. Institute of Mathematics, Siberian Federal University, Krasnoyarsk, Russia

    August Tsikh

Authors
  1. Mikael Passare
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  2. Dmitry Pochekutov
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  3. August Tsikh
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Corresponding author

Correspondence to Dmitry Pochekutov.

Additional information

The second author was supported by RFBR grant 09-09-00762 and “Möbius Competition” fund for support to young scientists. The third author was supported by the Russian Presidential grant NŠ-7347.2010.1 and by RFBR 11-01-00852.

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Passare, M., Pochekutov, D. & Tsikh, A. Amoebas of Complex Hypersurfaces in Statistical Thermodynamics. Math Phys Anal Geom 16, 89–108 (2013). https://doi.org/10.1007/s11040-012-9122-x

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  • DOI: https://doi.org/10.1007/s11040-012-9122-x

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