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The zero-temperature limit of grand canonical ensembles via tropical geometry

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Abstract

The zero temperature limit of statistical mechanical systems in the grand canonical ensemble is calculated using the methods of tropical algebraic geometry. Particularly, the grand potentials are interpreted as tropical Laurent polynomials in two variables which parametrises the energy and chemical potential of the system. Plotting the roots of these tropical polynomials gives a visual representation of the states of systems at zero temperature. These curves satisfies the balancing condition obeyed by all tropical polynomials. Furthermore, it is proven that the balancing condition continues to hold for tropical ‘polynomials’ with rational exponents. These methods are then applied to various simple physical systems and toy models.

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  • 04 June 2021

    The original online version of this article was revised to update reference 20.

Notes

  1. We shall denote vectors in \(\mathbb {R}^2\) by the notation \(\mathbf {u}=\langle x,y \rangle =x\,\hat{\imath }+y\,\hat{\jmath }\).

  2. A more precise mathematical definition for \(\hat{u}\) is a vector such that there exists a positive real number k with \(\mathbf {u}=k\hat{u}\), where the integer length of \(\hat{u}\) is equal to one (i.e., the cardinality of \([Oz_{\tilde{u}}]\cap \mathbb {Z}^2\)).

  3. Note that \(\varPsi _{\mathrm {trop}}=-\mathcal {N}\psi \) has an overall prefactor \(\mathcal {N}\). Then one has to multiply \(\epsilon _ix\) and \(N_j\) by \(\mathcal {N}\) to obtain the correct total energy and number of particles.

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  1. Department of Mathematics, Xiamen University Malaysia, 43900, Sepang, Malaysia

    Mounir Nisse & Yen-Kheng Lim

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  1. Mounir Nisse
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MN is supported by Xiamen University Malaysia Research Fund (Grant No. XMUMRF/ 2020-C5/IMAT/0013). Y-KL is supported by Xiamen University Malaysia Research Fund (Grant No. XMUMRF/ 2019-C3/IMAT/0007)

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Nisse, M., Lim, YK. The zero-temperature limit of grand canonical ensembles via tropical geometry. Anal.Math.Phys. 11, 113 (2021). https://doi.org/10.1007/s13324-021-00555-8

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