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Charge-Current Correlation Identities for Stochastic Interacting Particle Systems

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From Particle Systems to Partial Differential Equations (ICPS 2019, ICPS 2018, ICPS 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 352))

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Abstract

We adapt a recent proof of generic quantum-mechanical charge-current correlation identities with local conservation laws to stochastic interacting particle systems. Unlike in earlier proofs no translation invariance is required. We clarify the validity of an Onsager-type current symmetry that generally appears in hyperbolic systems of conservation laws that arise as hydrodynamic limits of stochastic interacting particle systems.

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Notes

  1. 1.

    Solutions of (1) generically develop shocks after a finite time that depends on the initial data [6, 7].

  2. 2.

    (a) Notice the negative sign and the appearance of the time-reversed current in the definition of \(\tilde{C}^{\alpha \beta }_L(k,l,t)\), in contrast to the quantum case [14]. (b) For the distance \(r=l-k\) between two sites the periodicity implies that distance \(r=L-1\) is the same as distance \(r=-1\).

  3. 3.

    We have redefined \(A^{\alpha \beta }_L(k,t)\) by a factor of L compared to [14]. This makes formulas that appear in the proofs neater.

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Acknowledgements

It is a pleasure to acknowledge stimulating discussions with B. Doyon and A. Klümper on charge-current correlations in quantum systems.

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Authors and Affiliations

  1. CNRS, Laboratoire de Physique et Chimie Théoriques, Université de Lorraine, 54000, Nancy, France

    Dragi Karevski

  2. Institut for Biologische Informationsprozesse 5, Theoretische Physik der Lebenden Materie, 52425, Forschungszentrum Jülich, Jülich, Germany

    Gunter M. Schütz

Authors
  1. Dragi Karevski
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  2. Gunter M. Schütz
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Correspondence to Dragi Karevski .

Editor information

Editors and Affiliations

  1. Laboratoire J.A. Dieudonné, UMR CNRS-UNS, Université Côte d’Azur, Nice Cedex 02, France

    Cédric Bernardin

  2. Centre de Math. Laurent Schwartz (CMLS), École Polytechnique, CMLS, Palaiseau Cedex, France

    François Golse

  3. Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal

    Patrícia Gonçalves

  4. Dipartimento di Matematica e Informatica, Università di Palermo, Palermo, Italy

    Valeria Ricci

  5. Centro de Matemática, Universidade do Minho, Braga, Portugal

    Ana Jacinta Soares

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Karevski, D., Schütz, G.M. (2021). Charge-Current Correlation Identities for Stochastic Interacting Particle Systems. In: Bernardin, C., Golse, F., Gonçalves, P., Ricci, V., Soares, A.J. (eds) From Particle Systems to Partial Differential Equations. ICPS ICPS ICPS 2019 2018 2017. Springer Proceedings in Mathematics & Statistics, vol 352. Springer, Cham. https://doi.org/10.1007/978-3-030-69784-6_15

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