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Transport Properties of the Classical Toda Chain: Effect of a Pinning Potential

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Abstract

We consider energy transport in the classical Toda chain in the presence of an additional pinning potential. The pinning potential is expected to destroy the integrability of the system and an interesting question is to see the signatures of this breaking of integrability on energy transport. We investigate this by a study of the non-equilibrium steady state of the system connected to heat baths as well as the study of equilibrium correlations. Typical signatures of integrable systems are a size-independent energy current, a flat bulk temperature profile and ballistic scaling of equilibrium dynamical correlations, these results being valid in the thermodynamic limit. We find that, as expected, these properties change drastically on introducing the pinning potential in the Toda model. In particular, we find that the effect of a harmonic pinning potential is drastically smaller at low temperatures, compared to a quartic pinning potential. We explain this by noting that at low temperatures the Toda potential can be approximated by a harmonic inter-particle potential for which the addition of harmonic pinning does not destroy integrability.

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Acknowledgements

We thank Cédric Bernardin, Stefano Olla, Herbert Spohn, Ovidiu Costin, Rodica Costin and Panayotis Kevrekidis for very useful comments. JAS thanks Mitchell Dorrell for his generous computer programming guidance. The work of JLL was supported by AFOSR Grant FA9550-16-1-0037. JAS was supported by a Rutgers University Bevier Fellowship. AD would like to thank the support from the grant EDNHS ANR-14-CE25-0011 of the French National Research Agency (ANR) and from Indo-French Centre for the Promotion of Advanced Research (IFCPAR) under Project 5604-2.

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Authors and Affiliations

  1. International Centre for Theoretical Sciences - Tata Institute of Fundamental Research Bengaluru, Bengaluru, 560089, India

    Abhishek Dhar & Aritra Kundu

  2. Department of Mathematics, Rutgers University, Piscataway, NJ, 08854, USA

    Joel L. Lebowitz

  3. Department of Physics and Astronomy, Rutgers University, Piscataway, NJ, 08854, USA

    Joel L. Lebowitz & Jasen A. Scaramazza

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  1. Abhishek Dhar
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  2. Aritra Kundu
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  3. Joel L. Lebowitz
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  4. Jasen A. Scaramazza
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Correspondence to Aritra Kundu.

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Communicated by Giulio Biroli.

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Dhar, A., Kundu, A., Lebowitz, J.L. et al. Transport Properties of the Classical Toda Chain: Effect of a Pinning Potential. J Stat Phys 175, 1298–1310 (2019). https://doi.org/10.1007/s10955-019-02284-6

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  • DOI: https://doi.org/10.1007/s10955-019-02284-6

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