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Thermalization of Local Observables in the \(\alpha \)-FPUT Chain

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Abstract

Most studies on the problem of equilibration of the Fermi–Pasta–Ulam–Tsingou (FPUT) system have focused on equipartition of energy being attained amongst the normal modes of the corresponding harmonic system. In the present work, we instead discuss the equilibration problem in terms of local variables, and consider initial conditions corresponding to spatially localized energy. We estimate the time-scales for equipartition of space localized degrees of freedom and find significant differences with the times scales observed for normal modes. Measuring thermalization in classical systems necessarily requires some averaging, and this could involve one over initial conditions or over time or spatial averaging. Here we consider averaging over initial conditions chosen from a narrow distribution in phase space. We examine in detail the effect of the width of the initial phase space distribution, and of integrability and chaos, on the time scales for thermalization. We show how thermalization properties of the system, quantified by its equilibration time, defined in this work, can be related to chaos, given by the maximal Lyapunov exponent. Somewhat surprisingly we also find that the ensemble averaging can lead to thermalization of the integrable Toda chain, though on much longer time scales.

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Notes

  1. Note that for each fixed time t, the set \(\left\{ f_i(t) \right\} \) defines a discrete probability distribution over the set \(\left\{ 0,1,\dots ,N-1\right\} \), and then S(t) is just the information entropy.

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Acknowledgements

SG would like to thank Hemanta Kumar for help with computations and Varun Dubey for useful discussions. The numerical simulations were done on Contra, Mario and Tetris computing clusters of ICTS-TIFR.

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Authors and Affiliations

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

    Santhosh Ganapa, Amit Apte & Abhishek Dhar

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  1. Santhosh Ganapa
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  2. Amit Apte
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  3. Abhishek Dhar
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Correspondence to Santhosh Ganapa.

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Communicated by David Ruelle.

Dedicated to Joel Lebowitz, to thank him for being a constant source of new ideas, for explaining Boltzmann, and for his warmth and kindness.

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Ganapa, S., Apte, A. & Dhar, A. Thermalization of Local Observables in the \(\alpha \)-FPUT Chain. J Stat Phys 180, 1010–1030 (2020). https://doi.org/10.1007/s10955-020-02576-2

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