Advertisement

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

  • Abhishek Dhar
  • Aritra KunduEmail author
  • Joel L. Lebowitz
  • Jasen A. Scaramazza
Article
  • 16 Downloads

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.

Keywords

Equilibrium and non-equilibrium thermal transport Integrability Non-linearity Lyapunov spectra Heat conduction Thermalization 

Notes

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.

References

  1. 1.
    Lepri, S.: Thermal Transport in Low Dimensions. Springer International Publishing, Berlin (2016)CrossRefGoogle Scholar
  2. 2.
    Bonetto, F., Lebowitz, J.L., Rey-Bellet, L.: Fourier’s law: a challenge to theorists. In: Fokas, A., Grigoryan, A., Kibble, T., Zegarlinski, B. (eds.) Mathematical Physics 2000, pp. 128–150. Imperial College Press, London (2000)CrossRefGoogle Scholar
  3. 3.
    Lepri, S., Livi, R., Politi, A.: Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377, 1 (2003)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Dhar, A.: Heat transport in low-dimensional systems. Adv. Phys. 57, 457 (2008)ADSCrossRefGoogle Scholar
  5. 5.
    Rieder, Z., Lebowitz, J.L., Lieb, E.: Properties of a harmonic crystal in a stationary nonequilibrium state. J. Math. Phys. 8, 1073 (1967)ADSCrossRefGoogle Scholar
  6. 6.
    Spohn, H.: Large Scale Dynamics of Interacting Particles. Springer, Berlin (1991)CrossRefzbMATHGoogle Scholar
  7. 7.
    Zotos, X.: Ballistic transport in classical and quantum integrable systems. J. Low Temp. Phys. 126, 1185 (2002)ADSCrossRefGoogle Scholar
  8. 8.
    Mazur, P.: Non-ergodicity of phase functions in certain systems. Physics 43, 533 (1969)MathSciNetGoogle Scholar
  9. 9.
    Suzuki, M.: Ergodicity, constants of motion, and bounds for susceptibilities. Physica 51, 277 (1971)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Toda, M.: Solitons and heat conduction. Phys. Scr. 20, 424 (1979)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Shastry, B.S., Young, A.P.: Dynamics of energy transport in a Toda ring. Phys. Rev. B 82, 104306 (2010)ADSCrossRefGoogle Scholar
  12. 12.
    Chen, S., Wang, J., Casati, G., Benenti, G.: Nonintegrability and the Fourier heat conduction law. Phys. Rev. E 90, 032134 (2014)ADSCrossRefGoogle Scholar
  13. 13.
    Lepri, S., Livi, R., Politi, A.: Heat conduction in Chains of nonlinear oscillators. Phys. Rev. Lett. 78, 1896 (1997)ADSCrossRefGoogle Scholar
  14. 14.
    Mai, T., Dhar, A., Narayan, O.: Equilibration and universal heat conduction in fermi-pasta-ulam chains. Phys. Rev. Lett. 98, 184301 (2007)ADSCrossRefGoogle Scholar
  15. 15.
    Zhong, Y., Zhang, Y., Wang, J., Zhao, H.: Normal heat conduction in one-dimensional momentum conserving lattices with asymmetric interactions. Phys. Rev. E 85, 060102(R) (2012)ADSCrossRefGoogle Scholar
  16. 16.
    Das, S.G., Dhar, A., Narayan, O.: Heat conduction in the \(\alpha -\beta \) fermi-pasta-ulam chain. J. Stat. Phys. 154, 204 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hatano, T.: Heat conduction in the diatomic Toda lattice revisited. Phys. Rev. E 59, R1 (1999)ADSCrossRefGoogle Scholar
  18. 18.
    Dhar, A.: Heat conduction in a one-dimensional gas of elastically colliding particles of unequal masses. Phys. Rev. Lett. 86, 3554 (2001)ADSCrossRefGoogle Scholar
  19. 19.
    Grassberger, P., Nadler, W., Yang, L.: Heat conduction and entropy production in a one-dimensional hard-particle gas. Phys. Rev. Lett. 89, 180601 (2002)ADSCrossRefGoogle Scholar
  20. 20.
    Casati, G., Prosen, T.: Anomalous heat conduction in a one-dimensional ideal gas. Phys. Rev. E 67, 015203 (2003)ADSCrossRefGoogle Scholar
  21. 21.
    Narayan, O., Ramaswamy, S.: Anomalous heat conduction in one-dimensional momentum-conserving systems. Phys. Rev. Lett. 89, 20 (2002)CrossRefGoogle Scholar
  22. 22.
    Van Beijeren, H.: Exact results for anomalous transport in one-dimensional hamiltonian systems. Phys. Rev. Lett. 108(18), 180601 (2012)CrossRefGoogle Scholar
  23. 23.
    Mendl, C.B., Spohn, H.: Dynamic correlators of fermi-pasta-ulam chains and nonlinear fluctuating hydrodynamics. Phys. Rev. Lett. 111, 230601 (2013)ADSCrossRefGoogle Scholar
  24. 24.
    Spohn, H.: Nonlinear fluctuating hydrodynamics for anharmonic chains. J. Stat. Phys. 154(5), 1191–1227 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Doyon, B., Spohn, H., Yoshimura, T.: A geometric viewpoint on generalized hydrodynamics. Nucl. Phys. B 926, 570–583 (2018)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kundu, A., Dhar, A.: Equilibrium dynamical correlations in the Toda chain and other integrable models. Phys. Rev. E 94, 062130 (2016)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Prosen, T., Zunkovic, B.: Macroscopic diffusive transport in a microscopically integrable Hamiltonian system. Phys. Rev. Lett. 111, 040602 (2013)ADSCrossRefGoogle Scholar
  28. 28.
    Zhang, Z., Tang, C., Kang, J., Tong, P.: Dynamical energy equipartition of the Toda model with additional on-site potentials. Chin. Phys. B 26, 100505 (2017)ADSCrossRefGoogle Scholar
  29. 29.
    Cao, X., Bulchandani, V.B., Moore, J.E.: Incomplete thermalization from trap-induced integrability breaking: lessons from classical hard rods. Phys. Rev. Lett. 120(16), 164101 (2018)ADSCrossRefGoogle Scholar
  30. 30.
    Lebowitz, J.L., Scaramazza, J.A.: Ballistic transport in the classical Toda chain with harmonic pinning, arXiv:1801.07153
  31. 31.
    Di Cintio, P., Lubini, S., Lepri, S., Livi, R.: Transport in perturbed classical integrable systems: the pinned Toda chain. Chaos Solitons Fractals 117, 249–254 (2018)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Toda, M.: Waves in nonlinear lattice. Supp. Prog. Theor. Phys. 45, 174 (1970)ADSCrossRefGoogle Scholar
  33. 33.
    Hénon, M.: Integrals of the Toda lattice. Phys. Rev. B 9, 1921 (1974)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Flaschka, H.: The Toda lattice II: existence of integrals. Phys. Rev. B 9, 1924 (1974)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Lichtenberg, A.J., Lieberman, M.A.: Regular and Chaotic Dynamics, vol. 38. Springer Science Business Media, Berlin (2013)zbMATHGoogle Scholar
  36. 36.
    Das, Avijit, et al.: Light-cone spreading of perturbations and the butterfly effect in a classical spin chain. Phys. Rev. Lett. 121(2), 024101 (2018)ADSCrossRefGoogle Scholar
  37. 37.
    Allen, M.P., Tildesley, D.J.: Computer Simulation of Liquids. Oxford University Press, Oxford (2017)CrossRefzbMATHGoogle Scholar
  38. 38.
    Roy, D., Dhar, A.: Heat transport in ordered harmonic lattices. J. Stat. Phys. 131, 535 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.International Centre for Theoretical Sciences - Tata Institute of Fundamental Research BengaluruBengaluruIndia
  2. 2.Department of MathematicsRutgers UniversityPiscatawayUSA
  3. 3.Department of Physics and AstronomyRutgers UniversityPiscatawayUSA

Personalised recommendations