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The Einstein Relation for the KPZ Equation

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Abstract

We compute the non-universal constants in the KPZ equation in one dimension, in terms of the thermodynamical quantities associated to the underlying microscopic dynamics. In particular, we derive the second-order Einstein relation \(\lambda = \frac{a}{2}\frac{d^2}{d\rho ^2} \chi (\rho ) D(\rho )\) for the transport coefficient \(\lambda \) of the KPZ equation, in terms of the conserved quantity \(\rho \), the diffusion coefficient \(D\), the strength of the asymmetry \(a\) and the static compressibility of the system \(\chi \).

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Notes

  1. We point out that in [1], the authors use generic constants in front of the three terms of this equation, and they do not discuss their meanings in terms of thermodynamical quantities of the underlying systems.

References

  1. Kardar, M., Parisi, G., Zhang, Y.C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889–892 (1986)

    Article  ADS  MATH  Google Scholar 

  2. van Beijeren, H., Kutner, R., Spohn, H.: Excess noise for driven diffusive systems. Phys. Rev. Lett. 54, 2026–2029 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  3. Krug, H., Spohn, H. (eds.): Solids Far from Equilibrium. Cambridge University Press, Cambridge (1991)

  4. Bertini, L., Giacomin, G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183(3), 571–607 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209(2), 437–476 (2000)

    Article  ADS  MATH  Google Scholar 

  6. Baik, J., Deift, P., Johansson, K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. Am. Math. Soc. 12(4), 1119–1178 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Prahofer, M., Spohn, H.: Exact scaling functions for one-dimensional stationary KPZ growth. J. Stat. Phys. 115(1–2), 255–279 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  8. Tracy, C., Widom, H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290(1), 129–154 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. Amir, G., Corwin, I., Quastel, J.: Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions. Commun. Pure Appl. Math. 64(4), 466–537 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Sasamoto, T., Spohn, H.: The one-dimensional KPZ equation: an exact solution and its universality. Phys. Rev. Lett. 104(23), 230602 (2010).

  11. Borodin, A., Petrov, L.: Integrable probability: from representation theory to Macdonald processes. Probab. Surv. 11, 1–58 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  12. I. Corwin, The Kardar-Parisi-Zhang equation and Universality class. Random Matrices. 1, 1–76 (2012).

  13. Spohn, H.: Stochastic integrability and the KPZ equation. Int. Assoc. Math. Phys. News Bull. 5–10 (2012). http://www.iamp.org/bulletins/old-bulletins/201207.pdf.

  14. Nagahata, Y.: The gradient condition for one-dimensional symmetric exclusion process. J. Stat. Phys. 91, 587–602 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  15. Brox, T., Rost, H.: Equilibrium fluctuations of stochastic particle systems: the role of conserved quantities. Ann. Prob. 12(3), 742–759 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  16. De Masi, A., Presutti, E., Spohn, H., Wick, W.D.: Asymptotic equivalence of fluctuation fields for reversible exclusion processes with speed change. Ann. Prob. 14(2), 409–423 (1986)

    Article  MATH  Google Scholar 

  17. Gonçalves, P., Jara, M.: Scaling limits of additive functionals of interacting particle systems. Commun. Pure Appl. Math. 66(5), 649–677 (2013)

    Article  MATH  Google Scholar 

  18. Gonçalves, P., Jara, M.: Crossover to the KPZ equation. Annales Henri Poincaré 13(4), 813–826 (2012)

    Article  ADS  MATH  Google Scholar 

  19. Gonçalves, P., Jara, M.: Nonlinear fluctuations of weakly asymmetric interacting particle systems. Arch. Ration. Mech. Anal. 212(2), 597–644 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  20. P. Gonçalves, M. Jara and S. Sethuraman, A stochastic Burgers equation from a class of microscopic interactions, to appear in Annals of Probability, arXiv:1210.0017.

  21. Bertini, L., Faggionato, A., Gabrielli, D.: Large deviation principles for nongradient weakly asymmetric stochastic lattice gases. Ann. Appl. Probab. 23(1), 1–65 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

PG thanks CNPq (Brazil) for support through the research Project “Additive functionals of particle systems” 480431/2013-2. PG thanks CMAT for support by “FEDER” through the “Programa Operacional Factores de Competitividade COMPETE” and by FCT through the project PEst-C/MAT/UI0013/2011. MJ was funded by FAPERJ “Jovem Cientista do Nosso Estado” with the grant E-26/103.051/2012

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Authors and Affiliations

  1. Departamento de Matemática, PUC-RIO, Rua Marquês de São Vicente, No. 225, Rio de Janeiro, RJ, 22453-900, Brazil

    Patrícia Gonçalves

  2. CMAT, Centro de Matemática da Universidade do Minho, Campus de Gualtar, 4710-057, Braga, Portugal

    Patrícia Gonçalves

  3. IMPA, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, CEP, 22460-340, Brazil

    Milton Jara

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  1. Patrícia Gonçalves
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  2. Milton Jara
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Correspondence to Patrícia Gonçalves.

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Gonçalves, P., Jara, M. The Einstein Relation for the KPZ Equation. J Stat Phys 158, 1262–1270 (2015). https://doi.org/10.1007/s10955-014-1158-9

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  • DOI: https://doi.org/10.1007/s10955-014-1158-9

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