Journal of Statistical Physics

, Volume 164, Issue 5, pp 1167–1182 | Cite as

Characteristic Sign Renewals of Kardar–Parisi–Zhang Fluctuations

  • Kazumasa A. Takeuchi
  • Takuma Akimoto


Tracking the sign of fluctuations governed by the \((1+1)\)-dimensional Kardar–Parisi–Zhang (KPZ) universality class, we show, both experimentally and numerically, that its evolution has an unexpected link to a simple stochastic model called the renewal process, studied in the context of aging and ergodicity breaking. Although KPZ and the renewal process are fundamentally different in many aspects, we find remarkable agreement in some of the time correlation properties, such as the recurrence time distributions and the persistence probability, while the two systems can be different in other properties. Moreover, we find inequivalence between long-time and ensemble averages in the fraction of time occupied by a specific sign of the KPZ-class fluctuations. The distribution of its long-time average converges to nontrivial broad functions, which are found to differ significantly from that of the renewal process, but instead be characteristic of KPZ. Thus, we obtain a new type of ergodicity breaking for such systems with many-body interactions. Our analysis also detects qualitative differences in time-correlation properties of circular and flat KPZ-class interfaces, which were suggested from previous experiments and simulations but still remain theoretically unexplained.


Growth phenomenon Scaling laws KPZ universality class Renewal theory Stochastic process Weak ergodicity breaking 



We acknowledge fruitful discussions with E. Barkai, I. Dornic, C. Godrèche, and S. N. Majumdar. This work is supported in part by KAKENHI from JSPS (No. JP25707033 and No. JP25103004), the JSPS Core-to-Core Program “Non-equilibrium dynamics of soft matter and information”, and the National Science Foundation under Grant No. NSF PHY11-25915.


  1. 1.
    Akimoto, T., Miyaguchi, T.: Distributional ergodicity in stored-energy-driven Lévy flights. Phys. Rev. E 87, 062134 (2013)ADSCrossRefGoogle Scholar
  2. 2.
    Barabási, A.L., Stanley, H.E.: Fractal Concepts in Surface Growth. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bouchaud, J.P.: Weak ergodicity breaking and aging in disordered systems. J. Phys. I (France) 2, 1705–1713 (1992)CrossRefGoogle Scholar
  4. 4.
    Bray, A.J., Majumdar, S.N., Schehr, G.: Persistence and first-passage properties in nonequilibrium systems. Adv. Phys. 62, 225–361 (2013)CrossRefGoogle Scholar
  5. 5.
    Brokmann, X., Hermier, J.P., Messin, G., Desbiolles, P., Bouchaud, J.P., Dahan, M.: Statistical aging and nonergodicity in the fluorescence of single nanocrystals. Phys. Rev. Lett. 90, 120601 (2003)ADSCrossRefGoogle Scholar
  6. 6.
    Cakir, R., Grigolini, P., Krokhin, A.A.: Dynamical origin of memory and renewal. Phys. Rev. E 74, 021108 (2006)ADSCrossRefGoogle Scholar
  7. 7.
    Carrasco, I.S.S., Takeuchi, K.A., Ferreira, S.C., Oliveira, T.J.: Interface fluctuations for deposition on enlarging flat substrates. New J. Phys. 16, 123057 (2014)ADSCrossRefGoogle Scholar
  8. 8.
    Corwin, I.: The Kardar–Parisi–Zhang equation and universality class. Random Matrices Theory Appl. 1, 1130001 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cox, D.R.: Renewal Theory. Methuen, London (1962)zbMATHGoogle Scholar
  10. 10.
    Ding, M., Yang, W.: Distribution of the first return time in fractional brownian motion and its application to the study of on-off intermittency. Phys. Rev. E 52, 207–213 (1995)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Dornic, I., Godrèche, C.: Large deviations and nontrivial exponents in coarsening systems. J. Phys. A 31, 5413–5429 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dotsenko, V.: Two-time free energy distribution function in (1+1) directed polymers. J. Stat. Mech. 2013, P06017 (2013)MathSciNetGoogle Scholar
  13. 13.
    Dotsenko, V.: Two-time free energy distribution function in the kpz problem. arXiv:1507.06135 (2015)
  14. 14.
    Dynkin, E.B.: Some limit theorems for sums of independent random variables with infinite mathematical expectations. Izv. Akad. Nauk SSSR Ser. Mat. 19, 247–266 (1955). Selected Translations Math. Stat. Prob 1, 171–189 (1961)Google Scholar
  15. 15.
    Feller, W.: An Introduction to Probability Theory and Its Applications. Wiley, New York (1968)zbMATHGoogle Scholar
  16. 16.
    Ferrari, P.L., Spohn, H.: On the current time correlations for one-dimensional exclusion processes. arXiv:1602.00486 (2016)
  17. 17.
    García-García, R., Rosso, A., Schehr, G.: Longest excursion of fractional brownian motion: numerical evidence of non-Markovian effects. Phys. Rev. E 81, 010102 (2010)ADSCrossRefGoogle Scholar
  18. 18.
    Godrèche, C., Luck, J.M.: Statistics of the occupation time of renewal processes. J. Stat. Phys. 104, 489–524 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hansen, A., Engøy, T., Måløy, K.J.: Measuring hurst exponents with the first return method. Fractals 02(04), 527–533 (1994)CrossRefzbMATHGoogle Scholar
  20. 20.
    He, Y., Burov, S., Metzler, R., Barkai, E.: Random time-scale invariant diffusion and transport coefficients. Phys. Rev. Lett. 101, 058101 (2008)ADSCrossRefGoogle Scholar
  21. 21.
    Herault, J., Pétrélis, F., Fauve, S.: \(1/f^\alpha \) low frequency fluctuations in turbulent flows. J. Stat. Phys. 161, 1379–1389 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Herault, J., Pétrélis, F., Fauve, S.: Experimental observation of \(1/f\) noise in quasi-bidimensional turbulent flows. Europhys. Lett. 111, 44002 (2015)ADSCrossRefzbMATHGoogle Scholar
  23. 23.
    Jeon, J.H., Tejedor, V., Burov, S., Barkai, E., Selhuber-Unkel, C., Berg-Sørensen, K., Oddershede, L., Metzler, R.: In vivo anomalous diffusion and weak ergodicity breaking of lipid granules. Phys. Rev. Lett. 106, 048103 (2011)ADSCrossRefGoogle Scholar
  24. 24.
    Johansson, K.: Two time distribution in brownian directed percolation. arXiv:1502.00941 (2015)
  25. 25.
    Kallabis, H., Krug, J.: Persistence of Kardar–Parisi–Zhang interfaces. Europhys. Lett. 45, 20–25 (1999)ADSCrossRefGoogle Scholar
  26. 26.
    Kardar, M., Parisi, G., Zhang, Y.C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986)ADSCrossRefzbMATHGoogle Scholar
  27. 27.
    Kriecherbauer, T., Krug, J.: A pedestrian’s view on interacting particle systems, KPZ universality and random matrices. J. Phys. A 43, 403001 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Krug, J., Kallabis, H., Majumdar, S.N., Cornell, S.J., Bray, A.J., Sire, C.: Persistence exponents for fluctuating interfaces. Phys. Rev. E 56, 2702–2712 (1997)ADSCrossRefGoogle Scholar
  29. 29.
    Kuno, M., Fromm, D.P., Hamann, H.F., Gallagher, A., Nesbitt, D.J.: Nonexponential “blinking” kinetics of single cdse quantum dots: a universal power law behavior. J. Chem. Phys. 112, 3117 (2000)ADSCrossRefGoogle Scholar
  30. 30.
    Lamperti, J.: An occupation time theorem for a class of stochastic processes. Trans. Am. Math. Soc. 88, 380–387 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Manzo, C., Torreno-Pina, J.A., Massignan, P., Lapeyre, G.J., Lewenstein, M., Garcia Parajo, M.F.: Weak ergodicity breaking of receptor motion in living cells stemming from random diffusivity. Phys. Rev. X 5, 011021 (2015)Google Scholar
  32. 32.
    Margolin, G., Barkai, E.: Nonergodicity of blinking nanocrystals and other Lévy-walk processes. Phys. Rev. Lett. 94, 080601 (2005)ADSCrossRefGoogle Scholar
  33. 33.
    Margolin, G., Barkai, E.: Nonergodicity of a time series obeying Lévy statistics. J. Stat. Phys. 122, 137–167 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Metzler, R., Jeon, J.H., Cherstvy, A.G., Barkai, E.: Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys. 16, 24128–24164 (2014)CrossRefGoogle Scholar
  35. 35.
    Miyaguchi, T., Akimoto, T.: Intrinsic randomness of transport coefficient in subdiffusion with static disorder. Phys. Rev. E 83, 031926 (2011)ADSCrossRefGoogle Scholar
  36. 36.
    Miyaguchi, T., Akimoto, T.: Anomalous diffusion in a quenched-trap model on fractal lattices. Phys. Rev. E 91, 010102 (2015)ADSCrossRefGoogle Scholar
  37. 37.
    Prähofer, M., Spohn, H.: Universal distributions for growth processes in \(1+1\) dimensions and random matrices. Phys. Rev. Lett. 84, 4882–4885 (2000)ADSCrossRefGoogle Scholar
  38. 38.
    Schulz, J.H.P., Barkai, E., Metzler, R.: Aging effects and population splitting in single-particle trajectory averages. Phys. Rev. Lett. 110, 020602 (2013)ADSCrossRefGoogle Scholar
  39. 39.
    Schulz, J.H.P., Barkai, E., Metzler, R.: Aging renewal theory and application to random walks. Phys. Rev. X 4, 011028 (2014)Google Scholar
  40. 40.
    Singha, S.B.: Persistence of surface fluctuations in radially growing surfaces. J. Stat. Mech. 2005, P08006 (2005)CrossRefGoogle Scholar
  41. 41.
    Spohn, H.: Fluctuating hydrodynamics approach to equilibrium time correlations for anharmonic chains. arXiv:1505.05987 (2015)
  42. 42.
    Stefani, F.D., Hoogenboom, J.P., Barkai, E.: Beyond quantum jumps: blinking nanoscale light emitters. Phys. Today 62, 34 (2009)CrossRefGoogle Scholar
  43. 43.
    Stefani, F.D., Zhong, X., Knoll, W., Han, M., Kreiter, M.: Memory in quantum-dot photoluminescence blinking. New J. Phys. 7, 197 (2005)ADSCrossRefGoogle Scholar
  44. 44.
    Tabei, S.M.A., Burov, S., Kim, H.Y., Kuznetsov, A., Huynh, T., Jureller, J., Philipson, L.H., Dinner, A.R., Scherer, N.F.: Intracellular transport of insulin granules is a subordinated random walk. Proc. Natl. Acad. Sci. USA 110(13), 4911–4916 (2013)ADSCrossRefGoogle Scholar
  45. 45.
    Takeuchi, K.A.: Statistics of circular interface fluctuations in an off-lattice eden model. J. Stat. Mech. 2012, P05007 (2012)CrossRefGoogle Scholar
  46. 46.
    Takeuchi, K.A.: Experimental approaches to universal out-of-equilibrium scaling laws: turbulent liquid crystal and other developments. J. Stat. Mech. 2014, P01006 (2014)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Takeuchi, K.A., Sano, M.: Universal fluctuations of growing interfaces: evidence in turbulent liquid crystals. Phys. Rev. Lett. 104, 230601 (2010)ADSCrossRefGoogle Scholar
  48. 48.
    Takeuchi, K.A., Sano, M.: Evidence for geometry-dependent universal fluctuations of the Kardar–Parisi–Zhang interfaces in liquid-crystal turbulence. J. Stat. Phys. 147, 853–890 (2012)ADSCrossRefzbMATHGoogle Scholar
  49. 49.
    Takeuchi, K.A., Sano, M., Sasamoto, T., Spohn, H.: Growing interfaces uncover universal fluctuations behind scale invariance. Sci. Rep. 1, 34 (2011)ADSCrossRefGoogle Scholar
  50. 50.
    Weigel, A.V., Simon, B., Tamkun, M.M., Krapf, D.: Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking. Proc. Natl. Acad. Sci. USA 108, 6438–6443 (2011)ADSCrossRefGoogle Scholar
  51. 51.
    Wong, I.Y., Gardel, M.L., Reichman, D.R., Weeks, E.R., Valentine, M.T., Bausch, A.R., Weitz, D.A.: Anomalous diffusion probes microstructure dynamics of entangled f-actin networks. Phys. Rev. Lett. 92, 178101 (2004)ADSCrossRefGoogle Scholar
  52. 52.
    Yamamoto, E., Akimoto, T., Yasui, M., Yasuoka, K.: Origin of 1/f noise in hydration dynamics on lipid membrane surfaces. Sci. Rep. 5, 8876 (2015)ADSCrossRefGoogle Scholar
  53. 53.
    Yamamoto, E., Kalli, A.C., Akimoto, T., Yasuoka, K., Sansom, M.S.P.: Anomalous dynamics of a lipid recognition protein on a membrane surface. Sci. Rep. 5, 18,245 (2015)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of PhysicsTokyo Institute of TechnologyTokyoJapan
  2. 2.Department of Mechanical EngineeringKeio UniversityYokohamaJapan

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