# Characteristic Sign Renewals of Kardar–Parisi–Zhang Fluctuations

- 150 Downloads
- 5 Citations

## Abstract

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.

## Keywords

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

### Acknowledgments

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.

## References

- 1.Akimoto, T., Miyaguchi, T.: Distributional ergodicity in stored-energy-driven Lévy flights. Phys. Rev. E
**87**, 062134 (2013)ADSCrossRefGoogle Scholar - 2.Barabási, A.L., Stanley, H.E.: Fractal Concepts in Surface Growth. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
- 3.Bouchaud, J.P.: Weak ergodicity breaking and aging in disordered systems. J. Phys. I (France)
**2**, 1705–1713 (1992)CrossRefGoogle Scholar - 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.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.Cakir, R., Grigolini, P., Krokhin, A.A.: Dynamical origin of memory and renewal. Phys. Rev. E
**74**, 021108 (2006)ADSCrossRefGoogle Scholar - 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.Corwin, I.: The Kardar–Parisi–Zhang equation and universality class. Random Matrices Theory Appl.
**1**, 1130001 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Cox, D.R.: Renewal Theory. Methuen, London (1962)zbMATHGoogle Scholar
- 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.Dornic, I., Godrèche, C.: Large deviations and nontrivial exponents in coarsening systems. J. Phys. A
**31**, 5413–5429 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 12.Dotsenko, V.: Two-time free energy distribution function in (1+1) directed polymers. J. Stat. Mech.
**2013**, P06017 (2013)MathSciNetGoogle Scholar - 13.Dotsenko, V.: Two-time free energy distribution function in the kpz problem. arXiv:1507.06135 (2015)
- 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.Feller, W.: An Introduction to Probability Theory and Its Applications. Wiley, New York (1968)zbMATHGoogle Scholar
- 16.Ferrari, P.L., Spohn, H.: On the current time correlations for one-dimensional exclusion processes. arXiv:1602.00486 (2016)
- 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.Godrèche, C., Luck, J.M.: Statistics of the occupation time of renewal processes. J. Stat. Phys.
**104**, 489–524 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 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.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.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.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.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.Johansson, K.: Two time distribution in brownian directed percolation. arXiv:1502.00941 (2015)
- 25.Kallabis, H., Krug, J.: Persistence of Kardar–Parisi–Zhang interfaces. Europhys. Lett.
**45**, 20–25 (1999)ADSCrossRefGoogle Scholar - 26.Kardar, M., Parisi, G., Zhang, Y.C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett.
**56**, 889–892 (1986)ADSCrossRefzbMATHGoogle Scholar - 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.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.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.Lamperti, J.: An occupation time theorem for a class of stochastic processes. Trans. Am. Math. Soc.
**88**, 380–387 (1958)MathSciNetCrossRefzbMATHGoogle Scholar - 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.Margolin, G., Barkai, E.: Nonergodicity of blinking nanocrystals and other Lévy-walk processes. Phys. Rev. Lett.
**94**, 080601 (2005)ADSCrossRefGoogle Scholar - 33.Margolin, G., Barkai, E.: Nonergodicity of a time series obeying Lévy statistics. J. Stat. Phys.
**122**, 137–167 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 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.Miyaguchi, T., Akimoto, T.: Intrinsic randomness of transport coefficient in subdiffusion with static disorder. Phys. Rev. E
**83**, 031926 (2011)ADSCrossRefGoogle Scholar - 36.Miyaguchi, T., Akimoto, T.: Anomalous diffusion in a quenched-trap model on fractal lattices. Phys. Rev. E
**91**, 010102 (2015)ADSCrossRefGoogle Scholar - 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.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.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.Singha, S.B.: Persistence of surface fluctuations in radially growing surfaces. J. Stat. Mech.
**2005**, P08006 (2005)CrossRefGoogle Scholar - 41.Spohn, H.: Fluctuating hydrodynamics approach to equilibrium time correlations for anharmonic chains. arXiv:1505.05987 (2015)
- 42.Stefani, F.D., Hoogenboom, J.P., Barkai, E.: Beyond quantum jumps: blinking nanoscale light emitters. Phys. Today
**62**, 34 (2009)CrossRefGoogle Scholar - 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.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.Takeuchi, K.A.: Statistics of circular interface fluctuations in an off-lattice eden model. J. Stat. Mech.
**2012**, P05007 (2012)CrossRefGoogle Scholar - 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.Takeuchi, K.A., Sano, M.: Universal fluctuations of growing interfaces: evidence in turbulent liquid crystals. Phys. Rev. Lett.
**104**, 230601 (2010)ADSCrossRefGoogle Scholar - 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.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.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.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.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.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