Abstract
Brownian motion and viscoelastic anomalous diffusion in homogeneous environments are intrinsically Gaussian processes. In a growing number of systems, however, non-Gaussian displacement distributions of these processes are being reported. The physical cause of the non-Gaussianity is typically seen in different forms of disorder. These include, for instance, imperfect “ensembles” of tracer particles, the presence of local variations of the tracer mobility in heteroegenous environments, or cases in which the speed or persistence of moving nematodes or cells are distributed. From a theoretical point of view stochastic descriptions based on distributed (“superstatistical”) transport coefficients as well as time-dependent generalisations based on stochastic transport parameters with built-in finite correlation time are invoked. After a brief review of the history of Brownian motion and the famed Gaussian displacement distribution, we here provide a brief introduction to the phenomenon of non-Gaussianity and the stochastic modelling in terms of superstatistical and diffusing-diffusivity approaches.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Einstein, Ann. Phys. (Leipzig) 322, 549 (1905)
W. Sutherland, Philos. Mag. 9, 781 (1905)
M. von Smoluchowski, Ann. Phys. (Leipzig) 21, 756 (1906)
P. Langevin, C.R. Acad. Sci. Paris 146, 530 (1908)
J. Perrin, Compt. Rend. (Paris) 146, 967 (1908)
J. Perrin, Ann. Chim. Phys. 18, 5 (1909)
K. Pearson, Nature 72, 294 (1905)
Rayleigh, Nature 72, 318 (1905)
N. van Kampen,Stochastic processes in physics and chemistry (North Holland, Amsterdam, 1981)
P. Hänggi, F. Marchesoni, Chaos 15, 026101 (2005)
E. Kappler, Ann. Phys. (Leipzig) 11, 233 (1931)
G.E. Uhlenbeck, L.S. Ornstein, Phys. Rev. 36, 823 (1930)
A.G. Cherstvy, S. Thapa, Y. Mardoukhi, A.V. Chechkin, R. Metzler, Phys. Rev. E 98, 022134 (2018)
J.-P. Bouchaud, A. Georges, Phys. Rep. 195, 127 (1990)
R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)
R. Metzler, J. Klafter, J. Phys. A 37, R161 (2004)
S.C. Lim, S.V. Muniandy, Phys. Rev. E 66, 021114 (2002)
M.J. Saxton, Biophys. J. 81, 2226 (2001)
P.P. Mitra, P.N. Sen, L.M. Schwartz, P. Le Doussal, Phys. Rev. Lett. 68, 3555 (1992)
J.F. Lutsko, J.P. Boon, Phys. Rev. Lett. 88, 022108 (2013)
A. Fuliński, J. Chem. Phys. 138, 021101 (2013)
A. Fuliński, Phys. Rev. E 83, 061140 (2011)
F. Thiel, I.M. Sokolov, Phys. Rev. E 89, 012115 (2014)
H. Safdari, A.G. Cherstvy, A.V. Chechkin, F. Thiel, I.M. Sokolov, R. Metzler, J. Phys. A 48, 375002 (2015)
J.-H. Jeon, A.V. Chechkin, R. Metzler, Phys. Chem. Chem. Phys. 16, 15811 (2014)
N.V. Brilliantov, T. Pöschel,Kinetic Theory of Granular Gases (Oxford University Press, Oxford, UK, 2004)
D. De Walle, A. Rango,Principles of Snow Hydrology (Cambridge University Press, Cambridge, UK, 2008)
A. Molini, P. Talkner, G.G. Katul, A. Porporato, Physica A 390, 1841 (2011)
G.K. Batchelor, Math. Proc. Cambridge Philos. Soc. 48, 345 (1952)
D.S. Novikov, J.H. Jensen, J.A. Helpern, E. Fieremans, Proc. Natl. Acad. Sci. USA 111, 5088 (2014)
A. Bodrova, A.V. Chechkin, A.G. Cherstvy, R. Metzler, New J. Phys. 17, 063038 (2015)
A. Bodrova, A.V. Chechkin, A.G. Cherstvy, R. Metzler, Phys. Chem. Chem. Phys. 17, 21791 (2015)
A.S. Bodrova, A.V. Chechkin, A.G. Cherstvy, H. Safdari, I.M. Sokolov, R. Metzler, Sci. Rep. 6, 30520 (2016)
B.B. Mandelbrot, J.W. van Ness, SIAM Rev. 10, 422 (1968)
See also A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 26, 115 (1940)
R. Metzler, J.-H. Jeon, A.G. Cherstvy, E. Barkai, Phys. Chem. Chem. Phys. 16, 24128 (2014)
I. Goychuk, Phys. Rev. E 80, 046125 (2009)
I. Goychuk, Adv. Chem. Phys. 150, 187 (2012)
E. Lutz, Phys. Rev. E 64, 051106 (2001)
W. Deng, E. Barkai, Phys. Rev. E 79, 011112 (2009)
R. Zwanzig,Nonequilibrium Statistical Mechanics (Oxford University Press, Oxford, UK, 2001)
S.C. Weber, A.J. Spakowitz, J.A. Theriot, Phys. Rev. Lett. 104, 238102 (2010)
J.-H. Jeon, N. Leijnse, L.B. Oddershede, R. Metzler, New J. Phys. 15, 045011 (2013)
J. Szymanski, M. Weiss, Phys. Rev. Lett. 103, 038102 (2009)
G. Guigas, C. Kalla, M. Weiss, Biophys. J. 93, 316 (2007)
J.-H. Jeon, V. Tejedor, S. Burov, E. Barkai, C. Selhuber-Unkel, K. Berg-Sørensen, L. Oddershede, R. Metzler, Phys. Rev. Lett. 106, 048103 (2011)
J.F. Reverey, J.-H. Jeon, H. Bao, M. Leippe, R. Metzler, C. Selhuber-Unkel, Sci. Rep. 5,11690 (2015)
S. Thapa, N. Lukat, C. Selhuber-Unkel, A. Cherstvy, R. Metzler, J. Chem. Phys. 150, 144901 (2019)
J.-H. Jeon, H. Martinez-Seara Monne, M. Javanainen, R. Metzler, Phys. Rev. Lett. 109, 188103 (2012)
J.-H. Jeon, M. Javanainen, H. Martinez-Seara, R. Metzler, I. Vattulainen, Phys. Rev. X 6, 021006 (2016)
D. Molina-Garcia, T. Sandev, H. Safdari, G. Pagnini, A. Chechkin, R. Metzler, New J. Phys. 20, 103027 (2018)
B. Wang, J. Kuo, S.C. Bae, S. Granick, Nat. Mater. 11, 481 (2012)
B. Wang, S.M. Anthony, S.C. Bae, S. Granick, Proc. Natl. Acad. Sci. USA 106, 15160 (2009)
J. Guan, B. Wang, S. Granick, ACS Nano 8, 3331 (2014)
K. He, F.B. Khorasani, S.T. Retterer, D.K. Tjomasn, J.C. Conrad, R. Krishnamoorti, ACS Nano 7, 5122 (2013)
C. Xue, X. Zheng, K. Chen, Y. Tian, G. Hu, J. Phys. Chem. Lett. 7, 514 (2016)
D. Wang, R. Hu, M.J. Skaug, D. Schwartz, J. Phys. Chem. Lett. 6 54 (2015)
S. Dutta, J. Chakrabarti, Europhys. Lett. 116, 38001 (2016)
K.C. Leptos, J.S. Guasto, J.P. Gollub, A.I. Pesci, R.E. Goldstein, Phys. Rev. Lett. 103, 198103 (2009)
S. Hapca, J.W. Crawford, I.M. Young, J.R. Soc. Interface 6, 111 (2009)
W. Kob, H.C. Andersen, Phys. Rev. E 51, 4626 (1995)
P. Chaudhuri, L. Berthier, W. Kob, Phys. Rev. Lett. 99, 060604 (2007)
S. Roldan-Vargas, L. Rovigatti, F. Sciortino, Soft Matter 13, 514 (2017)
N. Samanta, R. Chakrabarti, Soft Matter 12, 8554 (2016)
M.J. Skaug, L. Wang, Y. Ding, D.K. Schwartz, ACS Nano 9, 2148 (2015)
D. Krapf, G. Campagnola, K. Nepal, O.B. Peerson, Soft Matter 18, 12633 (2016)
M. Soares e Silva, B. Stuhrmann, T. Betz, G.H. Koenderink, New J. Phys. 16, 075010 (2014)
B. Stuhrmann, M. Soares e Silva, M. Depken, F.C. MacKintosh, G.H. Koenderink, Phys. Rev. E 86, 020901(R) (2012)
C. Beck, E.G.D. Cohen, Physica A 332, 267 (2003)
C. Beck, E.G.D. Cohen, H.L. Swinney, Phys. Rev. E 72, 056133 (2005)
C. Beck, Prog. Theoret. Phys. Suppl. 162, 29 (2006)
A.V. Chechkin, F. Seno, R. Metzler, I.M. Sokolov, Phys. Rev. X 7, 021002 (2017)
V. Sposini, A.V. Chechkin, F. Seno, G. Pagnini, R. Metzler, New J. Phys. 20, 043044 (2018)
R. Jain, K.L. Sebastian, Phys. Rev. E 95, 032135 (2017)
C. Beck, Phys. Rev. Lett. 98, 064502 (2007)
C. Beck, Europhys. Lett. 64, 151 (2003)
C. Beck, Eur. Phys. J. A 40, 267 (2009)
B. Schafer, C. Beck, K. Aihara, D. Witthaut, M. Timme, Nat. Energy 3, 119 (2018)
K. Briggs, C. Beck, Physica A 378, 498 (2007)
B.I. Shraiman, E.D. Siggia, Phys. Rev. E 49, 2912 (1994)
B.I. Shraiman, E.D. Siggia, Nature 405, 639 (2000)
C. Beck, Philos. Trans. Roy. Soc. A 369, 453 (2011)
S. Abe, C. Beck, E.G.D. Cohen, Phys. Rev. E 76, 031102 (2007)
E. van der Straeten, C. Beck, Phys. Rev. E 80, 036108 (2009)
A. Mura, M.S. Taqqu, F. Mainardi, Physica A 387, 5033 (2008)
A. Mura, G. Pagnini, J. Phys. A: Math. Theor. 41, 285003 (2008)
A. Mura, F. Mainardi, Int. Transf. Spe. Funct. 20, 185 (2009)
D. Molina-García, T.M. Pham, P. Paradisi, G. Pagnini, Phys. Rev. E 94, 052147 (2016)
H. Scher, E.W. Montroll, Phys. Rev. B 12, 2455 (1975)
S. Havlin, D. Ben-Avraham, Adv. Phys. 36, 695 (1987)
A.G. Cherstvy, A.V. Chechkin, R. Metzler, New J. Phys. 15, 083039 (2013)
A.G. Cherstvy, R. Metzler, Phys. Chem. Chem. Phys. 15, 20220 (2013)
I. Munguira, I. Casuso, H. Takahashi, F. Rico, A. Miyagi, M. Chami, S. Scheuring, ACS Nano 10, 2584 (2016)
S. Gupta, J.U. de Mel, R.M. Perera, P. Zolnierczuk, M. Bleuel, A. Faraone, G.J. Schneider, J. Phys. Chem. Lett. 9, 2956 (2018)
W. He, H. Song, Y. Su, L. Geng, B.J. Ackerson, H.B. Peng, P. Tong, Nat. Commun. 7, 11701 (2016)
D. Krapf, R. Metzler, Phys. Today 72, 48 (2019)
S. Hanot, S. Lyonnard, S. Mossa, Nanoscale 8, 3314 (2016)
D. Wang, C. He, M.P. Stoykovich, D.K. Schwartz, ACS Nano 9, 1656 (2015)
M.J. Skaug, J. Mabry, D.K. Schwartz, Phys. Rev. Lett. 110, 256101 (2013)
O.V. Bychuk, B. O’Shaughnessy, Phys. Rev. Lett. 74, 1795 (1995)
O.V. Bychuk, B. O’Shaugnessy, J. Chem. Phys. 101, 772 (1994)
A.V. Chechkin, I.M. Zaid, M.A. Lomholt, I.M. Sokolov, R. Metzler, Phys. Rev. E 79 040105(R) (2009)
A.V. Chechkin, I.M. Zaid, M.A. Lomholt, I.M. Sokolov, R. Metzler, Phys. Rev. E 86, 041101 (2012)
G. Kwon, B.J. Sung, A. Yethiraj, J. Phys. Chem. B 118, 8128 (2014)
E.R. Weeks, J.C. Crocker, A.C. Levitt, A. Schofield, D.A. Weitz, Science 287, 627 (2000)
P. Charbonneau, Y. Jin, G. Parisi, F. Zamponi, Proc. Natl. Acad. Sci. USA 111, 15025 (2014)
C.E. Wagner, B.S. Turner, M. Rubinstein, G.H. McKinley, K. Ribbeck, Biomacromol. 18, 3654 (2017)
A.G. Cherstvy, S. Thapa, C.E. Wagner, R. Metzler, Soft Matter 15, 2526 (2019)
R.R.L. Aure, C.C. Bernido, M.V. Carpio-Bernido, R.G. Bacabac, Biophys. J. 117, 1029 (2019)
I. Chakraborty, Y. Roichman, https://arXiv:1909.11364
S. Ghosh, A.G. Cherstvy, R. Metzler, Phys. Chem. Chem. Phys. 17, 1847 (2015)
S. Ghosh, A.G. Cherstvy, D. Grebenkov, R. Metzler, New J. Phys. 18, 013027 (2016)
L. Luo, M. Yi, Phys. Rev. E 97, 042122 (2018)
A. Cuetos, N. Morillo, A. Patti, Phys. Rev. E 98, 042129 (2018)
T.J. Lampo, S. Stylianido, M.P. Backlund, P.A. Wiggins, A.J. Spakowitz, Biophys. J. 112, 532 (2017)
R. Metzler, Biophys. J. 112, 413 (2017)
E. van der Straeten, C. Beck, Physica A 390, 951 (2011)
J. Ślȩzak, R. Metzler, M. Magdziarz, New J. Phys. 20, 023026 (2018)
A.G. Cherstvy, R. Metzler, Phys. Chem. Chem. Phys. 18, 23840 (2016)
S. Stylianido, T.J. Lampo, A.J. Spakowitz, P.A. Wiggins, Phys. Rev. E 97, 062410 (2018)
E.B. Postnikov, A. Chechkin, I.M. Sokolov, https://arXiv:1810.02605
M. Hidalgo-Soria, E. Barkai, https://arXiv:1909.07189
S. Burov, E. Barkai, https://arXiv:1907.10002
J. Ślȩzak, K. Burnecki, R. Metzler, New J. Phys. 21, 073056 (2019)
A. Wyłomańska, A. Chechkin, J. Gajda, I.M. Sokolov, Physica A 421, 412 (2015)
J. Ślȩzak, R. Metzler, M. Magdziarz, New J. Phys. 21, 053008 (2019)
M.V. Chubynsky, G.W. Slater, Phys. Rev. Lett. 113, 098302 (2014)
R. Jain, K.L. Sebastian, J. Phys. Chem. B 120, 3988 (2016)
R. Jain, K.L. Sebastian, J. Chem. Sci. 129, 929 (2017)
N. Tyagi, B.J. Cherayil, J. Phys. Chem. B 121, 7204 (2017)
Y. Lanoiselée, D.S. Grebenkov, J. Phys. A 51, 145602 (2018)
S. Thapa, M.A. Lomholt, J. Krog, A.G. Cherstvy, R. Metzler, Phys. Chem. Chem. Phys. 20, 29018 (2018)
Y. Lanoiselée, N. Moutal, D.S. Grebenkov, Nat. Commun. 9, 4398 (2018)
J.-P. Fouqué, G. Papanicolaou, K.R. Sircar,Derivatives in financial markets with stochastic volatility (Cambridge University Press, Cambridge, UK, 2000)
J.C. Cox, J.E. Ingersoll, S.A. Ross, Econometrica 53, 385 (1985)
S.L. Heston, Rev. Financ. Studies 6, 327 (1993)
A. Dragulescu, V. Yakovenko, Quantit. Finance 2, 443 (2002)
V. Sposini, A.V. Chechkin, R. Metzler, J. Phys. A 52, 04LT01 (2019)
O. Pulkkinen, R. Metzler, Phys. Rev. Lett. 110, 198101 (2013)
M. Bauer, R. Metzler, PLoS ONE 8, e53956 (2013)
G. Kolesov, Z. Wunderlich, O.N. Laikova, M.S. Gelfand, L.A. Mirny, Proc. Natl. Acad. Sci. USA 104, 13948 (2007)
L. Liu, A.G. Cherstvy, R. Metzler, J. Phys. Chem. 121, 1284 (2017)
V. Sposini, D.S. Grebenkov, R. Metzler, G. Oshanin, F. Seno, https://arXiv:1911.11661
D. Krapf, E. Marinari, R. Metzler, G. Oshanin, A. Squarcini, X. Xu, New J. Phys. 20, 023029 (2018)
D. Krapf, N. Lukat, E. Marinari, R. Metzler, G. Oshanin, C. Selhuber-Unkel, A. Squarcini, L. Stadler, M. Weiss, X. Xu, Phys. Rev. X 9, 011019 (2019)
V. Sposini, R. Metzler, G. Oshanin, New J. Phys. 21, 073043 (2019)
A.G. Cherstvy, O. Nagel, C. Beta, R. Metzler, Phys. Chem. Chem. Phys. 20, 23034 (2018)
T. Miyaguchi, T. Akimoto, E. Yamamoto, Phys. Rev. E 94, 012109 (2016)
D.S. Grebenkov, Phys. Rev. E 99, 032133 (2019)
M. Montero, J. Masoliver, Phys. Rev. E 76, 061115 (2007)
J.H.P. Schulz, A.V. Chechkin, R. Metzler, J. Phys. A. 46, 475001 (2013)
V. Tejedor, R. Metzler, J. Phys. A 43, 082002 (2010)
A.V. Chechkin, M. Hofmann, I.M. Sokolov, Phys. Rev. E 80, 031112 (2008)
M. Magdziarz, R. Metzler, W. Szczotka, P. Zebrowski, Phys. Rev. E 85, 051103 (2012)
P. Massignan, C. Manzo, J.A. Torrena-Pina, M.F. García-Parajo, M. Lewenstein, G.J. Lapeyre, Jr., Phys. Rev. Lett. 112, 150603 (2014)
E. Teomy, Y. Roichman, Y. Shokef, https://arXiv:1908.07242
Acknowledgments
Open access funding provided by Projekt DEAL.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Metzler, R. Superstatistics and non-Gaussian diffusion. Eur. Phys. J. Spec. Top. 229, 711–728 (2020). https://doi.org/10.1140/epjst/e2020-900210-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2020-900210-x