Abstract
Brownian particles suspended in disordered crowded environments often exhibit non-Gaussian normal diffusion (NGND), whereby their displacements grow with mean square proportional to the observation time and non-Gaussian statistics. Their distributions appear to decay almost exponentially according to “universal” laws largely insensitive to the observation time. This effect is generically attributed to slow environmental fluctuations, which perturb the local configuration of the suspension medium. To investigate the microscopic mechanisms responsible for the NGND phenomenon, we study Brownian diffusion in low dimensional systems, like the free diffusion of ellipsoidal and active particles, the diffusion of colloidal particles in fluctuating corrugated channels and Brownian motion in arrays of planar convective rolls. NGND appears to be a transient effect related to the time modulation of the instantaneous particle’s diffusivity, which can occur even under equilibrium conditions. Consequently, we propose to generalize the definition of NGND to include transient displacement distributions which vary continuously with the observation time. To this purpose, we provide a heuristic one-parameter function, which fits all time-dependent transient displacement distributions corresponding to the same diffusion constant. Moreover, we reveal the existence of low dimensional systems where the NGND distributions are not leptokurtic (fat exponential tails), as often reported in the literature, but platykurtic (thin sub-Gaussian tails), i.e., with negative excess kurtosis. The actual nature of the NGND transients is related to the specific microscopic dynamics of the diffusing particle.
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References
B. Wang, S. M. Anthony, S. C. Bae, and S. Granick, Anomalous yet Brownian, Proc. Natl. Acad. Sci. USA 106(36), 15160 (2009)
B. Wang, J. Kuo, C. Bae, and S. Granick, When Brownian diffusion is not Gaussian, Nat. Mater. 11(6), 481 (2012)
S. Bhattacharya, D. K. Sharma, S. Saurabh, S. De, A. Sain, A. Nandi, and A. Chowdhury, Plasticization of poly(vinylpyrrolidone) thin films under ambient humidity: Insight from single-molecule tracer diffusion dynamics, J. Phys. Chem. B 117(25), 7771 (2013)
J. Kim, C. Kim, and B. J. Sung, Simulation study of seemingly Fickian but heterogeneous dynamics of two dimensional colloids, Phys. Rev. Lett. 110(4), 047801 (2013)
G. Kwon, B. J. Sung, and A. Yethiraj, Dynamics in crowded environments: Is non-Gaussian Brownian diffusion normal? J. Phys. Chem. B 118(28), 8128 (2014)
J. Guan, B. Wang, and S. Granick, Even hard-sphere colloidal suspensions display Fickian yet non-Gaussian diffusion, ACS Nano 8(4), 3331 (2014)
C. Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences, Berlin: Springer, 2009
E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schofield, and D. A. Weitz, Three-dimensional direct imaging of structural relaxation near the colloidal glass transition, Science 287(5453), 627 (2000)
J. D. Eaves, and D. R. Reichman, Spatial dimension and the dynamics of supercooled liquids, Proc. Natl. Acad. Sci. USA 106(36), 15171 (2009)
K. C. Leptos, J. S. Guasto, J. P. Gollub, A. I. Pesci, and R. E. Goldstein, Dynamics of enhanced tracer diffusion in suspensions of swimming eukaryotic microorganisms, Phys. Rev. Lett. 103(19), 198103 (2009)
W. K. Kegel and A. van Blaaderen, Direct observation of dynamical heterogeneities in colloidal hard-sphere suspensions, Science 287(5451), 290 (2000)
P. Chaudhuri, L. Berthier, and W. Kob, Universal nature of particle displacements close to glass and jamming transitions, Phys. Rev. Lett. 99(6), 060604 (2007)
S. K. Ghosh, A. G. Cherstvy, D. S. Grebenkov, and R. Metzler, Anomalous non-Gaussian tracer diffusion in crowded two-dimensional environments, New J. Phys. 18(1), 013027 (2016)
W. He, H. Song, Y. Su, L. Geng, B. J. Ackerson, H. B. Peng, and P. Tong, Dynamic heterogeneity and non-Gaussian statistics for acetylcholine receptors on live cell membrane, Nat. Commun. 7(1), 11701 (2016)
K. Białas, J. Łuczka, P. Hänggi, and J. Spiechowicz, Colossal Brownian yet non-Gaussian diffusion induced by nonequilibrium noise, Phys. Rev. E 102, 042121 (2020)
M. V. Chubynsky and G. W. Slater, Diffusing diffusivity: A model for anomalous, yet Brownian, diffusion, Phys. Rev. Lett. 113(9), 098302 (2014)
A. G. Cherstvy and R. Metzler, Anomalous diffusion in time-fluctuating non-stationary diffusivity landscapes, Phys. Chem. Chem. Phys. 18(34), 23840 (2016)
R. Jain and K. L. Sebastian, Diffusion in a crowded, rearranging environment, J. Phys. Chem. B 120(16), 3988 (2016)
R. Jain and K. L. Sebastian, Diffusing diffusivity: A new derivation and comparison with simulations, J. Chem. Sci. 129(7), 929 (2017)
N. Tyagi and B. J. Cherayil, Non-Gaussian Brownian diffusion in dynamically disordered thermal environments, J. Phys. Chem. B 121(29), 7204 (2017)
L. Luo and M. Yi, Non-Gaussian diffusion in static disordered media, Phys. Rev. E 97(4), 042122 (2018)
A. V. Chechkin, F. Seno, R. Metzler, and I. M. Sokolov, Brownian yet non-Gaussian diffusion: From superstatistics to subordination of diffusing diffusivities, Phys. Rev. X 7(2), 021002 (2017)
J. Ślçzak, R. Metzler, and M. Magdziarz, Superstatistical generalised Langevin equation: Non-Gaussian viscoelastic anomalous diffusion, New J. Phys. 20(2), 023026 (2018)
Y. Li, F. Marchesoni, D. Debnath, and P. K. Ghosh, Non-Gaussian normal diffusion in a fluctuating corrugated channel, Phys. Rev. Res. 1(3), 033003 (2019)
P. Hänggi and F. Marchesoni, Artificial Brownian motors: Controlling transport on the nanoscale, Rev. Mod. Phys. 81(1), 387 (2009)
R. Lipowsky, Generic interactions of flexible membranes, in: Handbook of Biological Physics, Eds. R. Lipowsky and E. Sackmann, Vol. 1, Ch. 11, Elsevier, 1995
P. S. Burada, P. Hänggi, F. Marchesoni, G. Schmid, and P. Talkner, Diffusion in confined geometries, ChemPhysChem 10(1), 45 (2009)
X. Yang, C. Liu, Y. Li, F. Marchesoni, P. Hänggi, and H. P. Zhang, Hydrodynamic and entropic effects on colloidal diffusion in corrugated channels, Proc. Natl. Acad. Sci. USA 114(36), 9564 (2017)
V. Sposini, A. V. Chechkin, F. Seno, G. Pagnini, and R. Metzler, Random diffusivity from stochastic equations: Comparison of two models for Brownian yet non-Gaussian diffusion, New J. Phys. 20(4), 043044 (2018)
L. Luo and M. Yi, Quenched trap model on the extreme landscape: The rise of subdiffusion and non-Gaussian diffusion, Phys. Rev. E 100(4), 042136 (2019)
L. Luo and M. Yi, Ergodicity recovery of random walk in heterogeneous disordered media, Chin. Phys. B 29(5), 050503 (2020)
For a review, see: H. C. Berg, Random Walk in Biology, Princeton University Press, 1984
F. Perrin, Mouvement brownien d’un ellipsoide (I): Dispersion diélectrique pour des molécules ellipsoidales, J. Phys. Radium 5(10), 497 (1934)
F. Perrin, Mouvement Brownien d’un ellipsoide (II): Rotation libre et dépolarisation des fluorescences (Translation et diffusion de molécules ellipsoidales), J. Phys. Radium VII, 1 (1936)
Y. Han, A. M. Alsayed, M. Nobili, J. Zhang, T. C. Lubensky, and A. G. Yodh, Brownian motion of an ellipsoid, Science 314(5799), 626 (2006)
P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992
S. R. Aragón and R. Pecora, General theory of dynamic light scattering from cylindrically symmetric particles with translational-rotational coupling, J. Chem. Phys. 82(12), 5346 (1985)
S. Leitmann, F. Höfling, and T. Franosch, Dynamically crowded solutions of infinitely thin Brownian needles, Phys. Rev. E 96(1), 012118 (2017)
S. Prager, Interaction of rotational and translational diffusion, J. Chem. Phys. 23(12), 2404 (1955)
S. Jiang and S. Granick (Eds.), Janus particle synthesis, self-assembly and applications, RSC Publishing, Cambridge, 2012
A. Walther and A. H. E. Müller, Janus particles: Synthesis, self-assembly, physical properties, and applications, Chem. Rev. 113(7), 5194 (2013)
M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrodynamics of soft active matter, Rev. Mod. Phys. 85(3), 1143 (2013)
J. Elgeti, R. G. Winkler, and G. Gompper, Physics of microswimmers, single particle motion and collective behavior: A review, Rep. Prog. Phys. 78(5), 056601 (2015)
see: e.g., Smart Drug Delivery System, edited by A. D. Sezer, IntechOpen, 2016
J. Wang, Nanomachines: Fundamentals and Applications, Wiley-VCH, Weinheim, 2013
G. Volpe, I. Buttinoni, D. Vogt, H. J. Kümmerer, and C. Bechinger, Microswimmers in patterned environments, Soft Matter 7(19), 8810 (2011)
P. K. Ghosh, V. R. Misko, F. Marchesoni, and F. Nori, Self-propelled Janus particles in a ratchet: Numerical simulations, Phys. Rev. Lett. 110(26), 268301 (2013)
S. van Teeffelen and H. Löwen, Dynamics of a Brownian circle swimmer, Phys. Rev. E 78, 020101 (2008)
D. Debnath, P. K. Ghosh, Y. Li, F. Marchesoni, and B. Li, Diffusion of eccentric microswimmers, Soft Matter 12(7), 2017 (2016)
C. Kurzthaler, S. Leitmann, and T. Franosch, Intermediate scattering function of an anisotropic active Brownian particle, Sci. Rep. 6(1), 36702 (2016)
J. R. Howse, R. A. L. Jones, A. J. Ryan, T. Gough, R. Vafabakhsh, and R. Golestanian, Self-motile colloidal particles: From directed propulsion to random walk, Phys. Rev. Lett. 99(4), 048102 (2007)
B. ten Hagen, S. van Teeffelen, and H. Löwen, Non-Gaussian behaviour of a self-propelled particle on a substrate, Condens. Matter Phys. 12(4), 725 (2009)
X. Ao, P. K. Ghosh, Y. Li, G. Schmid, P. Hä nggi, and F. Marchesoni, Diffusion of chiral Janus particles in a sinusoidal channel, EPL 109(1), 10003 (2015)
X. Zheng, B. ten Hagen, A. Kaiser, M. Wu, H. Cui, Z. Silber-Li, and H. Löwen, Non-Gaussian statistics for the motion of self-propelled Janus particles: Experiment versus theory, Phys. Rev. E 88(3), 032304 (2013)
D. Debnath, P. K. Ghosh, V. R. Misko, Y. Li, F. Marchesoni, and F. Nori, Enhanced motility in a binary mixture of active nano/microswimmers, Nanoscale 12(17), 9717 (2020)
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, New York: Wiley, 1991
L. Bosi, P. K. Ghosh, and F. Marchesoni, Analytical estimates of free Brownian diffusion times in corrugated narrow channels, J. Chem. Phys. 137(17), 174110 (2012)
T. H. Solomon and J. P. Gollub, Chaotic particle transport in time-dependent Rayleigh-Bénard convection, Phys. Rev. A 38(12), 6280 (1988)
T. H. Solomon and I. Mezić, Uniform resonant chaotic mixing in fluid flows, Nature 425(6956), 376 (2003)
M. N. Rosenbluth, H. L. Berk, I. Doxas, and W. Horton, Effective diffusion in laminar convective flows, Phys. Fluids 30(9), 2636 (1987)
W. Young, A. Pumir, and Y. Pomeau, Anomalous diffusion of tracer in convection rolls, Phys. Fluids A 1(3), 462 (1989)
Y. N. Young and M. J. Shelley, Stretch-coil transition and transport of fibers in cellular flows, Phys. Rev. Lett. 99(5), 058303 (2007)
H. Manikantan and D. Saintillan, Subdiffusive transport of fluctuating elastic filaments in cellular flows, Phys. Fluids 25(7), 073603 (2013)
A. Sarracino, F. Cecconi, A. Puglisi, and A. Vulpiani, Nonlinear response of inertial tracers in steady laminar flows: Differential and absolute negative mobility, Phys. Rev. Lett. 117(17), 174501 (2016)
C. Torney and Z. Neufeld, Transport and aggregation of self-propelled particles in fluid flows, Phys. Rev. Lett. 99(7), 078101 (2007)
N. O. Weiss, The expulsion of magnetic flux by eddies, Proc. R. Soc. Lond. A 293(1434), 310 (1966)
Y. Li, L. Li, F. Marchesoni, D. Debnath, and P. K. Ghosh, Diffusion of chiral janus particles in convection rolls., Physical Review Research 2(1), 013250 (2020)
Q. Yin, Y. Li, F. Marchesoni, T. Debnath, and P. K. Ghosh, Exit times of a Brownian particle out of a convection roll, Phys. Fluids 32(9), 092010 (2020)
J. Feng and T. G. Kurtz, Large Deviations for Stochastic processes, Mathematical Surveys and Monographs, Vol. 131, Am. Math. Society, 2006
Q. Yin, Y. Li, B. Li, F. Marchesoni, S. Nayak, and P. K. Ghosh, Diffusion transients in convection rolls, J. Fluid Mech., Doi: https://doi.org/10.1017/jfm.2020.1127 (2021)
Acknowledgements
Y.L. was supported by the NSF China under Grant Nos. 11875201 and 11935010. P.K.G. was supported by SERB Start-up Research Grant (Young Scientist) No. YSS/2014/000853, and the UGC-BSR Start-Up Grant No. F.30-92/2015.
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arXiv: 2101.06875. This article can also be found at https://doi.org/10.1007/s11467-020-1022-0.
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Yin, Q., Li, Y., Marchesoni, F. et al. Non-Gaussian normal diffusion in low dimensional systems. Front. Phys. 16, 33203 (2021). https://doi.org/10.1007/s11467-020-1022-0
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DOI: https://doi.org/10.1007/s11467-020-1022-0