Ageing and confinement in non-ergodic heterogeneous diffusion processes

We study the effects of ageing—the time delay between initiation of the physical process at t = 0 and start of observation at some time t a > 0 ?> —and spatial confinement on the properties of heterogeneous diffusion processes (HDPs) with deterministic power-law space-dependent diffusivities, D ( x ) = D 0 | x | &agr; ?> . From analysis of the ensemble and time averaged mean squared displacements and the ergodicity breaking parameter quantifying the inherent degree of irreproducibility of individual realizations of the HDP we obtain striking similarities to ageing subdiffusive continuous time random walks with scale-free waiting time distributions. We also explore how both processes can be distinguished. For confined HDPs we study the long-time saturation of the ensemble and time averaged particle displacements as well as the magnitude of the inherent scatter of time averaged displacements and contrast the outcomes to the results known for other anomalous diffusion processes under confinement.

[1]  V. Dragović,et al.  Pseudo-integrable billiards and double reflection nets , 2015 .

[2]  Andrey G. Cherstvy,et al.  Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. , 2014, Physical chemistry chemical physics : PCCP.

[3]  Ralf Metzler,et al.  Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous diffusion. , 2014, Physical chemistry chemical physics : PCCP.

[4]  Andrey G. Cherstvy,et al.  Nonergodicity, fluctuations, and criticality in heterogeneous diffusion processes. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  M. Weiss,et al.  Probing the type of anomalous diffusion with single-particle tracking. , 2014, Physical chemistry chemical physics : PCCP.

[6]  M. Lewenstein,et al.  Nonergodic subdiffusion from Brownian motion in an inhomogeneous medium. , 2014, Physical review letters.

[7]  Leonardo Dagdug,et al.  Discriminating between anomalous diffusion and transient behavior in microheterogeneous environments. , 2014, Biophysical journal.

[8]  J. M. Sancho,et al.  Single-molecule diffusion in a periodic potential at a solid-liquid interface. , 2014, Soft matter.

[9]  S. Granick,et al.  Even hard-sphere colloidal suspensions display Fickian yet non-Gaussian diffusion. , 2013, ACS nano.

[10]  I. Sokolov,et al.  Scaled Brownian motion as a mean-field model for continuous-time random walks. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Andrey G. Cherstvy,et al.  Particle invasion, survival, and non-ergodicity in 2D diffusion processes with space-dependent diffusivity. , 2013, Soft matter.

[12]  R. Metzler,et al.  Aging renewal theory and application to random walks , 2013, 1310.1058.

[13]  Andrey G. Cherstvy,et al.  Population splitting, trapping, and non-ergodicity in heterogeneous diffusion processes. , 2013, Physical chemistry chemical physics : PCCP.

[14]  Ralf Metzler,et al.  Noisy continuous time random walks. , 2013, The Journal of chemical physics.

[15]  G. Radons,et al.  Subdiffusive continuous time random walks and weak ergodicity breaking analyzed with the distribution of generalized diffusivities , 2013 .

[16]  R. Metzler,et al.  Anomalous diffusion and power-law relaxation of the time averaged mean squared displacement in worm-like micellar solutions , 2013 .

[17]  Andrey G. Cherstvy,et al.  Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes , 2013, 1303.5533.

[18]  A. Kuznetsov,et al.  Intracellular transport of insulin granules is a subordinated random walk , 2013, Proceedings of the National Academy of Sciences.

[19]  T. Franosch,et al.  Anomalous transport in the crowded world of biological cells , 2013, Reports on progress in physics. Physical Society.

[20]  U. Scherf,et al.  Mobility relaxation and electron trapping in a donor/acceptor copolymer , 2013 .

[21]  A. Fuliński.  Communication: how to generate and measure anomalous diffusion in simple systems. , 2013, The Journal of chemical physics.

[22]  R. Metzler,et al.  Aging effects and population splitting in single-particle trajectory averages. , 2013, Physical review letters.

[23]  Yuval Garini,et al.  Improved estimation of anomalous diffusion exponents in single-particle tracking experiments. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Kevin Burrage,et al.  Inferring diffusion in single live cells at the single-molecule level , 2012, Philosophical Transactions of the Royal Society B: Biological Sciences.

[25]  R. Metzler,et al.  Anomalous diffusion of phospholipids and cholesterols in a lipid bilayer and its origins. , 2012, Physical review letters.

[26]  A. Godec,et al.  Finite-time effects and ultraweak ergodicity breaking in superdiffusive dynamics. , 2012, Physical review letters.

[27]  Igor M. Sokolov,et al.  Models of anomalous diffusion in crowded environments , 2012 .

[28]  R. Metzler,et al.  Strange kinetics of single molecules in living cells , 2012 .

[29]  S. Granick,et al.  When Brownian diffusion is not Gaussian. , 2012, Nature materials.

[30]  R. Metzler,et al.  Inequivalence of time and ensemble averages in ergodic systems: exponential versus power-law relaxation in confinement. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  I. Goychuk.  Viscoelastic Subdiffusion: Generalized Langevin Equation Approach , 2012 .

[32]  G. Kneller,et al.  Communication: consistent picture of lateral subdiffusion in lipid bilayers: molecular dynamics simulation and exact results. , 2011, The Journal of chemical physics.

[33]  Jussi Timonen,et al.  Protein Diffusion in Mammalian Cell Cytoplasm , 2011, PloS one.

[34]  N. Seeman,et al.  Subdiffusion of a sticky particle on a surface. , 2011, Physical review letters.

[35]  A. Fulínski,et al.  Anomalous diffusion and weak nonergodicity. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  Aubrey V. Weigel,et al.  Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking , 2011, Proceedings of the National Academy of Sciences.

[37]  R. Metzler,et al.  In vivo anomalous diffusion and weak ergodicity breaking of lipid granules. , 2010, Physical review letters.

[38]  Ralf Metzler,et al.  Single particle tracking in systems showing anomalous diffusion: the role of weak ergodicity breaking. , 2010, Physical chemistry chemical physics : PCCP.

[39]  K. Lindenberg,et al.  Weak disorder: anomalous transport and diffusion are normal yet again. , 2010, Physical review letters.

[40]  R. Metzler,et al.  Aging and nonergodicity beyond the Khinchin theorem , 2010, Proceedings of the National Academy of Sciences.

[41]  Ralf Metzler,et al.  Analysis of short subdiffusive time series: scatter of the time-averaged mean-squared displacement , 2010 .

[42]  I. Sokolov,et al.  Kramers-like escape driven by fractional Gaussian noise. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  Ralf Metzler,et al.  Fractional Brownian motion and motion governed by the fractional Langevin equation in confined geometries. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  Ralf Jungmann,et al.  Quantitative analysis of single particle trajectories: mean maximal excursion method. , 2010, Biophysical journal.

[45]  M. Weiss,et al.  Elucidating the origin of anomalous diffusion in crowded fluids. , 2009, Physical review letters.

[46]  Jeremy C. Smith,et al.  Subdiffusion in time-averaged, confined random walks. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[47]  I. Goychuk.  Viscoelastic subdiffusion: from anomalous to normal. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[48]  P. Hänggi,et al.  Universal fluctuations in subdiffusive transport , 2008, 0811.4738.

[49]  E. Barkai,et al.  Ergodic properties of fractional Brownian-Langevin motion. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[50]  R. Metzler,et al.  Random time-scale invariant diffusion and transport coefficients. , 2008, Physical review letters.

[51]  J. Klafter,et al.  Nonergodicity mimics inhomogeneity in single particle tracking. , 2008, Physical review letters.

[52]  Huan‐Xiang Zhou,et al.  Macromolecular crowding and confinement: biochemical, biophysical, and potential physiological consequences. , 2008, Annual review of biophysics.

[53]  J. Klafter,et al.  Probing microscopic origins of confined subdiffusion by first-passage observables , 2008, Proceedings of the National Academy of Sciences.

[54]  Bruce J. West,et al.  Linear response and Fluctuation-Dissipation Theorem for non-Poissonian renewal processes , 2007, 0807.1305.

[55]  E. Barkai,et al.  Distribution of time-averaged observables for weak ergodicity breaking. , 2007, Physical review letters.

[56]  Ralf Metzler,et al.  Subdiffusion and weak ergodicity breaking in the presence of a reactive boundary. , 2007, Physical review letters.

[57]  E. Cox,et al.  Physical nature of bacterial cytoplasm. , 2006, Physical review letters.

[58]  E. Barkai,et al.  Random walk to a nonergodic equilibrium concept. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[59]  Golan Bel,et al.  Weak Ergodicity Breaking in the Continuous-Time Random Walk , 2005 .

[60]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[61]  D. Reichman,et al.  Anomalous diffusion probes microstructure dynamics of entangled F-actin networks. , 2004, Physical review letters.

[62]  Denis Le Bihan,et al.  Looking into the functional architecture of the brain with diffusion MRI , 2003, Nature Reviews Neuroscience.

[63]  E. Barkai.  Aging in subdiffusion generated by a deterministic dynamical system. , 2003, Physical review letters.

[64]  S. C. Lim,et al.  Self-similar Gaussian processes for modeling anomalous diffusion. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[65]  E. Lutz.  Fractional Langevin equation. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[66]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[67]  Ralf Metzler,et al.  Boundary value problems for fractional diffusion equations , 2000 .

[68]  P. M. Lee,et al.  Random Walks and Random Environments: Volume 1: Random Walks , 1995 .

[69]  J. Bouchaud.  Weak ergodicity breaking and aging in disordered systems , 1992 .

[70]  J. Bouchaud,et al.  Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications , 1990 .

[71]  Sergei M. Rytov,et al.  Principles of Statistical Radiophysics 1: Elements of Random Process Theory , 1987 .

[72]  E. Montroll,et al.  Anomalous transit-time dispersion in amorphous solids , 1975 .

[73]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[74]  M. Bisset,et al.  Electronic addresses: , 1995 .