Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking

Handbook of Mathematical Functions

McQuarrie, D. A. (1966) · American Journal of Physics · doi:10.1119/1.1972842 · Cited by 40,433

Fractional Differential Equations

(2025) · MDPI eBooks · doi:10.3390/books978-3-7258-4742-6 · Cited by 20,278

The random walk's guide to anomalous diffusion: a fractional dynamics approach

Metzler, R., & Klafter, J. (2000) · Physics Reports · doi:10.1016/s0370-1573(00)00070-3 · Cited by 9,026

Fractional Brownian Motions, Fractional Noises and Applications

Mandelbrot, B. B., & Van Ness, J. W. (1968) · SIAM Review · doi:10.1137/1010093 · Cited by 7,645

The Fractal Geometry of Nature

Mandelbrot, B. B., & Longley, P. (1982) · CERN Document Server (European Organization for Nuclear Research) · Cited by 7,293

Long-Term Storage Capacity of Reservoirs

Hurst, H. E. (1951) · Transactions of the American Society of Civil Engineers · doi:10.1061/taceat.0006518 · Cited by 6,212

Understanding individual human mobility patterns

González, M. C., Hidalgo, C. A., & Barabási, A.-L. (2008) · Nature · doi:10.1038/nature06958 · Cited by 6,059

The fluctuation-dissipation theorem

Kubo, R. (1966) · Reports on Progress in Physics · doi:10.1088/0034-4885/29/1/306 · Cited by 5,094

Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications

Bouchaud, J.-P., & Georges, A. (1990) · Physics Reports · doi:10.1016/0370-1573(90)90099-n · Cited by 4,162

How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension

Mandelbrot, B. (1967) · Science · doi:10.1126/science.156.3775.636 · Cited by 4,090

Fractals

Feder, J. (1988) · Springer eBooks · doi:10.1007/978-1-4899-2124-6 · Cited by 3,595

Anomalous transit-time dispersion in amorphous solids

Scher, H., & Montroll, E. W. (1975) · Physical Review B · doi:10.1103/physrevb.12.2455 · Cited by 3,107

Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics

Berne, B. J., & Pecora, R. (1976) · CERN Document Server (European Organization for Nuclear Research) · Cited by 2,856

Mobility measurement by analysis of fluorescence photobleaching recovery kinetics

Axelrod, D., Koppel, D. E., Schlessinger, J., Elson, E., & Webb, W. W. (1976) · Biophysical Journal · doi:10.1016/s0006-3495(76)85755-4 · Cited by 2,714

Log-normal distributions across the sciences: keys and clues

LIMPERT, E., STAHEL, W. A., & ABBT, M. (2001) · BioScience · doi:10.1641/0006-3568(2001)051[0341:lndats]2.0.co;2 · Cited by 2,621

Théorie de la spéculation

Bachelier, L. (1900) · Annales scientifiques de l'École normale supérieure · doi:10.24033/asens.476 · Cited by 2,539

Elasticity and Anelasticity of Metals.

Zener, C. M., & Siegel, S. (1949) · The Journal of Physical Chemistry · doi:10.1021/j150474a017 · Cited by 2,369

The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics

Metzler, R., & Klafter, J. (2004) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/37/31/r01 · Cited by 2,354

Ueber Diffusion

Fick, A. (1855) · Annalen der Physik · doi:10.1002/andp.18551700105 · Cited by 2,304

Self-Motile Colloidal Particles: From Directed Propulsion to Random Walk

Howse, J. R., Jones, R. A. L., Ryan, A. J., Gough, T., Vafabakhsh, R., & Golestanian, R. (2007) · Physical Review Letters · doi:10.1103/physrevlett.99.048102 · Cited by 2,230

Nonequilibrium Statistical Mechanics

Zwanzig, R. (2001) · doi:10.1093/oso/9780195140187.001.0001 · Cited by 2,051

Investigations on the Theory of the Brownian Movement

Topping, J. (1956) · Physics Bulletin · doi:10.1088/0031-9112/7/10/012 · Cited by 2,033

Physical aging in amorphous polymers and other materials

Haward, R. N. (1979) · Polymer · doi:10.1016/0032-3861(79)90237-4 · Cited by 1,873

Investigations on the theory of the Brownian movement

Einstein, A. (1926) · BURJC Digital (King Juan Carlos University) · Cited by 1,861

SINGLE-PARTICLE TRACKING:Applications to Membrane Dynamics

Saxton, M. J., & Jacobson, K. (1997) · Annual Review of Biophysics and Biomolecular Structure · doi:10.1146/annurev.biophys.26.1.373 · Cited by 1,810

Atmospheric diffusion shown on a distance-neighbour graph

Richardson, L. F. (1926) · Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character · doi:10.1098/rspa.1926.0043 · Cited by 1,673

Physical aging in amorphous polymers and other materials

Struik, L. C. E. (1978) · Research Repository (Delft University of Technology) · Cited by 1,664

Chaos, fractional kinetics, and anomalous transport

ZASLAVSKY, G. (2002) · Physics Reports · doi:10.1016/s0370-1573(02)00331-9 · Cited by 1,612

Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen

von Smoluchowski, M. (1906) · Annalen der Physik · doi:10.1002/andp.19063261405 · Cited by 1,495

Zur Elektronentheorie der Metalle

Drude, P. (1900) · Annalen der Physik · doi:10.1002/andp.19003060312 · Cited by 1,494

Lévy flight search patterns of wandering albatrosses

Viswanathan, G. M., Afanasyev, V., Buldyrev, S. V., Murphy, E. J., Prince, P. A., & Stanley, H. E. (1996) · Nature · doi:10.1038/381413a0 · Cited by 1,410

Optimizing the success of random searches.

Viswanathan, G. M., Buldyrev, S. V., Havlin, S., da Luz, M. G. E., Raposo, E. P., & Stanley, H. E. (1999) · Nature · doi:10.1038/44831 · Cited by 1,359

Nonequilibrium Statistical Mechanics

(1990) · doi:10.1142/1009 · Cited by 1,322

Modelling the scaling properties of human mobility

Song, C., Koren, T., Wang, P., & Barabási, A.-L. (2010) · Nature Physics · doi:10.1038/nphys1760 · Cited by 1,284

Anomalous transport in the crowded world of biological cells

Höfling, F., & Franosch, T. (2013) · Reports on Progress in Physics · doi:10.1088/0034-4885/76/4/046602 · Cited by 1,247

High-density mapping of single-molecule trajectories with photoactivated localization microscopy

Manley, S., Gillette, J. M., Patterson, G. H., Shroff, H., Hess, H. F., Betzig, E., & Lippincott-Schwartz, J. (2008) · Nature Methods · doi:10.1038/nmeth.1176 · Cited by 1,199

Modeling non‐Fickian transport in geological formations as a continuous time random walk

Berkowitz, B., Cortis, A., Dentz, M., & Scher, H. (2006) · Reviews of Geophysics · doi:10.1029/2005rg000178 · Cited by 1,168

Illuminating Single Molecules in Condensed Matter

Moerner, W. E., & Orrit, M. (1999) · Science · doi:10.1126/science.283.5408.1670 · Cited by 1,129

Diffusion in regular and disordered lattices

Haus, J. W., & Kehr, K. W. (1987) · Physics Reports · doi:10.1016/0370-1573(87)90005-6 · Cited by 1,122

Random Walks And Random Environments

Hughes, B. D. (1995) · doi:10.1093/oso/9780198537885.001.0001 · Cited by 1,083

XXVII. A brief account of microscopical observations made in the months of June, July and August 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies

Brown, R. (1828) · The Philosophical Magazine · doi:10.1080/14786442808674769 · Cited by 1,060

Stochastic Calculus for Fractional Brownian Motion and Applications

Biagini, F., Hu, Y., Øksendal, B., & Zhang, T. (2008) · Probability and its applications · doi:10.1007/978-1-84628-797-8 · Cited by 1,054

Strange kinetics

Shlesinger, M. F., Zaslavsky, G. M., & Klafter, J. (1993) · Nature · doi:10.1038/363031a0 · Cited by 1,051

Anomalous Diffusion on Percolating Clusters

Gefen, Y., Aharony, A., & Alexander, S. (1983) · Physical Review Letters · doi:10.1103/physrevlett.50.77 · Cited by 1,029

The Problem of the Random Walk

PEARSON, K. (1905) · Nature · doi:10.1038/072294b0 · Cited by 960

A Modern Course in Statistical Physics

Brereton, M. (1981) · Physics Bulletin · doi:10.1088/0031-9112/32/3/049 · Cited by 955

Environmental context explains Lévy and Brownian movement patterns of marine predators

Humphries, N. E., Queiroz, N., Dyer, J. R. M., Pade, N. G., Musyl, M. K., Schaefer, K. M., Fuller, D. W., Brunnschweiler, J. M., Doyle, T. K., Houghton, J. D. R., et al. (2010) · Nature · doi:10.1038/nature09116 · Cited by 943

Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer

Edwards, A. M., Phillips, R. A., Watkins, N. W., Freeman, M. P., Murphy, E. J., Afanasyev, V., Buldyrev, S. V., da Luz, M. G. E., Raposo, E. P., Stanley, H. E., et al. (2007) · Nature · doi:10.1038/nature06199 · Cited by 925

Physical Nature of Bacterial Cytoplasm

Golding, I., & Cox, E. C. (2006) · Physical Review Letters · doi:10.1103/physrevlett.96.098102 · Cited by 901

Stochastic Process with Ultraslow Convergence to a Gaussian: The Truncated Lévy Flight

Mantegna, R. N., & Stanley, H. E. (1994) · Physical Review Letters · doi:10.1103/physrevlett.73.2946 · Cited by 880

An Introduction to Probability Theory and Its Application.

Blackwell, D., & Feller, W. (1951) · Journal of the American Statistical Association · doi:10.2307/2280517 · Cited by 866

Protein Conformational Dynamics Probed by Single-Molecule Electron Transfer

Yang, H., Luo, G., Karnchanaphanurach, P., Louie, T.-M., Rech, I., Cova, S., Xun, L., & Xie, X. S. (2003) · Science · doi:10.1126/science.1086911 · Cited by 843

Real-Time Single-Molecule Imaging of the Infection Pathway of an Adeno-Associated Virus

Seisenberger, G., Ried, M. U., Endreß, T., Büning, H., Hallek, M., & Bräuchle, C. (2001) · Science · doi:10.1126/science.1064103 · Cited by 820

Anomalous Diffusion and Relaxation Close to Thermal Equilibrium: A Fractional Fokker-Planck Equation Approach

Metzler, R., Barkai, E., & Klafter, J. (1999) · Physical Review Letters · doi:10.1103/physrevlett.82.3563 · Cited by 777

Anomalous transport in the crowded world of biological cells

Höfling, F., & Franosch, T. (2013) · MPG.PuRe (Max Planck Society) · Cited by 773

Correlation theory of stationary and related random functions

Iaglom, A. M. (1987) · Springer eBooks · Cited by 769

Intermittent search strategies

Bénichou, O., Loverdo, C., Moreau, M., & Voituriez, R. (2011) · Reviews of Modern Physics · doi:10.1103/revmodphys.83.81 · Cited by 765

A Lévy flight for light

Barthelemy, P., Bertolotti, J., & Wiersma, D. S. (2008) · Nature · doi:10.1038/nature06948 · Cited by 743

An introduction to probability theory and its application

Charp, S. (1950) · Journal of the Franklin Institute · doi:10.1016/0016-0032(50)90714-9 · Cited by 736

Hamiltonian Chaos and Fractional Dynamics

Zaslavsky, G. M. (2004) · doi:10.1093/oso/9780198526049.001.0001 · Cited by 726

Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow

Solomon, T., Weeks, E. R., & Swinney, H. L. (1993) · Physical Review Letters · doi:10.1103/physrevlett.71.3975 · Cited by 721

Stochastic pathway to anomalous diffusion

Klafter, J., Blumen, A., & Shlesinger, M. F. (1987) · Physical Review A · doi:10.1103/physreva.35.3081 · Cited by 719

Strange kinetics of single molecules in living cells

Barkai, E., Garini, Y., & Metzler, R. (2012) · Physics Today · doi:10.1063/pt.3.1677 · Cited by 683

Diffusion in a field of homogeneous turbulence

Batchelor, G. K. (1952) · Mathematical Proceedings of the Cambridge Philosophical Society · doi:10.1017/s0305004100027687 · Cited by 659

LXXV. A dynamical theory of diffusion for non-electrolytes and the molecular mass of albumin

Sutherland, W. (1905) · The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science · doi:10.1080/14786440509463331 · Cited by 658

In Vivo Anomalous Diffusion and Weak Ergodicity Breaking of Lipid Granules

Jeon, J.-H., Tejedor, V., Burov, S., Barkai, E., Selhuber-Unkel, C., Berg-Sørensen, K., Oddershede, L., & Metzler, R. (2011) · Physical Review Letters · doi:10.1103/physrevlett.106.048103 · Cited by 652

Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking

Weigel, A. V., Simon, B., Tamkun, M. M., & Krapf, D. (2011) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1016325108 · Cited by 648

Bacterial Chromosomal Loci Move Subdiffusively through a Viscoelastic Cytoplasm

Weber, S. C., Spakowitz, A. J., & Theriot, J. A. (2010) · Physical Review Letters · doi:10.1103/physrevlett.104.238102 · Cited by 633

Stochastic models of intracellular transport

Bressloff, P. C., & Newby, J. M. (2013) · Reviews of Modern Physics · doi:10.1103/revmodphys.85.135 · Cited by 629

Anomalous Diffusion Probes Microstructure Dynamics of Entangled F-Actin Networks

Wong, I. Y., Gardel, M. L., Reichman, D. R., Weeks, E. R., Valentine, M. T., Bausch, A. R., & Weitz, D. A. (2004) · Physical Review Letters · doi:10.1103/physrevlett.92.178101 · Cited by 621

The Glass Transition: Relaxation Dynamics in Liquids and Disordered Materials

Donth, E.-J. (2001) · Springer Series in MATERIALS SCIENCE · doi:10.1007/978-3-662-04365-3 · Cited by 611

Fractals in the Physical Sciences

Hideki Takayasu, C. (1990) · Cited by 606

When Brownian diffusion is not Gaussian

Wang, B., Kuo, J., Bae, S. C., & Granick, S. (2012) · Nature Materials · doi:10.1038/nmat3308 · Cited by 581

Biological measurement beyond the quantum limit

Taylor, M. A., Janousek, J., Daria, V., Knittel, J., Hage, B., Bachor, H.-A., & Bowen, W. P. (2013) · Nature Photonics · doi:10.1038/nphoton.2012.346 · Cited by 572

Models of anomalous diffusion in crowded environments

Sokolov, I. M. (2012) · Soft Matter · doi:10.1039/c2sm25701g · Cited by 565

The Limiting Behavior of a One-Dimensional Random Walk in a Random Medium

Sinaǐ, Y. G. (1983) · Theory of Probability and Its Applications · doi:10.1137/1127028 · Cited by 560

Generalized Langevin Equation with Fractional Gaussian Noise: Subdiffusion within a Single Protein Molecule

Kou, S. C., & Xie, X. S. (2004) · Physical Review Letters · doi:10.1103/physrevlett.93.180603 · Cited by 560

Random Time-Scale Invariant Diffusion and Transport Coefficients

He, Y., Burov, S., Metzler, R., & Barkai, E. (2008) · Physical Review Letters · doi:10.1103/physrevlett.101.058101 · Cited by 554

Random Walks on Lattices. III. Calculation of First-Passage Times with Application to Exciton Trapping on Photosynthetic Units

Montroll, E. W. (1969) · Journal of Mathematical Physics · doi:10.1063/1.1664902 · Cited by 550

Single-molecule microscopy on model membranes reveals anomalous diffusion

Schütz, G. J., Schindler, H., & Schmidt, T. (1997) · Biophysical Journal · doi:10.1016/s0006-3495(97)78139-6 · Cited by 540

The Physics of Foraging: An Introduction to Random Searches and Biological Encounters

Viswanathan, G. M., da Luz, M. G. E., Raposo, E. P., & Stanley, H. E. (2011) · Cambridge University Press eBooks · doi:10.1017/cbo9780511902680 · Cited by 533

Boundary value problems for fractional diffusion equations

Metzler, R., & Klafter, J. (2000) · Physica A Statistical Mechanics and its Applications · doi:10.1016/s0378-4371(99)00503-8 · Cited by 516

Models of traps and glass phenomenology

Monthus, C., & Bouchaud, J.-P. (1996) · Journal of Physics A: Mathematical and General · doi:10.1088/0305-4470/29/14/012 · Cited by 499

Transient Anomalous Diffusion of Telomeres in the Nucleus of Mammalian Cells

Bronstein, I., Israel, Y., Kepten, E., Mai, S., Shav-Tal, Y., Barkai, E., & Garini, Y. (2009) · Physical Review Letters · doi:10.1103/physrevlett.103.018102 · Cited by 495

Constrained diffusion or immobile fraction on cell surfaces: a new interpretation

Feder, T. J., Brust-Mascher, I., Slattery, J. P., Baird, B., & Webb, W. W. (1996) · Biophysical Journal · doi:10.1016/s0006-3495(96)79846-6 · Cited by 474

Fractional Fokker-Planck equation, solution, and application

Barkai, E. (2001) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.63.046118 · Cited by 464

Correlation Theory of Stationary and Related Random Functions

Yaglom, A. M. (1987) · Springer series in statistics · doi:10.1007/978-1-4612-4628-2 · Cited by 447

Slow relaxation and compaction of granular systems

Richard, P., Nicodemi, M., Delannay, R., Ribière, P., & Bideau, D. (2005) · Nature Materials · doi:10.1038/nmat1300 · Cited by 440

Light Scattering from Polymer Solutions and Nanoparticle Dispersions

(2007) · Springer Laboratory · doi:10.1007/978-3-540-71951-9 · Cited by 431

Ultrafine Membrane Compartments for Molecular Diffusion as Revealed by Single Molecule Techniques

Murase, K., Fujiwara, T., Umemura, Y., Suzuki, K., Iino, R., Yamashita, H., Saito, M., Murakoshi, H., Ritchie, K., & Kusumi, A. (2004) · Biophysical Journal · doi:10.1529/biophysj.103.035717 · Cited by 431

Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid

Huang, R., Chavez, I., Taute, K. M., Lukić, B., Jeney, S., Raizen, M. G., & Florin, E.-L. (2011) · Nature Physics · doi:10.1038/nphys1953 · Cited by 424

Fluorescence Correlation Spectroscopy: Theory and Applications

Rigler, R., & Elson, E. S. (2001) · Springer Series in Chemical Physics · doi:10.1007/978-3-642-59542-4 · Cited by 413

Diffusing Diffusivity: A Model for Anomalous, yet Brownian, Diffusion

Chubynsky, M. V., & Slater, G. W. (2014) · Physical Review Letters · doi:10.1103/physrevlett.113.098302 · Cited by 413

Lévy flights in external force fields: Langevin and fractional Fokker-Planck equations and their solutions

Jespersen, S. N., Metzler, R., & Fogedby, H. C. (1999) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.59.2736 · Cited by 399

Resonances arising from hydrodynamic memory in Brownian motion

Franosch, T., Grimm, M., Belushkin, M., Mor, F. M., Foffi, G., Forró, L., & Jeney, S. (2011) · Nature · doi:10.1038/nature10498 · Cited by 392

Statistical Aging and Nonergodicity in the Fluorescence of Single Nanocrystals

Brokmann, X., Hermier, J.-P., Messin, G., Desbiolles, P., Bouchaud, J.-P., & Dahan, M. (2003) · Physical Review Letters · doi:10.1103/physrevlett.90.120601 · Cited by 391

Single particle tracking in systems showing anomalous diffusion: the role of weak ergodicity breaking

Burov, S., Jeon, J.-H., Metzler, R., & Barkai, E. (2011) · Physical Chemistry Chemical Physics · doi:10.1039/c0cp01879a · Cited by 389

Random Walks and Random Environments: Volume 1: Random Walks

Lee, P. M. (1995) · Cited by 388

Scale-invariant motion in intermittent chaotic systems

Zumofen, G., & Klafter, J. (1993) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.47.851 · Cited by 366

Intracellular transport of insulin granules is a subordinated random walk

Tabei, S. M. A., Burov, S., Kim, H. Y., Kuznetsov, A., Huynh, T., Jureller, J., Philipson, L. H., Dinner, A. R., & Scherer, N. F. (2013) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1221962110 · Cited by 359

Accelerated Diffusion in Josephson Junctions and Related Chaotic Systems

Geisel, T., Nierwetberg, J., & Zacherl, A. (1985) · Physical Review Letters · doi:10.1103/physrevlett.54.616 · Cited by 358

Single-Molecule Approach to Molecular Biology in Living Bacterial Cells

Xie, X. S., Choi, P. J., Li, G.-W., Lee, N. K., & Lia, G. (2008) · Annual Review of Biophysics · doi:10.1146/annurev.biophys.37.092607.174640 · Cited by 351

Single-molecule spectral fluctuations at room temperature

Lu, H. P., & Xie, X. S. (1997) · Nature · doi:10.1038/385143a0 · Cited by 342

The Physics of Foraging: An Introduction to Random Searches and Biological Encounters

Viswanathan, G. M., da Luz, M. G. E., Raposo, E. P., & Stanley, E. (2011) · Cited by 341

Langevin equations for continuous time Lévy flights

Fogedby, H. C. (1994) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.50.1657 · Cited by 335

Optimizing the Encounter Rate in Biological Interactions: Lévy versus Brownian Strategies

Bartumeus, F., Catalan, J., Fulco, U. L., Lyra, M. L., & Viswanathan, G. M. (2002) · Physical Review Letters · doi:10.1103/physrevlett.88.097901 · Cited by 333

Nonergodicity Mimics Inhomogeneity in Single Particle Tracking

Lubelski, A., Sokolov, I. M., & Klafter, J. (2008) · Physical Review Letters · doi:10.1103/physrevlett.100.250602 · Cited by 328

Statistics of the Occupation Time of Renewal Processes

Godrèche, C., & Luck, J. M. (2001) · Journal of Statistical Physics · doi:10.1023/a:1010364003250 · Cited by 322

Deriving fractional Fokker-Planck equations from a generalised master equation

Metzler, R., Barkai, E., & Klafter, J. (1999) · Europhysics Letters (EPL) · doi:10.1209/epl/i1999-00279-7 · Cited by 300

Cajal Body dynamics and association with chromatin are ATP-dependent

Platani, M., Goldberg, I., Lamond, A. I., & Swedlow, J. R. (2002) · Nature Cell Biology · doi:10.1038/ncb809 · Cited by 285

Correlation Theory of Stationary and Related Random Functions

Yaglom, A. M. (1987) · Springer Series in Statistics · doi:10.1007/978-1-4612-4620-6 · Cited by 282

Lévy strategies in intermittent search processes are advantageous

Lomholt, M. A., Tal, K., Metzler, R., & Joseph, K. (2008) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0803117105 · Cited by 281

Probing the Nanoscale Viscoelasticity of Intracellular Fluids in Living Cells

Guigas, G., Kalla, C., & Weiss, M. (2007) · Biophysical Journal · doi:10.1529/biophysj.106.099267 · Cited by 281

Anomalous Subdiffusion in Fluorescence Photobleaching Recovery: A Monte Carlo Study

Saxton, M. J. (2001) · Biophysical Journal · doi:10.1016/s0006-3495(01)75870-5 · Cited by 275

The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering

Coffey, W. T., & Kalmykov, Y. P. (2012) · World Scientific Series in Contemporary Chemical Physics · doi:10.1142/8195 · Cited by 256

Anomalous Diffusion in Intermittent Chaotic Systems

Geisel, T., & Thomae, S. (1984) · Physical Review Letters · doi:10.1103/physrevlett.52.1936 · Cited by 248

Anomalous diffusion and power-law relaxation of the time averaged mean squared displacement in worm-like micellar solutions

Jeon, J.-H., Leijnse, N., Oddershede, L. B., & Metzler, R. (2013) · New Journal of Physics · doi:10.1088/1367-2630/15/4/045011 · Cited by 248

Beyond quantum jumps: Blinking nanoscale light emitters

Stefani, F. D., Hoogenboom, J. P., & Barkai, E. (2009) · Physics Today · doi:10.1063/1.3086100 · Cited by 241

Thermally activated escape rate in presence of long-time memory

Hanggi, P., & Mojtabai, F. (1982) · Physical Review A · doi:10.1103/physreva.26.1168 · Cited by 239

Viscoelastic subdiffusion: From anomalous to normal

Goychuk, I. (2009) · Physical Review E · doi:10.1103/physreve.80.046125 · Cited by 234

Flow Intermittency, Dispersion, and Correlated Continuous Time Random Walks in Porous Media

de Anna, P., Borgne, T. L., Dentz, M., Tartakovsky, A. M., Bolster, D., & Davy, P. (2013) · Physical Review Letters · doi:10.1103/physrevlett.110.184502 · Cited by 234

On strong anomalous diffusion

Castiglione, P., Mazzino, A., Muratore-Ginanneschi, P., & Vulpiani, A. (1999) · Physica D Nonlinear Phenomena · doi:10.1016/s0167-2789(99)00031-7 · Cited by 228

Sampling the Cell with Anomalous Diffusion—The Discovery of Slowness

Guigas, G., & Weiss, M. (2008) · Biophysical Journal · doi:10.1529/biophysj.107.117044 · Cited by 220

Space- and time-fractional diffusion and wave equations, fractional Fokker–Planck equations, and physical motivation

Metzler, R., & Nonnenmacher, T. F. (2002) · Chemical Physics · doi:10.1016/s0301-0104(02)00537-2 · Cited by 218

Probing microscopic origins of confined subdiffusion by first-passage observables

Condamin, S., Tejedor, V., Voituriez, R., Bénichou, O., & Klafter, J. (2008) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0712158105 · Cited by 218

Subrecoil laser cooling and Lévy flights

Bardou, F., Bouchaud, J. P., Emile, O., Aspect, A., & Cohen-Tannoudji, C. (1994) · Physical Review Letters · doi:10.1103/physrevlett.72.203 · Cited by 216

Nonergodic Subdiffusion from Brownian Motion in an Inhomogeneous Medium

Massignan, P., Manzo, C., Torreno-Pina, J. A., Parajo, M. F. G., Lewenstein, M., & Lapeyre, G. J. (2014) · Physical Review Letters · doi:10.1103/physrevlett.112.150603 · Cited by 215

Crumpling a Thin Sheet

Matan, K., Williams, R. B., Witten, T. A., & Nagel, S. R. (2002) · Physical Review Letters · doi:10.1103/physrevlett.88.076101 · Cited by 215

Anomalous and normal diffusion of proteins and lipids in crowded lipid membranes

Javanainen, M., Hammaren, H., Monticelli, L., Jeon, J.-H., Miettinen, M. S., Martinez-Seara, H., Metzler, R., & Vattulainen, I. (2013) · Faraday Discuss. · doi:10.1039/c2fd20085f · Cited by 209

Principles of Statistical Radiophysics 1

Rytov, S. M., Kravtsov, Y. A., & Tatarskii, V. I. (1987) · Springer eBooks · doi:10.1007/978-3-642-69201-7 · Cited by 209

Encounter success of free-ranging marine predator movements across a dynamic prey landscape

Sims, D. W., Witt, M. J., Richardson, A. J., Southall, E. J., & Metcalfe, J. D. (2006) · Proceedings of the Royal Society B: Biological Sciences · doi:10.1098/rspb.2005.3444 · Cited by 201

Stochastic modeling in nanoscale biophysics: Subdiffusion within proteins

Kou, S. C. (2008) · The Annals of Applied Statistics · doi:10.1214/07-aoas149 · Cited by 199

Single-particle tracking: effects of corrals

Saxton, M. J. (1995) · Biophysical Journal · doi:10.1016/s0006-3495(95)79911-8 · Cited by 198

Lévy Flights in Random Environments

Fogedby, H. C. (1994) · Physical Review Letters · doi:10.1103/physrevlett.73.2517 · Cited by 198

Importance of Particle Tracking and Calculating the Mean-Squared Displacement in Distinguishing Nanopropulsion from Other Processes

Dunderdale, G., Ebbens, S., Fairclough, P., & Howse, J. (2012) · Langmuir · doi:10.1021/la301370y · Cited by 197

Correlation functions and masterequations of generalized (non-Markovian) Langevin equations

H�nggi, P. (1978) · The European Physical Journal B · doi:10.1007/bf01351552 · Cited by 196

Aging and nonergodicity beyond the Khinchin theorem

Burov, S., Metzler, R., & Barkai, E. (2010) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1003693107 · Cited by 195

Some properties of a random walk on a comb structure

Weiss, G. H., & Havlin, S. (1986) · Physica A Statistical Mechanics and its Applications · doi:10.1016/0378-4371(86)90060-9 · Cited by 191

Observation of Anomalous Diffusion and Fractional Self-Similarity in One Dimension

Sagi, Y., Brook, M., Almog, I., & Davidson, N. (2012) · Physical Review Letters · doi:10.1103/physrevlett.108.093002 · Cited by 186

Protein Diffusion in Mammalian Cell Cytoplasm

Kühn, T., Ihalainen, T. O., Hyväluoma, J., Dross, N., Willman, S. F., Langowski, J., Vihinen-Ranta, M., & Timonen, J. (2011) · PLoS ONE · doi:10.1371/journal.pone.0022962 · Cited by 182

Motion Analysis of Self-Propelled Pt−Silica Particles in Hydrogen Peroxide Solutions

Ke, H., Ye, S., Carroll, R. L., & Showalter, K. (2010) · The Journal of Physical Chemistry A · doi:10.1021/jp101193u · Cited by 180

Particle tracking in living cells: a review of the mean square displacement method and beyond

Gal, N., Lechtman-Goldstein, D., & Weihs, D. (2013) · Rheologica Acta · doi:10.1007/s00397-013-0694-6 · Cited by 179

Beyond Boltzmann–Gibbs statistical mechanics in optical lattices

Lutz, E., & Renzoni, F. (2013) · Nature Physics · doi:10.1038/nphys2751 · Cited by 176

Aging in Subdiffusion Generated by a Deterministic Dynamical System

Barkai, E. (2003) · Physical Review Letters · doi:10.1103/physrevlett.90.104101 · Cited by 168

Universal Algorithm for Identification of Fractional Brownian Motion. A Case of Telomere Subdiffusion

Burnecki, K., Kepten, E., Janczura, J., Bronshtein, I., Garini, Y., & Weron, A. (2012) · Biophysical Journal · doi:10.1016/j.bpj.2012.09.040 · Cited by 167

Aging continuous time random walks

Barkai, E., & Cheng, Y. (2003) · The Journal of Chemical Physics · doi:10.1063/1.1559676 · Cited by 160

Fractional Brownian motion in crowded fluids

Ernst, D., Hellmann, M., Köhler, J. R., & Weiß, M. (2012) · Soft Matter · doi:10.1039/c2sm25220a · Cited by 152

Leapover Lengths and First Passage Time Statistics for Lévy Flights

Koren, T., Lomholt, M. A., Chechkin, A. V., Klafter, J., & Metzler, R. (2007) · Physical Review Letters · doi:10.1103/physrevlett.99.160602 · Cited by 146

Distribution of Time-Averaged Observables for Weak Ergodicity Breaking

Rebenshtok, A., & Barkai, E. (2007) · Physical Review Letters · doi:10.1103/physrevlett.99.210601 · Cited by 142

Visualization and Tracking of Single Protein Molecules in the Cell Nucleus

Kues, T., Peters, R., & Kubitscheck, U. (2001) · Biophysical Journal · doi:10.1016/s0006-3495(01)76261-3 · Cited by 142

Fractional Langevin equation: Overdamped, underdamped, and critical behaviors

Burov, S., & Barkai, E. (2008) · Physical Review E · doi:10.1103/physreve.78.031112 · Cited by 139

In Vivo Determination of Fluctuating Forces during Endosome Trafficking Using a Combination of Active and Passive Microrheology

Robert, D., Nguyen, T.-H., Gallet, F., & Wilhelm, C. (2010) · PLoS ONE · doi:10.1371/journal.pone.0010046 · Cited by 133

Exact and Explicit Probability Densities for One-Sided Lévy Stable Distributions

Penson, K. A., & Górska, K. (2010) · Physical Review Letters · doi:10.1103/physrevlett.105.210604 · Cited by 131

Population splitting, trapping, and non-ergodicity in heterogeneous diffusion processes

Cherstvy, A. G., & Metzler, R. (2013) · Physical Chemistry Chemical Physics · doi:10.1039/c3cp53056f · Cited by 128

Viscoelasticity in Homogeneous Protein Solutions

Pan, W., Filobelo, L. F., Pham, N., Galkin, O., Uzunova, V., & Vekilov, P. G. (2009) · Physical Review Letters · doi:10.1103/physrevlett.102.058101 · Cited by 120

Subdiffusion and Weak Ergodicity Breaking in the Presence of a Reactive Boundary

Lomholt, M. A., Zaid, I. M., & Metzler, R. (2007) · Physical Review Letters · doi:10.1103/physrevlett.98.200603 · Cited by 119

Lineshape theory and photon counting statistics for blinking quantum dots: a Lévy walk process

Jung, Y., Barkai, E., & Silbey, R. (2002) · Chemical Physics · doi:10.1016/s0301-0104(02)00547-5 · Cited by 118

Infinite Covariant Density for Diffusion in Logarithmic Potentials and Optical Lattices

Kessler, D. A., & Barkai, E. (2010) · Physical Review Letters · doi:10.1103/physrevlett.105.120602 · Cited by 118

Critical Exponent of the Fractional Langevin Equation

Burov, S., & Barkai, E. (2008) · Physical Review Letters · doi:10.1103/physrevlett.100.070601 · Cited by 117

Particle invasion, survival, and non-ergodicity in 2D diffusion processes with space-dependent diffusivity

Cherstvy, A. G., Chechkin, A. V., & Metzler, R. (2014) · Soft Matter · doi:10.1039/c3sm52846d · Cited by 115

Wanted: A Positive Control for Anomalous Subdiffusion

Saxton, M. J. (2012) · Biophysical Journal · doi:10.1016/j.bpj.2012.10.038 · Cited by 112

Time scale of diffusion in molecular and cellular biology

Holcman, D., & Schuss, Z. (2014) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/47/17/173001 · Cited by 109

Nonergodicity, fluctuations, and criticality in heterogeneous diffusion processes

Cherstvy, A. G., & Metzler, R. (2014) · Physical Review E · doi:10.1103/physreve.90.012134 · Cited by 109

Improved estimation of anomalous diffusion exponents in single-particle tracking experiments

Kepten, E., Bronshtein, I., & Garini, Y. (2013) · Physical Review E · doi:10.1103/physreve.87.052713 · Cited by 107

Quantifying Hopping and Jumping in Facilitated Diffusion of DNA-Binding Proteins

Loverdo, C., Bénichou, O., Voituriez, R., Biebricher, A. S., Bonnet, I., & Desbiolles, P. (2009) · Physical Review Letters · doi:10.1103/physrevlett.102.188101 · Cited by 107

Noisy continuous time random walks

Jeon, J., Barkai, E., & Metzler, R. (2013) · The Journal of Chemical Physics · doi:10.1063/1.4816635 · Cited by 106

Theory of Fractional Lévy Kinetics for Cold Atoms Diffusing in Optical Lattices

Kessler, D. A., & Barkai, E. (2012) · Physical Review Letters · doi:10.1103/physrevlett.108.230602 · Cited by 105

Inferring diffusion in single live cells at the single-molecule level

Robson, A., Burrage, K., & Leake, M. C. (2012) · Philosophical Transactions of the Royal Society B Biological Sciences · doi:10.1098/rstb.2012.0029 · Cited by 104

Subdiffusion of a Sticky Particle on a Surface

Xu, Q., Feng, L., Sha, R., Seeman, N. C., & Chaikin, P. M. (2011) · Physical Review Letters · doi:10.1103/physrevlett.106.228102 · Cited by 101

Fluctuations of 1/f Noise and the Low-Frequency Cutoff Paradox

Niemann, M., Kantz, H., & Barkai, E. (2013) · Physical Review Letters · doi:10.1103/physrevlett.110.140603 · Cited by 101

Mechanical Properties of Solid Polymers

Krevelen, D. W. V., & Nijenhuis, K. T. (2009) · Elsevier eBooks · doi:10.1016/b978-0-08-054819-7.00013-3 · Cited by 101

Subdiffusion and localization in the one-dimensional trap model

Bertin, É., & Bouchaud, J. (2003) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.67.026128 · Cited by 99

Non-Normalizable Densities in Strong Anomalous Diffusion: Beyond the Central Limit Theorem

Rebenshtok, A., Denisov, S., Peter Hänggi, A., & Barkai, E. (2014) · Physical Review Letters · doi:10.1103/physrevlett.112.110601 · Cited by 99

Levy Statistics and Laser Cooling

Bardou, F., Bouchaud, J., Aspect, A., & Cohen-Tannoudji, C. N. (2001) · Cambridge University Press eBooks · doi:10.1017/cbo9780511755668 · Cited by 98

Diffusion on random-site percolation clusters: Theory and NMR microscopy experiments with model objects

Klemm, A., Metzler, R., & Kimmich, R. (2002) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.65.021112 · Cited by 94

Joint probability distributions for a class of non-Markovian processes

Baule, A., & Friedrich, R. (2005) · Physical Review E · doi:10.1103/physreve.71.026101 · Cited by 93

How Molecular Motors Work in the Crowded Environment of Living Cells: Coexistence and Efficiency of Normal and Anomalous Transport

Goychuk, I., Kharchenko, V. O., & Metzler, R. (2014) · PLoS ONE · doi:10.1371/journal.pone.0091700 · Cited by 87

Continuous-time random walk with correlated waiting times

Chechkin, A. V., Hofmann, M., & Sokolov, I. M. (2009) · Physical Review E · doi:10.1103/physreve.80.031112 · Cited by 86

Occupation Time Statistics in the Quenched Trap Model

Burov, S., & Barkai, E. (2007) · Physical Review Letters · doi:10.1103/physrevlett.98.250601 · Cited by 84

Fractional diffusion, waiting-time distributions, and Cattaneo-type equations

Metzler, R., & Nonnenmacher, T. F. (1998) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.57.6409 · Cited by 84

Subdiffusion in time-averaged, confined random walks

Neusius, T., Sokolov, I. M., & Smith, J. C. (2009) · Physical Review E · doi:10.1103/physreve.80.011109 · Cited by 84

Time-averaged Einstein relation and fluctuating diffusivities for the Lévy walk

Froemberg, D., & Barkai, E. (2013) · Physical Review E · doi:10.1103/physreve.87.030104 · Cited by 83

Power spectra and random walks in intermittent chaotic systems

Zumofen, G., & Klafter, J. (1993) · Physica D Nonlinear Phenomena · doi:10.1016/0167-2789(93)90105-a · Cited by 81

Nonergodicity of a Time Series Obeying Lévy Statistics

Margolin, G., & Barkai, E. (2005) · Journal of Statistical Physics · doi:10.1007/s10955-005-8076-9 · Cited by 81

Anomalous diffusion: Testing ergodicity breaking in experimental data

Magdziarz, M., & Weron, A. (2011) · Physical Review E · doi:10.1103/physreve.84.051138 · Cited by 77

Dynamics in Crowded Environments: Is Non-Gaussian Brownian Diffusion Normal?

Kwon, G., Sung, B. J., & Yethiraj, A. (2014) · The Journal of Physical Chemistry B · doi:10.1021/jp5011617 · Cited by 77

Distribution- Versus Correlation-Induced Anomalous Transport in Quenched Random Velocity Fields

Dentz, M., & Bolster, D. (2010) · Physical Review Letters · doi:10.1103/physrevlett.105.244301 · Cited by 76

Population splitting, trapping, and non-ergodicity in heterogeneous diffusion processes

Cherstvy, A. G., & Metzler, R. (2013) · publish.UP (University of Potsdam) · Cited by 76

Kramers-like escape driven by fractional Gaussian noise

Sliusarenko, O. Y., Gonchar, V. Y., Chechkin, A. V., Sokolov, I. M., & Metzler, R. (2010) · Physical Review E · doi:10.1103/physreve.81.041119 · Cited by 72

Analysis of short subdiffusive time series: scatter of the time-averaged mean-squared displacement

Jeon, J., & Metzler, R. (2010) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/43/25/252001 · Cited by 72

Photon Counting Statistics for Blinking CdSe−ZnS Quantum Dots: A Lévy Walk Process

Margolin, G., Protasenko, V., Kuno, M. K., & Barkai, E. (2006) · The Journal of Physical Chemistry B · doi:10.1021/jp061487m · Cited by 71

Lévy walks and scaling in quenched disordered media

Burioni, R., Caniparoli, L., & Vezzani, A. (2010) · Physical Review E · doi:10.1103/physreve.81.060101 · Cited by 68

Mikroskopische Beobachtungen über die im Pollen der Pflanzen enthaltenen Partikeln, und über das allgemeine Vorkommen activer Molecüle in organischen und unorganischen Körpern

Brown, R. L. (1828) · Annalen der Physik · doi:10.1002/andp.18280901016 · Cited by 66

Anomalous diffusion on a random comblike structure

Havlin, S., Kiefer, J. E., & Weiss, G. H. (1987) · Physical Review A · doi:10.1103/physreva.36.1403 · Cited by 66

Probability distribution of the time-averaged mean-square displacement of a Gaussian process

Grebenkov, D. S. (2011) · Physical Review E · doi:10.1103/physreve.84.031124 · Cited by 62

Microscopic Origin of the Logarithmic Time Evolution of Aging Processes in Complex Systems

Lomholt, M. A., Lizana, L., Metzler, R., & Ambjörnsson, T. (2013) · Physical Review Letters · doi:10.1103/physrevlett.110.208301 · Cited by 62

Intrinsic randomness of transport coefficient in subdiffusion with static disorder

Miyaguchi, T., & Akimoto, T. (2011) · Physical Review E · doi:10.1103/physreve.83.031926 · Cited by 59

Molecular motors pulling cargos in the viscoelastic cytosol: how power strokes beat subdiffusion

Goychuk, I., Kharchenko, V. O., & Metzler, R. (2014) · Physical Chemistry Chemical Physics · doi:10.1039/c4cp01234h · Cited by 57

Natural cutoff in Lévy flights caused by dissipative nonlinearity

Chechkin, A. V., Gonchar, V. Y., Klafter, J., & Metzler, R. (2005) · Physical Review E · doi:10.1103/physreve.72.010101 · Cited by 55

Distributional Response to Biases in Deterministic Superdiffusion

Akimoto, T. (2012) · Physical Review Letters · doi:10.1103/physrevlett.108.164101 · Cited by 55

Quantifying brain microstructure with diffusion MRI: Theory and parameter estimation

Novikov, D. S., Fieremans, E., Jespersen, S. N., & Kiselev, V. G. (2019) · NMR in Biomedicine · doi:10.1002/nbm.3998 · Cited by 581

Brownian yet Non-Gaussian Diffusion: From Superstatistics to Subordination of Diffusing Diffusivities

Chechkin, A. V., Seno, F., Metzler, R., & Sokolov, I. M. (2017) · Physical Review X · doi:10.1103/physrevx.7.021002 · Cited by 432

Molecular Dynamics Simulations of Membrane Permeability

Venable, R. M., Krämer, A., & Pastor, R. W. (2019) · Chemical Reviews · doi:10.1021/acs.chemrev.8b00486 · Cited by 394

Age-related microstructural and physiological changes in normal brain measured by MRI γ-metrics derived from anomalous diffusion signal representation

Guerreri, M., Palombo, M., Caporale, A., Fasano, F., Macaluso, E., Bozzali, M., & Capuani, S. (2019) · NeuroImage · doi:10.1016/j.neuroimage.2018.12.044 · Cited by 31