Diffusion: Theory, Modeling, Applications


Classical diffusion

P. Levy, Processus stochastiques et movement brownien (Paris, Gauthier-Villard, 1965).
W. Feller, An Introduction to Probability Theory and Its Applications, Volumes I and II, Second Edition (John Wiley & Sons, New York, 1971).
F. Spitzer, Principles of Random Walk (New York: Springer, 1976).
G. H. Weiss, Aspects and Applications of the Random Walk (North-Holland, Amsterdam, 1994).
S. Redner, A Guide to First-Passage Processes (Cambridge University Press, Cambridge, England, 2001).
H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids, 2nd Ed. (Clarendon, Oxford, 1959).
J. Crank, The Mathematics of Diffusion, 2nd Ed. (Clarendon, Oxford, 1975).
M. Freidlin, Functional Integration and Partial Differential Equations, Annals of Mathematics Studies (Princeton University, Princeton, New Jersey, 1985).
R. F. Bass, Diffusions and Elliptic Operators (Springer, 1998).
J. Klafter and I. M. Sokolov, First Steps in Random Walks: From Tools to Applications (Oxford University Press, 2011).
C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (2nd Ed., Springer, Berlin, 1985).
H. Risken, The Fokker-Planck equation. Methods of solution and applications (2ed., Springer, Berlin, 1989).
Z. Schuss, Brownian Dynamics at Boundaries and Interfaces: In Physics, Chemistry, and Biology (Applied Mathematical Sciences, Springer, New York, 2013).


Diffusion in complex geometries

B. Mandelbrot, The Fractal Geometry of Nature (San Francisco, Freeman, 1982).
J. Feder, Fractals (New York, Plenum Press, 1988).
K. J. Falconer, The Geometry of Fractal Sets (Cambridge, England Cambridge University Press, 1986).
D. ben-Avraham and S. Havlin, Diffusion and reactions in fractals and disordered systems (CUP, 2000).
J.-P. Bouchaud, A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep. 195, 127-293 (1990).
J. W. Haus, K. W. Kehr, Diffusion in regular and disordered lattices, Phys. Rep. 150, 263-406 (1987).
S. Havlin, D. ben Avraham, Diffusion in disordered media, Adv. Phys. 51, 187-292 (2002).
B. Sapoval, Transport Across Irregular Interfaces: Fractal Electrodes, Membranes and Catalysts, in Fractals and Disordered Systems, Eds. A. Bunde, S. Havlin, pp. 233-261 (Springer-Verlag, Berlin, 1996).


Continuous-Time Random Walks, fractional diffusion equation

I. Podlubny, Fractional Differential Equations, (Academic Press, London, 1999).
R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339, 1 (2000).
R. Metzler, J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A 37, R161 (2004).
G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep. 371, 461-580 (2002).
M. F. Shlesinger, J. Klafter, G. Zumofen, Above, below and beyond Brownian motion, Am. J. Phys. 67, 1253 (1999).
J. Klafter, M. F. Shlesinger, G. Zumofen, Beyond Brownian Motion, Physics Today, February 33 (1996).
M. F. Shlesinger, G. M. Zaslavsky, J. Klafter, Strange kinetics, Nature 363, 31 (1993).


Generalized Langevin equation

E. Lutz, Fractional Langevin equation, Phys. Rev. E 64, 051106 (2001).
A. D. Vinales and M. A. Desposito, Anomalous diffusion: Exact solution of the generalized Langevin equation for harmonically bounded particle, Phys. Rev. E 73, 016111 (2006).
M. A. Desposito and A. D. Vinales, Subdiffusive behavior in a trapping potential: Mean square displacement and velocity autocorrelation function, Phys. Rev. E 80, 021111 (2009).

Diffusion in a scalar field, NMR applications

D. S. Grebenkov, NMR Survey of Reflected Brownian Motion, Rev. Mod. Phys. 79, 1077-1137 (2007).
D. S. Grebenkov, Laplacian eigenfunctions in NMR. I. A Numerical Tool, Conc. Magn. Reson. 32A, 277 (2008).
D. S. Grebenkov, Laplacian eigenfunctions in NMR. II. Theoretical advances, Conc. Magn. Reson. 34A, 264 (2009).


Searching problems and intermittent diffusion

O. Benichou, C. Loverdo, M. Moreau, R. Voituriez, Intermittent search strategies, Rev. Mod. Phys. 83, 81-130 (2011).
O. Benichou, D. S. Grebenkov, P. E. Levitz, C. Loverdo, R. Voituriez, Optimal Reaction Time for Surface-Mediated Diffusion, Phys. Rev. Lett. 105, 150606 (2010).
O. Benichou, D. S. Grebenkov, P. E. Levitz, C. Loverdo, R. Voituriez, Mean First-Passage Time of Surface-Mediated Diffusion in Spherical Domains, J. Stat. Phys. 142, 657-685 (2011).
G. Oshanin, H. S. Wio, K. Lindenberg, S. F. Burlatsky, Intermittent random walks for an optimal search strategy: one-dimensional case, J. Phys. Condens. Matter. 19, 065142 (2007).
G. Oshanin, K. Lindenberg, H. S Wio, S. Burlatsky Efficient search by optimized intermittent random walks, J. Phys. A: Math. Theor. 42, 434008 (2009).
C. Loverdo, O. Benichou, M. Moreau, R. Voituriez, Robustness of optimal intermittent search strategies in one, two, and three dimensions, Phys. Rev. E 80, 031146 (2009).
M. A. Lomholt, T. Koren, R. Metzler, J. Klafter, Are Levy strategies in intermittent search processes advantageous?, Proc. Nat. Acad. Sci 105, 11055 (2007).


Modeling of diffusion

K. K. Sabelfeld, Monte Carlo Methods in Boundary Value Problems (Springer-Verlag: New York - Heidelberg, Berlin, 1991).
K. K. Sabelfeld and N. A. Simonov, Random Walks on Boundary for Solving PDEs (Utrecht, The Netherlands, 1994).
G. N. Milshtein, Numerical Integration of Stochastic Differential Equations (Kluwer, Dordrecht, the Netherlands, 1995).
D. M. Ceperley, Microscopic simulations in physics, Rev. Mod. Phys. 71, S438 (1999).


Physics of respiration

E. R. Weibel, The Pathway for oxygen. Structure and function in the mammalian respiratory system (Harvard University, Cambridge, Massachusetts and London, England, 1984).
B. Mauroy, M. Filoche, E. Weibel, B. Sapoval, An Optimal Bronchial Tree May Be Dangerous, Nature 427, 633 (2004).
M. Felici, M. Filoche, B. Sapoval, Renormalized Random Walk Study of Oxygen Absorption in the Human Lung, Phys. Rev. Lett. 92, 068101 (2004).
W. R. Hendee, Physics and applications of medical imaging, Rev. Mod. Phys. 71, S444 (1999).


Physics of living cells

A. B. Fulton, How Crowded Is the Cytoplasm? Cell 30, 345 (1982).
R. J. Ellis, A. P. Minton, Join the crowd, Nature 425, 27 (2003).
I. M. Tolic-Norrelykke, E. L. Munteanu, G. Thon, L. Oddershede, K. Berg-Sorensen Anomalous Diffusion in Living Yeast Cells, Phys. Rev. Lett. 93, 078102 (2004).
M. Weiss, M. Elsner, F. Kartberg, T. Nilsson, Anomalous Subdiffusion Is a Measure for Cytoplasmic Crowding in Living Cells, Biophys. J. 87, 3518 (2004).


General statistical physics

P. L. Krapivsky, S. Redner, E. Ben-Naim, A Kinetic View of Statistical Physics (Cambridge University Press, Cambridge, 2010).


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Last modified 11/02/2011

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