Abstract
When formulated properly, most geophysical transport-type process involving passive scalars or motile particles may be described by the same space–time nonlocal field equation which consists of a classical mass balance coupled with a space–time nonlocal convective/dispersive flux. Specific examples employed here include stretched and compressed Brownian motion, diffusion in slit-nanopores, subdiffusive continuous-time random walks (CTRW), super diffusion in the turbulent atmosphere and dispersion of motile and passive particles in fractal porous media. Stretched and compressed Brownian motion, which may be thought of as Brownian motions run with nonlinear clocks, are defined as the limit processes of a special class of random walks possessing nonstationary increments. The limit process has a mean square displacement that increases as tα+1 where α > −1 is a constant. If α = 0 the process is classical Brownian, if α < 0 we say the process is compressed Brownian while if α > 0 it is stretched. The Fokker–Planck equations for these processes are classical ade’s with dispersion coefficient proportional to tα. The Brownian-type walks have fixed time step, but nonstationary spatial increments that are Gaussian with power law variance. With the CTRW, both the time increment and the spatial increment are random. The subdiffusive Fokker–Planck equation is fractional in time for the CTRW’s considered in this article. The second moments for a Levy spatial trajectory are infinite while the Fokker–Planck equation is an advective–dispersive equation, ade, with constant diffusion coefficient and fractional spatial derivatives. If the Lagrangian velocity is assumed Levy rather than the position, then a similar Fokker–Planck equation is obtained, but the diffusion coefficient is a power law in time. All these Fokker–Planck equations are special cases of the general non-local balance law.
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References
Adler PM (1992) Porous media: geometry and transports. Butterworth-Heinemann, Stoneham, MA
Berg HC (2000) Motile behavior of bacteria. Phys Today 53(1):24–29
Bhattacharya RN, Gupta VK (1990) Application of central limit theorems to solute dispersion in saturated porous media: From kinetic to field scales. In: Cushman JH (ed) Dynamics of fluids in hierarchical porous media, chap IV. Elsevier, New York, pp 61–95
Boone JP, Yip S (1980) Molecular hydrodynamics. McGraw-Hill, New York
Cushman JH (1990) Molecular scale lubrication. Nature 347(6290):227–228
Cushman JH (1991) On diffusion in fractal porous media. Water Resour Res 27(4):643–644
Cushman JH (1997) The physics of fluids in hierarchical porous media: angstroms to miles. Kluwer, Dordrecht, 490 pp
Cushman JH, Hu X (1995) A resumé of nonlocal transport theories. Stoch Hydrol Hydraul 9:105–116
Cushman JH, Moroni M (2001) Statistical mechanics with 3D-PTV experiments in the study of anomalous dispersion: part I. Theory. Phys Fluids 13(1):75–80
Cushman JH, Hu X, Ginn TR (1994) Nonequilibrium statistical mechanics of preasymptotic dispersion. J Stat Phys 75:859–878
Cushman JH, Bennethum LS, Hu BX (2002) A primer on upscaling methods for porous media. Adv Water Resour 25:1043–1067
Cushman JH, Park M, Kleinfelter N, Moroni M (2005) Super-diffusion via Levy Lagrangian velocity processes. Geophys Res Lett 32(19):L19816,1–L19816,4
Cushman JH, Park M, O’Malley D (2009a) The chaotic dynamics of super-diffusion revisited. Geophys Res Lett 36:L08812,1–L08812,4
Cushman JH, O’Malley D, Park M (2009b) Anomalous diffusion as modeled by a non-stationary extension of brownian motion. Phys Rev E 79(032101):1–4
Hu X, Cushman JH (1994) Nonequilibrium statistical mechanical derivation of a nonlocal Darcy’s law for unsaturated flow. Stoch Hydrol Hyd 8:109–116
Meerschaert MM, Benson DA, Baeumer B (1999) Multidimensional advection and fractional dispersion. Phys Rev E 59:5026–5028
Metzler R, Klafter J (2000) The random walks guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339:1–77
Park M, Cushman JH (2006) On upscaling operator-stable Levy motions in fractal porous media. J Comput Phys 217:159–165
Park M, Kleinfelter N, Cushman JH (2005) Scaling laws and dispersion equations for Levy particles in 1-dimensional fractal porous media. Phys Rev E 72(056305):1–7
Rhykerd CL Jr, Schoen M, Diestler DJ, Cushman JH (1987) Epitaxy in simple classical fluids in micropores and near solid surfaces. Nature 330(3):461–463
Richardson LF (1926) Atmospheric diffusion shown on a distance-neighborhood graph. Proc R Soc Lond Ser A 110:709–737
Sahimi M (1995) Flow and transport in porous media and fractured rock. VCH, Weinheim, Germany
Samorodnitsky G, Taqqu MS (1994) Stable Non-Gaussian random processes: stochastic models with infinite variance. CRC Press, Boca Raton, FL
Schoen M, Diestler DJ, Cushman JH (1993a) Isostress–isostrain ensemble Monte Carlo simulation of second-order phase transitions in a confined monolayer fluid. Mol Phys 78:1097–1115
Schoen M, Diestler DJ, Cushman JH (1993b) Shear melting of confined solid monolayer films. Phys Rev B 47(10):5603–5613
Schoen M, Cushman JH, Diestler DJ (1994a) Anomalous diffusion in molecularly thin films. Mol Phys 81(2):475–490
Schoen M, Diestler DJ, Cushman JH (1994b) Stratification-induced order-disorder phase transitions in molecularly thin confined films. J Chem Phys 101(8):6865–6873
Schoen M, Diestler DJ, Cushman JH (1994c) Fluids in micropores IV. The behavior of molecularly thin confined films in the Grand Isostress Ensemble. J Chem Phys 100(10):7707–7717
Tartakovsky DM, Neuman SP (2009) Perspective on theories of non-Fickian transport in heterogeneous media. Adv Water Res 32(5):670–680. doi:10.1016/j.advwatres.2008.08.005
Voth GA, La Port A, Crawford AM, Alexander J, Bodenschatz E (2002) Measurement of particle accelerations in fully developed turbulence. J Fluid Mech 469:121–160
Zwanzig R (1960) Ensemble method in the theory of irreversibility. J Chem Phys 33(5):1338–1341
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JHC wishes to thank the NSF for continuing support through grant EAR 0620460 and EAR 0838224.
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Cushman, J.H., Park, M., Moroni, M. et al. A universal field equation for dispersive processes in heterogeneous media. Stoch Environ Res Risk Assess 25, 1–10 (2011). https://doi.org/10.1007/s00477-010-0446-4
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DOI: https://doi.org/10.1007/s00477-010-0446-4