zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Front propagation in reactive systems with anomalous diffusion. (English) Zbl 1058.80004
Summary: We study front propagation of reactive fields in systems whose diffusive behavior is anomalous (both superdiffusive and subdiffusive). The features of the front propagation depend, not only on the scaling exponent $\nu (\langle x(t)^2\rangle \sim t^{2\nu})$, but also on the detailed shape of the probability distribution of the diffusive process. From the analysis of different systems we have three possible behavior of front propagation: the usual (Fisher-Kolmogorov like) scenario, i.e., the field has a spatial exponential tail moving with constant speed, $v_f$, and thickness, $\lambda$; the field has a spatial exponential tail but $v_f$ and $\lambda$ change in time (as a power law); and finally the field has a spatial power law tail and $v_f$ increases exponentially in time. A linear analysis of the front tail is in quantitative agreement with the numerical simulations. It is remarkable the fact that anomalous diffusion is neither necessary nor sufficient condition for the linear front propagation. Moreover, if the probability distribution of the transport process follows the scaling relation given by the Flory argument, the front propagation is standard (Fisher-Kolmogorov like) even in presence of super (or sub) diffusion.

80A30Chemical kinetics (thermodynamic aspects)
82C70Transport processes (time-dependent statistical mechanics)
Full Text: DOI
[1] J.D. Murray, Mathematical Biology, 2nd ed., Springer, Berlin, 1993. · Zbl 0779.92001
[2] Xin, J.: SIAM rev.. 42, 161 (2000)
[3] Z. Toroczkai, T. Tel (Eds.), Focus Issue: Active Chaotic Flow, Chaos 12 (2002).
[4] Ross, J.; Müller, S. C.; Vidal, C.: Science. 240, 460 (1988)
[5] N. Peters, Turbulent Combustion, Cambridge University Press, Cambridge, 2000. · Zbl 0955.76002
[6] Abraham, E. R.: Nature. 391, 577 (1998)
[7] Fisher, R. A.: Ann. eugenics. 7, 355 (1937)
[8] Ebert, U.; Van Saarloos, W.: Physica D. 146, 1 (2000)
[9] Constantin, P.; Kiselev, A.; Oberman, A.; Ryzhik, L.: Arch. rat. Mech.. 154, 53 (2000)
[10] Kiselev, A.; Ryzhik, L.: Ann. I.H. Poincaré. 18, 309 (2001)
[11] Abel, M.; Celani, A.; Vergni, D.; Vulpiani, A.: Phys. rev. E. 64, 46307 (2001)
[12] Bouchaud, J. P.; Georges, A.: Phys. rep.. 195, 127 (1990)
[13] Castiglione, P.; Mazzino, A.; Muratore-Ginanneschi, P.; Vulpiani, A.: Physica D. 134, 75 (1999)
[14] Zaslavsky, G. M.; Stevens, D.; Weitzener, A.: Phys. rev. E. 48, 1683 (1993)
[15] Klafter, J.; Shlensinger, M. F.; Zumofen, G.: Phys. today. 49, 33 (1996)
[16] Ishizaki, R.; Horita, T.; Kobayashi, T.; Mori, H.: Prog. theoret. Phys.. 85, 1013 (1991)
[17] Matheron, G.; De Marsily, G.: Water resour. Res.. 16, 901 (1980)
[18] Constantin, P.; Kiselev, A.; Oberman, A.; Ryzhik, L.: Arch. rat. Mech.. 154, 53 (2000)
[19] Majda, A. J.; Kramer, P. R.: Phys. rep.. 314, 237 (1999)
[20] A.J. Lichtenberg, M.A. Lieberman, Regular and Chaotic Dynamics, 2nd ed., Springer, New York, 1991. · Zbl 0748.70001
[21] Rechester, A. B.; White, R. B.: Phys. rev. Lett.. 44, 1586 (1980)
[22] Cocke, W. J.: Phys. fluids. 12, 2488 (1969)
[23] Richardson, L. P.: Proc. R. Soc. London A. 110, 709 (1926)
[24] U. Frisch, Turbulence: The Legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 1995. · Zbl 0832.76001
[25] Boffetta, G.; Celani, A.; Crisanti, A.; Vulpiani, A.: Phys. rev.. 60, 6734 (1999)
[26] Boffetta, G.; Sokolov, I. M.: Phys. fluids. 14, 3224 (2002)
[27] Cassi, D.; Sofia, R.: Mod. phys. Lett. B. 6, 1397 (1992)
[28] O’shaughnessy, B.; Procaccia, I.: Phys. rev. Lett.. 54, 455 (1985)
[29] B. Gnedenko, A.N. Kolmogorov, Limit distribution for Sums of Independent Random Variables, Addison-Wesley, Reading, MA, 1954. · Zbl 0056.36001
[30] Fisher, M. E.: J. chem. Phys.. 44, 616 (1966)
[31] Moffat, H. K.: Rep. prog. Phys.. 46, 621 (1983)
[32] M. Freidlin, Functional Integration and Partial Differential Equations, Princeton University Press, Princeton, NJ, 1985. · Zbl 0568.60057
[33] Biferale, L.; Crisanti, A.; Vergassola, M.; Vulpiani, A.: Phys. fluids. 7, 2725 (1995)
[34] Avellaneda, M.; Majda, A.: Commun. math. Phys.. 138, 339 (1991)
[35] Avellaneda, M.; Vergassola, M.: Phys. rev. E. 52, 3249 (1995)
[36] Constantin, P.; Kiselev, A.; Ryzhik, L.: Commun. pure appl. Math.. 54, 1320 (2001)
[37] A. Okubo, S.A. Levin, Diffusion and Ecological Problems, Springer, New York, 2001. · Zbl 1027.92022
[38] Mancinelli, R.; Vergni, D.; Vulpiani, A.: Eur. lett.. 60, 532 (2002)
[39] Paladin, G.; Vulpiani, A.: J. phys. A. 27, 4911 (1994) · Zbl 1194.76063
[40] D. del Castillo Negrete, B.A. Carreras, V. Lynch, Phys. Rev. Lett. 91 (2003) 018302.
[41] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes, Cambridge University Press, Cambridge, 1994. · Zbl 0587.65004