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August  2020, 40(8): 5079-5103. doi: 10.3934/dcds.2020212

On a discrete self-organized-criticality finite time result

Umberto Mosco 1, and  Maria Agostina Vivaldi 2,

1. 

Worcester Polytechnic Institute, MA, USA
2. 

Department of Basic and Applied Sciences for Engineering, Sapienza University of Rome, Italy

Received  February 2020 Published  May 2020

In this paper we deal with theoretical and numerical aspects of some nonlinear problems related to sandpile models. We introduce a purely discrete model for infinitely many particles interacting according to a toppling process on a uniform two-dimensional grid and prove the convergence of the solutions to a differential initial value problem.

Keywords: Sand pile models, self-organized-criticality, convergence of nonlinear finite difference schemes.
Mathematics Subject Classification: 35K55, 35K65, 35Q70, 37L05.
Citation: Umberto Mosco, Maria Agostina Vivaldi. On a discrete self-organized-criticality finite time result. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 5079-5103. doi: 10.3934/dcds.2020212
References:
[1]

G. Aronsson, L. C. Evans and Y. Wu, Fast/slow diffusion and growing sandpiles, J. Differential Equations, 131 (1996), 304-335.  doi: 10.1006/jdeq.1996.0166.  Google Scholar

[2]

P. Bak, C. Tang and K. Wiesenfeld, Self-organized criticality, Phys. Rev. A (3), 38 (1988), 364-374.  doi: 10.1103/PhysRevA.38.364.  Google Scholar

[3]

P. Bántay and I. M. Jánosi, Avalanche dynamics from anomalous diffusion, Phys. Rev. Lett., 68 (1992), 2058-2061.   Google Scholar

[4]

V. Barbu, Nonlinear Semi-Groups and Differential Equations in Banach Spaces, Noordhoff: Leyden, 1976.  Google Scholar

[5]

V. Barbu, Nonlinear Differential Equations of Monotone Type in Banach Spaces, Springer: NewYork, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

[6]

V. Barbu, Self-organized criticality and convergence to equilibrium of solutions to nonlinear diffusion problems, Annual Reviews in Control, JARAP, 340 (2010), 52-61.   Google Scholar

[7]

V. Barbu, Self-organized criticality of cellular automata model; absorbtion in finite-time of supercritical region into the critical one, Mathematical Methods in the Applied Sciences, 36 (2013), 1726-1733.  doi: 10.1002/mma.2718.  Google Scholar

[8]

V. Barbu, G. Da Prato and M. Röckner, Stochastic porous media equations and self- organized criticality, Comm. Math. Phys., 285 (2009), 901-923.  doi: 10.1007/s00220-008-0651-x.  Google Scholar

[9]

J. W. Barrett and L. Prigozhin, Sandpiles and superconductors: Nonconforming linear finite element approximations for mixed formulations of quasi-variational, IMA J. Numer. Anal., 35 (2015), 1-38.  doi: 10.1093/imanum/drt062.  Google Scholar

[10]

H. Brézis, Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Ed. L.Nachbin North-Hollands/ Americal Elsevier, 1973.  Google Scholar

[11]

H. Brézis, Analisi funzionale, Ed. Liguori 1986. Google Scholar

[12]

H. Brézis and A. Pazy, Accretive sets and differential equations in Banach spaces, Israel J. Math., 8 (1970), 367-383.  doi: 10.1007/BF02798683.  Google Scholar

[13]

J. M. Carlson, J. T. Chayes, E. R. Grannan and G. H. Swindle, Self-organized criticality in sandpiles: Nature of the critical phenomenon, Phys. Rev. A (3), 42 (1990), 2467-2470.  doi: 10.1103/PhysRevA.42.2467.  Google Scholar

[14]

J. M. Carlson and G. H. Swindle, Self-organized criticality: Sandpiles, singularities, and scaling, Proc. Nat. Acad. Sci. USA, 92 (1995), 6712-6719.  doi: 10.1073/pnas.92.15.6712.  Google Scholar

[15]

M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-298.  doi: 10.2307/2373376.  Google Scholar

[16]

M. Creutz, Abelian sandpiles, Nucl. Phys. B (Proc. Suppl.), 20 (1991), 758-761.   Google Scholar

[17]

D. Dahr, Self-organized critical state of sandpile automaton models, Phys. Rev. Lett., 64 (1990), 1613-1616.  doi: 10.1103/PhysRevLett.64.1613.  Google Scholar

[18]

L. C. Evans, M. Feldman and R. F. Gariepy, Fast/slow diffusion and collapsing sandpiles, J. Differential Equations, 137 (1997), 166-209.  doi: 10.1006/jdeq.1997.3243.  Google Scholar

[19]

B. Gess, Finite time extinction for stochastic sign fast diffusion and self-organized criticality, Comm. Math. Phys., 335 (2015), 309-344.  doi: 10.1007/s00220-014-2225-4.  Google Scholar

[20]

U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. in Math., 3 (1969), 510-585.  doi: 10.1016/0001-8708(69)90009-7.  Google Scholar

[21]

U. Mosco, An Introduction to the approximate solution of variational inequalities, In: Constructive Aspects of Functional Analysis. Cime Summer Schools Erice, (ed. G. Geymonat), Springer, 57 (1971), 497–685. Google Scholar

[22]

U. Mosco, Finite -time self - organized - criticality on synchronized infinite grids, SIAM J. Math. Anal., 50 (2018), 2409-2440.  doi: 10.1137/17M1122955.  Google Scholar

[23]

U. Mosco and M. A. Vivaldi, On the external approximation of Sobolev spaces by M–convergence, to appear. Google Scholar

show all references

References:
[1]

G. Aronsson, L. C. Evans and Y. Wu, Fast/slow diffusion and growing sandpiles, J. Differential Equations, 131 (1996), 304-335.  doi: 10.1006/jdeq.1996.0166.  Google Scholar

[2]

P. Bak, C. Tang and K. Wiesenfeld, Self-organized criticality, Phys. Rev. A (3), 38 (1988), 364-374.  doi: 10.1103/PhysRevA.38.364.  Google Scholar

[3]

P. Bántay and I. M. Jánosi, Avalanche dynamics from anomalous diffusion, Phys. Rev. Lett., 68 (1992), 2058-2061.   Google Scholar

[4]

V. Barbu, Nonlinear Semi-Groups and Differential Equations in Banach Spaces, Noordhoff: Leyden, 1976.  Google Scholar

[5]

V. Barbu, Nonlinear Differential Equations of Monotone Type in Banach Spaces, Springer: NewYork, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

[6]

V. Barbu, Self-organized criticality and convergence to equilibrium of solutions to nonlinear diffusion problems, Annual Reviews in Control, JARAP, 340 (2010), 52-61.   Google Scholar

[7]

V. Barbu, Self-organized criticality of cellular automata model; absorbtion in finite-time of supercritical region into the critical one, Mathematical Methods in the Applied Sciences, 36 (2013), 1726-1733.  doi: 10.1002/mma.2718.  Google Scholar

[8]

V. Barbu, G. Da Prato and M. Röckner, Stochastic porous media equations and self- organized criticality, Comm. Math. Phys., 285 (2009), 901-923.  doi: 10.1007/s00220-008-0651-x.  Google Scholar

[9]

J. W. Barrett and L. Prigozhin, Sandpiles and superconductors: Nonconforming linear finite element approximations for mixed formulations of quasi-variational, IMA J. Numer. Anal., 35 (2015), 1-38.  doi: 10.1093/imanum/drt062.  Google Scholar

[10]

H. Brézis, Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Ed. L.Nachbin North-Hollands/ Americal Elsevier, 1973.  Google Scholar

[11]

H. Brézis, Analisi funzionale, Ed. Liguori 1986. Google Scholar

[12]

H. Brézis and A. Pazy, Accretive sets and differential equations in Banach spaces, Israel J. Math., 8 (1970), 367-383.  doi: 10.1007/BF02798683.  Google Scholar

[13]

J. M. Carlson, J. T. Chayes, E. R. Grannan and G. H. Swindle, Self-organized criticality in sandpiles: Nature of the critical phenomenon, Phys. Rev. A (3), 42 (1990), 2467-2470.  doi: 10.1103/PhysRevA.42.2467.  Google Scholar

[14]

J. M. Carlson and G. H. Swindle, Self-organized criticality: Sandpiles, singularities, and scaling, Proc. Nat. Acad. Sci. USA, 92 (1995), 6712-6719.  doi: 10.1073/pnas.92.15.6712.  Google Scholar

[15]

M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-298.  doi: 10.2307/2373376.  Google Scholar

[16]

M. Creutz, Abelian sandpiles, Nucl. Phys. B (Proc. Suppl.), 20 (1991), 758-761.   Google Scholar

[17]

D. Dahr, Self-organized critical state of sandpile automaton models, Phys. Rev. Lett., 64 (1990), 1613-1616.  doi: 10.1103/PhysRevLett.64.1613.  Google Scholar

[18]

L. C. Evans, M. Feldman and R. F. Gariepy, Fast/slow diffusion and collapsing sandpiles, J. Differential Equations, 137 (1997), 166-209.  doi: 10.1006/jdeq.1997.3243.  Google Scholar

[19]

B. Gess, Finite time extinction for stochastic sign fast diffusion and self-organized criticality, Comm. Math. Phys., 335 (2015), 309-344.  doi: 10.1007/s00220-014-2225-4.  Google Scholar

[20]

U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. in Math., 3 (1969), 510-585.  doi: 10.1016/0001-8708(69)90009-7.  Google Scholar

[21]

U. Mosco, An Introduction to the approximate solution of variational inequalities, In: Constructive Aspects of Functional Analysis. Cime Summer Schools Erice, (ed. G. Geymonat), Springer, 57 (1971), 497–685. Google Scholar

[22]

U. Mosco, Finite -time self - organized - criticality on synchronized infinite grids, SIAM J. Math. Anal., 50 (2018), 2409-2440.  doi: 10.1137/17M1122955.  Google Scholar

[23]

U. Mosco and M. A. Vivaldi, On the external approximation of Sobolev spaces by M–convergence, to appear. Google Scholar

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