Non-Archimedean reaction-ultradiffusion equations and complex hierarchic systems. (English) Zbl 1397.80009
The article is about ultradiffusion in complex hierachical systems; it generalizes results obtained by V. A. Avetisov et al. [J. Phys. A, Math. Gen. 32, No. 50, 8785–8791 (1999; Zbl 0957.82034); Dokl. Math. 60, No. 2, 271–274 (1999; Zbl 1037.60089); J. Phys. A, Math. Gen. 35, No. 2, 177–189 (2002; Zbl 1038.82077); ibid. 36, No. 15, 4239–4246 (2003; Zbl 1049.82051); J. Phys. A, Math. Theor. 42, No. 8, Article ID 085003, 18 p. (2009; Zbl 1162.82010)]. To describe these (protein-like) systems from a mathematical point of view, \(p\)-adic analysis is used. The description of reaction-diffusion processes in such systems reduces to Markovian random walks. The author studies not only ultradiffusion equations but their finite-dimensional analogue as well and proves the convergence of this analogue to the initial problem. After a quite lengthy mathematical derivation, a theorem about the existence and uniqueness of the Cauchy problem for ultradiffusion-reaction equations is formulated and proved.
Reviewer: Aleksey Syromyasov (Saransk)
MSC:
| 80A99 | Thermodynamics and heat transfer |
| 11S80 | Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) |
| 12J12 | Formally \(p\)-adic fields |
| 45K05 | Integro-partial differential equations |
| 46S10 | Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis |
| 60J60 | Diffusion processes |
| 82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |
Keywords:
reaction-diffusion equations; ultradiffusion; ultrametric spaces; \(p\)-adic analysis; complex systems; free-energy functionals; Cauchy problemReferences:
| [1] | Albeverio S, Khrennikov A Yu and Shelkovich V M 2010 Theory of p-adic Distributions: Linear and Nonlinear Models(London Mathematical Society Lecture Note Series vol 370) (Cambridge: Cambridge University Press) · Zbl 1198.46001 |
| [2] | Alberti G and Bellettini G 1998 A non-local anisotropic model for phase transitions: asymptotic behaviour of rescaled energies Eur. J. Appl. Math.9 261-84 · Zbl 0932.49018 |
| [3] | Alberti G and Bellettini G 1998 A nonlocal anisotropic model for phase transitions. I. The optimal profile problem Math. Ann.310 527-60 · Zbl 0891.49021 |
| [4] | Avetisov V A, Bikulov A Kh and Zubarev A P 2009 First passage time distribution and the number of returns for ultrametric random walks J. Phys. A: Math. Theor.42 085003 · Zbl 1162.82010 |
| [5] | Avetisov V A, Bikulov A Kh and Osipov V A 2003 p-adic description of characteristic relaxation in complex systems J. Phys. A: Math. Gen.36 4239-46 · Zbl 1049.82051 |
| [6] | Avetisov V A, Bikulov A H, Kozyrev S V and Osipov V A 2002 p-adic models of ultrametric diffusion constrained by hierarchical energy landscapes J. Phys. A: Math. Gen.35 177-89 · Zbl 1038.82077 |
| [7] | Avetisov V A, Bikulov A Kh and Kozyrev S V 1999 Description of logarithmic relaxation by a model of a hierarchical random walk Dokl. Akad. Nauk 368 164-7 (Russian) · Zbl 1037.60089 |
| [8] | Avetisov V A, Bikulov A H and Kozyrev S V M 1999 Application of p-adic analysis to models of breaking of replica symmetry J. Phys. A: Math. Gen.32 8785-91 · Zbl 0957.82034 |
| [9] | Bachas C P and Huberman B A 1987 Complexity and ultradiffusion J. Phys. A: Math. Gen.20 4995-5014 |
| [10] | Bates P W, Fife P C, Ren X and Wang X 1997 Traveling waves in a convolution model for phase transitions Arch. Ration. Mech. Anal.138 105-36 · Zbl 0889.45012 |
| [11] | Bates P W and Chmaj A 1999 An integrodifferential model for phase transitions: stationary solutions in higher space dimensions J. Stat. Phys.95 1119-39 · Zbl 0958.82015 |
| [12] | Becker O M and Karplus M 1997 The topology of multidimensional protein energy surfaces: theory and application to peptide structure and kinetics J. Chem. Phys.106 1495-517 |
| [13] | Bikulov A Kh 2010 On solution properties of some types of p-adic kinetic equations of the form reaction-diffusion, p-adic numbers ultrametric Anal. Appl.2 187-206 · Zbl 1268.35076 |
| [14] | Cazenave T and Haraux A 1998 An Introduction to Semilinear Evolution Equations (Oxford: Oxford University Press) · Zbl 0926.35049 |
| [15] | Dellacherie C, Martinez S and San Martin J 2014 Inverse M-Matrices and Ultrametric Matrices(Lecture Notes in Mathematics vol 2118) (Berlin: Springer) · Zbl 1311.15002 |
| [16] | Dragovich B, Khrennikov A Yu, Kozyrev S V and Volovich I V 2009 On p-adic mathematical physics, p-adic numbers ultrametric Anal. Appl.1 1-17 · Zbl 1187.81004 |
| [17] | Fife P C 1979 Mathematical Aspects of Reacting and Diffusing Systems(Lecture Notes in Biomathematics vol 28) (Berlin: Springer) · Zbl 0403.92004 |
| [18] | Frauenfelder H, Chan S S and Chan W S (ed) 2010 The Physics of Proteins (Berlin: Springer) |
| [19] | Grindrod P 1991 Patterns and Waves. The Theory and Applications of Reaction-Diffusion Equations(Oxford Applied Mathematics and Computing Science Series) (Oxford: Clarendon) · Zbl 0743.35032 |
| [20] | Kochubei A N 2001 Pseudo-Differential Equations and Stochastics Over Non-Archimedean Fields (New York: Marcel Dekker) · Zbl 0984.11063 |
| [21] | Khrennikov A, Kozyrev S and Zúñiga-Galindo W A 2018 Ultrametric Equations and its Applications(Encyclopedia of Mathematics and its Applications vol 168) (Cambridge: Cambridge University Press) · Zbl 1397.35350 |
| [22] | Kozyrev S V 2008 Methods and applications of ultrametric and p-adic analysis: from wavelet theory to biophysics Proc. Steklov Inst. Math.12 3-168 · Zbl 1298.42038 |
| [23] | Khrennikov A Yu and Kozyrev S V 2006 Replica symmetry breaking related to a general ultrametric space I: replica matrices and functionals Physica A 359 222-40 |
| [24] | Khrennikov A Yu and Kozyrev S V 2006 Replica symmetry breaking related to a general ultrametric space II: RSB solutions and the n 0 limit Physica A 359 241-66 |
| [25] | Khrennikov A Yu and Kozyrev S V 2007 Replica symmetry breaking related to a general ultrametric space III: the case of general measure Physica A 378 283-98 |
| [26] | Mézard M, Parisi G and Virasoro M A 1987 Spin Glass Theory and Beyond (Singapore: World Scientific) · Zbl 0992.82500 |
| [27] | Miklavčič M 1998 Applied Functional Analysis and Partial differential Equations (Singapore: World Scientific) · Zbl 0913.35002 |
| [28] | Ogielski A T and Stein D L 1985 Dynamics on ultrametric spaces Phys. Rev. Lett.55 1634-7 |
| [29] | Rammal R, Toulouse G and Virasoro M A 1986 Ultrametricity for physicists Rev. Mod. Phys.58 765-88 |
| [30] | Smoller J 1994 Shock Waves and Reaction-Diffusion Equations (New York: Springer) · Zbl 0807.35002 |
| [31] | Taibleson M H 1975 Fourier Analysis on Local Fields (Princeton, NJ: Princeton University Press) · Zbl 0319.42011 |
| [32] | Torresblanca-Badillo A and Zúñiga-Galindo W A 2018 Ultrametric diffusion, exponential landscapes, and the first passage time problem Acta Appl. Math. (https://doi.org/10.1007/s10440-018-0165-2) · Zbl 1414.60065 |
| [33] | Ueyama E and Hosoe S 2012 Reaction-diffusion equation on a graph and phase transition on bistable media Proc. of SICE Annual Conf.(Akita University, Akita, Japan, 20-23 August 2012) pp 1798-801 |
| [34] | Vladimirov V S, Volovich I V and Zelenov E I 1994 p-adic analysis and mathematical physics (Singapore: World Scientific) · Zbl 0812.46076 |
| [35] | Yin G G and Zhang Q 2013 Continuous-Time Markov Chains and Applications: a Two-Time-Scale Approach 2nd edn (Berlin: Springer) · Zbl 1277.60127 |
| [36] | Zúñiga-Galindo W A 2016 Pseudodifferential Equations Over Non-Archimedean Spaces(Lectures Notes in Mathematics vol 2174) (Berlin: Springer) · Zbl 1367.35006 |
| [37] | Zwanzig R 1995 Simple model of protein folding kinetics Proc. Natl Acad. Sci. USA 92 9801-4 |
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