Lattice fractional calculus

Integration and differentiation of non-integer orders for N-dimensional physical lattices with long-range particle interactions are suggested. The proposed lattice fractional derivatives and integrals are represented by kernels of lattice long-range interactions, such that their Fourier series transformations have a power-law form with respect to components of wave vector. Continuous limits for these lattice fractional derivatives and integrals give the continuum derivatives and integrals of non-integer orders with respect to coordinates. Lattice analogs of fractional differential equations that include suggested lattice differential and integral operators can serve as an important element of microscopic approach to nonlocal continuum models in mechanics and physics.

[1]  Large Deviation Techniques Applied to Systems with Long-Range Interactions , 2004, cond-mat/0406358.

[2]  Vasily E. Tarasov,et al.  Lattice model with power-law spatial dispersion for fractional elasticity , 2013, 1501.01201.

[3]  F. Mainardi Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models , 2010 .

[4]  A. Cemal Eringen,et al.  Mechanics of continua , 1967 .

[5]  G. Zaslavsky,et al.  Fractional dynamics of systems with long-range interaction , 2006, 1107.5436.

[6]  Vasily E. Tarasov,et al.  No violation of the Leibniz rule. No fractional derivative , 2013, Commun. Nonlinear Sci. Numer. Simul..

[7]  Vasily E. Tarasov,et al.  Coupled oscillators with power-law interaction and their fractional dynamics analogues , 2007 .

[8]  Thierry Dauxois,et al.  Dynamics and thermodynamics of systems with long-range interactions , 2002 .

[9]  V. Uchaikin Fractional Derivatives for Physicists and Engineers , 2013 .

[10]  S. Ruffo,et al.  Abundance of regular orbits and nonequilibrium phase transitions in the thermodynamic limit for long-range systems. , 2008, Physical review letters.

[11]  M. Born,et al.  Dynamical Theory of Crystal Lattices , 1954 .

[12]  R. Sibatov,et al.  Fractional Kinetics in Solids: Anomalous Charge Transport in Semiconductors, Dielectrics and Nanosystems , 2012 .

[13]  Albert C. J. Luo,et al.  Long-range interactions, stochasticity and fractional dynamics , 2010 .

[14]  L. I. Sedov,et al.  A course in continuum mechanics , 1971 .

[15]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[16]  M. Ortigueira Fractional Central Differences and Derivatives , 2008 .

[17]  B. Moritz,et al.  Triangle lattice Green functions for vector fields , 2001 .

[18]  José António Tenreiro Machado,et al.  Fractional Order Calculus: Basic Concepts and Engineering Applications , 2010 .

[19]  Vasily E. Tarasov,et al.  Lattice Model of Fractional Gradient and Integral Elasticity: Long-Range Interaction of Grunwald-Letnikov-Riesz Type , 2014, 1502.06268.

[20]  O. Agrawal,et al.  Advances in Fractional Calculus , 2007 .

[21]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[22]  G. Zaslavsky,et al.  Fractional dynamics of coupled oscillators with long-range interaction. , 2005, Chaos.

[23]  Elizabeth L. Mansfield,et al.  Difference Forms , 2008, Found. Comput. Math..

[24]  D. Fanelli,et al.  Existence of quasi-stationary states at the long range threshold , 2010, 1007.2065.

[25]  V. E. Tarasov Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media , 2011 .

[26]  Yu. A. Brychkov,et al.  Integrals and series , 1992 .

[27]  V. E. Tarasov Continuous limit of discrete systems with long-range interaction , 2006, 0711.0826.

[28]  V. E. Tarasov General Lattice Model of Gradient Elasticity , 2015, 1501.01435.

[29]  Y. Levin,et al.  Nonequilibrium statistical mechanics of systems with long-range interactions , 2013, 1310.1078.

[30]  M. Shlesinger Book Review: Dynamics and Thermodynamics of Systems with Long-Range Interactions. T. Dauxois, S. Ruffo, E. Arimondo, and M. Wilkens (eds.). Springer-Verlag, Berlin, 2002 , 2004 .

[31]  Vasily E. Tarasov,et al.  Fractional Gradient Elasticity from Spatial Dispersion Law , 2013, 1306.2572.

[32]  Stationary states and fractional dynamics in systems with long-range interactions , 2009, 0912.3060.

[33]  M. Naber,et al.  Fractional differential forms , 2001, math-ph/0301013.

[34]  M. Riesz L'intégrale de Riemann-Liouville et le problème de Cauchy , 1949 .

[35]  Ralf Metzler,et al.  Fractional dynamics : recent advances , 2011 .

[36]  H. Böttger Principles of the Theory of Lattice Dynamics , 1983 .

[37]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[38]  R. Hilfer Applications Of Fractional Calculus In Physics , 2000 .

[39]  I. Podlubny Fractional differential equations , 1998 .

[40]  A. Kosevich The crystal lattice , 1999 .

[41]  O. Agrawal,et al.  Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering , 2007 .

[42]  Jacqueline Krim,et al.  Foundations of Nanomechanics: From Solid-State Theory to Device Applications , 2004 .

[43]  Ryosuke Ishiwata,et al.  Relationships between power-law long-range interactions and fractional mechanics , 2012 .

[44]  Francesco Mainardi,et al.  Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics , 2012, 1201.0863.

[45]  Kivshar,et al.  Kinks in the Frenkel-Kontorova model with long-range interparticle interactions. , 1990, Physical review. B, Condensed matter.

[46]  A. A. Maradudin,et al.  Theory of lattice dynamics in the harmonic approximation , 1971 .

[47]  Harold S. Park,et al.  Nano Mechanics and Materials: Theory, Multiscale Methods and Applications , 2006 .

[48]  Synchronization of extended chaotic systems with long-range interactions: an analogy to Lévy-flight spreading of epidemics. , 2006, Physical review letters.

[49]  CNRS,et al.  Statistical mechanics and dynamics of solvable models with long-range interactions , 2009, 0907.0323.

[50]  V. E. Tarasov Review of Some Promising Fractional Physical Models , 2013, 1502.07681.

[51]  V. Lakshmikantham,et al.  Basic theory of fractional differential equations , 2008 .

[52]  Brian Moritz,et al.  Vector difference calculus for physical lattice models , 1999 .

[53]  V. E. Tarasov Fractional Vector Calculus and Fractional Maxwell's Equations , 2008, 0907.2363.

[54]  Stevan Pilipović,et al.  Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles , 2014 .

[55]  Dominik Rogula,et al.  Nonlocal theory of material media , 1982 .

[56]  V. E. Tarasov Map of discrete system into continuous , 2006, 0711.2612.

[57]  M. Rivero,et al.  Fractional calculus: A survey of useful formulas , 2013, The European Physical Journal Special Topics.

[58]  A. Cleland Foundations of nanomechanics , 2002 .

[59]  D. Zorica,et al.  Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes , 2014 .

[60]  A. Eringen,et al.  Nonlocal Continuum Field Theories , 2002 .

[61]  Sauro Succi,et al.  Lattice differential operators for computational physics , 2013 .

[62]  Vasily E. Tarasov,et al.  Lattice with long-range interaction of power-law type for fractional non-local elasticity , 2014, 1502.05492.

[63]  Manuel Duarte Ortigueira,et al.  Fractional Calculus for Scientists and Engineers , 2011, Lecture Notes in Electrical Engineering.

[64]  F. Mainardi,et al.  Fractals and fractional calculus in continuum mechanics , 1997 .

[65]  G. Zaslavsky,et al.  Nonlinear fractional dynamics on a lattice with long range interactions , 2005, nlin/0512010.