February  2017, 10(1): 101-117. doi: 10.3934/dcdss.2017006

Discrete spin systems on random lattices at the bulk scaling

Zentrum Mathematik -M7, Technische Universität München, Boltzmannstrasse 3, 85748 Garching, Germany

Received  March 2015 Revised  June 2015 Published  December 2016

We study by Γ-convergence the stochastic homogenization of discrete energies on a class of random lattices as the lattice spacing vanishes. We consider general bounded spin systems at the bulk scaling and prove a homogenization result for stationary lattices. In the ergodic case we obtain a deterministic limit.

Citation: Marco Cicalese, Matthias Ruf. Discrete spin systems on random lattices at the bulk scaling. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 101-117. doi: 10.3934/dcdss.2017006
References:
[1]

M. A. Akcoglu and U. Krengel, Ergodic theorems for superadditive processes, J. Reine Ang. Math., 323 (1981), 53-67.  doi: 10.1515/crll.1981.323.53.  Google Scholar

[2]

R. Alicandro and M. Cicalese, A general integral representation result for continuum limits of discrete energies with superlinear growth, SIAM J. Math. Anal., 36 (2004), 1-37.  doi: 10.1137/S0036141003426471.  Google Scholar

[3]

R. Alicandro, A. Braides and M. Cicalese, book in preparation. Google Scholar

[4]

R. AlicandroA. Braides and M. Cicalese, Continuum limits of discrete thin films with superlinear growth densities, Calc. Var. and PDE, 33 (2008), 267-297.  doi: 10.1007/s00526-008-0159-4.  Google Scholar

[5]

R. AlicandroM. Cicalese and A. Gloria, Variational description of bulk energies for bounded and unbounded spin systems, Nonlinearity, 21 (2008), 1881-1910.  doi: 10.1088/0951-7715/21/8/008.  Google Scholar

[6]

R. AlicandroM. Cicalese and A. Gloria, Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity, Arch. Rat. Mech. Anal., 200 (2011), 881-943.  doi: 10.1007/s00205-010-0378-7.  Google Scholar

[7]

R. AlicandroM. Cicalese and M. Ruf, Domain formation in magnetic polymer composites: An approach via stochastic homogenization, Arch. Rat. Mech. Anal., 218 (2015), 945-984.  doi: 10.1007/s00205-015-0873-y.  Google Scholar

[8]

R. Alicandro and M. S. Gelli, Local and non local continuum limits of Ising type energies for spin systems, SIAM J. Math. Anal., 48 (2016), 895-931.  doi: 10.1137/140997373.  Google Scholar

[9]

A. Braides, Γ-convergence for Beginners, Oxford Lecture Series in Mathematics and its Applications 22, Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar

[10]

A. Braides and M. Cicalese, Interfaces, modulated phases and textures in lattice systems, Arch. Rat. Mech. Anal., (2016), 1-41.  doi: 10.1007/s00205-016-1050-7.  Google Scholar

[11]

A. BraidesM. Cicalese and F. Solombrino, Q-tensor continuum energies as limits of head-to-tail symmetric spin systems, SIAM J. Math. Anal., 47 (2015), 2832-2867.  doi: 10.1137/130941341.  Google Scholar

[12]

A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Oxford Lecture Series in Mathematics and its Applications 12, Oxford University Press, New York, 1998.  Google Scholar

[13]

A. Braides and L. Truskinovsky, Asymptotic expansions by Γ-convergence, Contin. Mech. Thermodyn., 20 (2008), 21-62.  doi: 10.1007/s00161-008-0072-2.  Google Scholar

[14]

G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations (Pitman Research Notes in Mathematics Ser. 207), 1989.  Google Scholar

[15]

M. CicaleseM. Ruf and F. Solombrino, Chirality transitions in frustrated S2-valued spin systems, Math. Models Methods Appl. Sci., 26 (2016), 1481-1529.  doi: 10.1142/S0218202516500366.  Google Scholar

[16]

M. Cicalese and F. Solombrino, Frustrated ferromagnetic spin chains: A variational approach to chirality transitions, Journal of Nonlinear Science, 25 (2015), 291-313.  doi: 10.1007/s00332-015-9230-4.  Google Scholar

[17]

G. Dal Maso and L. Modica, Integral functionals determined by their minima, Rend. Semin. Mat. Univ. Padova, 76 (1986), 255-267.   Google Scholar

[18]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine. Ang. Math., 368 (1986), 28-42.   Google Scholar

[19]

I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: Lp Spaces Springer, New York, 2007.  Google Scholar

[20]

D. GaleV. Klee and R. T. Rockafellar, Convex functions on convex polytopes, Proc. Amer. Math. Soc., 19 (1968), 867-873.  doi: 10.1090/S0002-9939-1968-0230219-6.  Google Scholar

show all references

References:
[1]

M. A. Akcoglu and U. Krengel, Ergodic theorems for superadditive processes, J. Reine Ang. Math., 323 (1981), 53-67.  doi: 10.1515/crll.1981.323.53.  Google Scholar

[2]

R. Alicandro and M. Cicalese, A general integral representation result for continuum limits of discrete energies with superlinear growth, SIAM J. Math. Anal., 36 (2004), 1-37.  doi: 10.1137/S0036141003426471.  Google Scholar

[3]

R. Alicandro, A. Braides and M. Cicalese, book in preparation. Google Scholar

[4]

R. AlicandroA. Braides and M. Cicalese, Continuum limits of discrete thin films with superlinear growth densities, Calc. Var. and PDE, 33 (2008), 267-297.  doi: 10.1007/s00526-008-0159-4.  Google Scholar

[5]

R. AlicandroM. Cicalese and A. Gloria, Variational description of bulk energies for bounded and unbounded spin systems, Nonlinearity, 21 (2008), 1881-1910.  doi: 10.1088/0951-7715/21/8/008.  Google Scholar

[6]

R. AlicandroM. Cicalese and A. Gloria, Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity, Arch. Rat. Mech. Anal., 200 (2011), 881-943.  doi: 10.1007/s00205-010-0378-7.  Google Scholar

[7]

R. AlicandroM. Cicalese and M. Ruf, Domain formation in magnetic polymer composites: An approach via stochastic homogenization, Arch. Rat. Mech. Anal., 218 (2015), 945-984.  doi: 10.1007/s00205-015-0873-y.  Google Scholar

[8]

R. Alicandro and M. S. Gelli, Local and non local continuum limits of Ising type energies for spin systems, SIAM J. Math. Anal., 48 (2016), 895-931.  doi: 10.1137/140997373.  Google Scholar

[9]

A. Braides, Γ-convergence for Beginners, Oxford Lecture Series in Mathematics and its Applications 22, Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar

[10]

A. Braides and M. Cicalese, Interfaces, modulated phases and textures in lattice systems, Arch. Rat. Mech. Anal., (2016), 1-41.  doi: 10.1007/s00205-016-1050-7.  Google Scholar

[11]

A. BraidesM. Cicalese and F. Solombrino, Q-tensor continuum energies as limits of head-to-tail symmetric spin systems, SIAM J. Math. Anal., 47 (2015), 2832-2867.  doi: 10.1137/130941341.  Google Scholar

[12]

A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Oxford Lecture Series in Mathematics and its Applications 12, Oxford University Press, New York, 1998.  Google Scholar

[13]

A. Braides and L. Truskinovsky, Asymptotic expansions by Γ-convergence, Contin. Mech. Thermodyn., 20 (2008), 21-62.  doi: 10.1007/s00161-008-0072-2.  Google Scholar

[14]

G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations (Pitman Research Notes in Mathematics Ser. 207), 1989.  Google Scholar

[15]

M. CicaleseM. Ruf and F. Solombrino, Chirality transitions in frustrated S2-valued spin systems, Math. Models Methods Appl. Sci., 26 (2016), 1481-1529.  doi: 10.1142/S0218202516500366.  Google Scholar

[16]

M. Cicalese and F. Solombrino, Frustrated ferromagnetic spin chains: A variational approach to chirality transitions, Journal of Nonlinear Science, 25 (2015), 291-313.  doi: 10.1007/s00332-015-9230-4.  Google Scholar

[17]

G. Dal Maso and L. Modica, Integral functionals determined by their minima, Rend. Semin. Mat. Univ. Padova, 76 (1986), 255-267.   Google Scholar

[18]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine. Ang. Math., 368 (1986), 28-42.   Google Scholar

[19]

I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: Lp Spaces Springer, New York, 2007.  Google Scholar

[20]

D. GaleV. Klee and R. T. Rockafellar, Convex functions on convex polytopes, Proc. Amer. Math. Soc., 19 (1968), 867-873.  doi: 10.1090/S0002-9939-1968-0230219-6.  Google Scholar

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