doi: 10.3934/dcdss.2021100
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Using random walks to establish wavelike behavior in a linear FPUT system with random coefficients

Department of Mathematics, Drexel University, 3141 Chestnut St., Philadelphia, PA 19104

Received  March 2021 Revised  July 2021 Early access August 2021

We consider a linear Fermi-Pasta-Ulam-Tsingou lattice with random spatially varying material coefficients. Using the methods of stochastic homogenization we show that solutions with long wave initial data converge in an appropriate sense to solutions of a wave equation. The convergence is strong and both almost sure and in expectation, but the rate is quite slow. The technique combines energy estimates with powerful classical results about random walks, specifically the law of the iterated logarithm.

Citation: Joshua A. McGinnis, J. Douglas Wright. Using random walks to establish wavelike behavior in a linear FPUT system with random coefficients. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021100
References:
[1]

M. Chirilus-BrucknerC. ChongO. Prill and G. Schneider, Rigorous description of macroscopic wave packets in infinite periodic chains of coupled oscillators by modulation equations, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 879-901.  doi: 10.3934/dcdss.2012.5.879.  Google Scholar

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G. Schneider and C. E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model, International Conference on Differential Equations, Vols. 1, 2 (Berlin, 1999), World Scientific, River Edge, NJ, 2000,390–404.  Google Scholar

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show all references

References:
[1]

M. Chirilus-BrucknerC. ChongO. Prill and G. Schneider, Rigorous description of macroscopic wave packets in infinite periodic chains of coupled oscillators by modulation equations, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 879-901.  doi: 10.3934/dcdss.2012.5.879.  Google Scholar

[2]

D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Ser. Math. Appl., 17, The Clarendon Press, Oxford University Press, New York, 1999.  Google Scholar

[3]

R. Durret, Probability, Theory and Examples, Cambride Ser. in Stat. and Prob. Math., Cambridge University Press, New York, 2010. doi: 10.1017/CBO9780511779398.  Google Scholar

[4]

W. Feller, The General Form of the So Called Law of the Iterated Logarithm, Trans. Amer. Math. Soc., 54 (1943), 373-402.  doi: 10.1090/S0002-9947-1943-0009263-7.  Google Scholar

[5]

J. GaisonS. MoskowJ. D. Wright and Q. Zhang, Approximation of Polyatomic FPU Lattices by KdV Equations, Mult. Scale Model. Simul., 12 (2014), 953-995.  doi: 10.1137/130941638.  Google Scholar

[6]

A. J. Martínez, P. G. Kevrekidis and M. A. Porter., Superdiffusive tansport and energy localization in disordered granular crystals, Phys Rev. E, 93 (2016), 022902. doi: 10.1103/physreve.93.022902.  Google Scholar

[7]

J. McNameeF. Stenger and E. L. Whitney, Whittaker's cardinal function in retrospect, Mathematics of Computation, 25 (1971), 141-154.  doi: 10.2307/2005140.  Google Scholar

[8]

A. Mielke, Macroscopic behavior of microscopic oscillations in harmonic lattices via Wigner-Husimi transforms, Arch. Rational Mech. Anal., 181 (2006), 401-448.  doi: 10.1007/s00205-005-0405-2.  Google Scholar

[9]

Y. OkadaS. Watanabe and H. Tanaca, Solitary wave in periodic nonlinear lattice, J. Phys. Soc. Jpn., 59 (1990), 2647-2658.  doi: 10.1143/JPSJ.59.2647.  Google Scholar

[10]

G. Schneider and C. E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model, International Conference on Differential Equations, Vols. 1, 2 (Berlin, 1999), World Scientific, River Edge, NJ, 2000,390–404.  Google Scholar

[11]

H. Yoshida, Construction of higher order symplectic integrators, Phys. Let. A, 150 (1990), 262-268.  doi: 10.1016/0375-9601(90)90092-3.  Google Scholar

Figure 1.  Figure 1 is a log-log plot of the relative error $ \rho $ divided by $ \sqrt{\log\log(1/\epsilon)}. $
Figure 2.  Figure 2 is 10 box plots of 40 different realization of masses at 10 various epsilons. It is also log-log
Figure 3.  Figure 3 is a log-log plot of the relative error masses chosen periodically
Figure 4.  In Figure 4 masses are chosen so that $ \chi(j) $ will grow like $ \sqrt{j} $
Figure 5.  In Figure 5 $ \epsilon $ is fixed and small while $ \sigma $ is varied and the absolute error is measured. When $ \sigma $ is smallest, the data is concentrated near the error for the constant coefficient case
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