# American Institute of Mathematical Sciences

September  2022, 15(9): 2581-2607. doi: 10.3934/dcdss.2021100

## 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 Published  September 2022 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 and Continuous Dynamical Systems - S, 2022, 15 (9) : 2581-2607. doi: 10.3934/dcdss.2021100
##### References:
 [1] M. Chirilus-Bruckner, C. Chong, O. 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. [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. [3] R. Durret, Probability, Theory and Examples, Cambride Ser. in Stat. and Prob. Math., Cambridge University Press, New York, 2010. doi: 10.1017/CBO9780511779398. [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. [5] J. Gaison, S. Moskow, J. 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. [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. [7] J. McNamee, F. Stenger and E. L. Whitney, Whittaker's cardinal function in retrospect, Mathematics of Computation, 25 (1971), 141-154.  doi: 10.2307/2005140. [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. [9] Y. Okada, S. Watanabe and H. Tanaca, Solitary wave in periodic nonlinear lattice, J. Phys. Soc. Jpn., 59 (1990), 2647-2658.  doi: 10.1143/JPSJ.59.2647. [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. [11] H. Yoshida, Construction of higher order symplectic integrators, Phys. Let. A, 150 (1990), 262-268.  doi: 10.1016/0375-9601(90)90092-3.

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##### References:
 [1] M. Chirilus-Bruckner, C. Chong, O. 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. [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. [3] R. Durret, Probability, Theory and Examples, Cambride Ser. in Stat. and Prob. Math., Cambridge University Press, New York, 2010. doi: 10.1017/CBO9780511779398. [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. [5] J. Gaison, S. Moskow, J. 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. [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. [7] J. McNamee, F. Stenger and E. L. Whitney, Whittaker's cardinal function in retrospect, Mathematics of Computation, 25 (1971), 141-154.  doi: 10.2307/2005140. [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. [9] Y. Okada, S. Watanabe and H. Tanaca, Solitary wave in periodic nonlinear lattice, J. Phys. Soc. Jpn., 59 (1990), 2647-2658.  doi: 10.1143/JPSJ.59.2647. [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. [11] H. Yoshida, Construction of higher order symplectic integrators, Phys. Let. A, 150 (1990), 262-268.  doi: 10.1016/0375-9601(90)90092-3.
Figure 1 is a log-log plot of the relative error $\rho$ divided by $\sqrt{\log\log(1/\epsilon)}.$
Figure 2 is 10 box plots of 40 different realization of masses at 10 various epsilons. It is also log-log
Figure 3 is a log-log plot of the relative error masses chosen periodically
In Figure 4 masses are chosen so that $\chi(j)$ will grow like $\sqrt{j}$
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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