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Using random walks to establish wavelike behavior in a linear FPUT system with random coefficients

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  • 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.

    Mathematics Subject Classification: 37L55, 37L60, 74J05.


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  • 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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