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Intermediate scattering function of Brownian particles in a tilted cosine potential
Regina Rusch1, Thomas Franosch1
1Universität Innsbruck, Institut für Theoretische Physik, Technikerstraße 21-A, A-6020 Innsbruck, Austria.
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We solve the Fokker-Planck equation for a Brownian particle in a tilted cosine potential and derive the intermediate scattering function (ISF), which captures the full spatiotemporal dynamics of the system. The model consists of a single overdamped Brownian particle in one dimension. We derive a generalized ISF comprising two wave vectors to describe correlations in the periodic potential. Exploiting the periodicity via Bloch's theorem, we formulate the problem within a spectral-theoretical framework and numerically compute the corresponding eigenfunctions and eigenvalues, from which we obtain the ISF and the probability density. Using time-dependent perturbation theory, we expand the ISF and derive low-order moments, including the mean-square displacement, time-dependent diffusivity, skewness, and the non-Gaussian parameter. Our analytical results are validated by Brownian-dynamics simulations and analyzed focusing on different regimes of the tilting force. The results are compared to a harmonic approximation and the deterministic limit.
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