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Intermediate scattering function of Brownian particles in a tilted cosine potential

Regina Rusch and Thomas Franosch

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Phys. Rev. E 114, 014127 – Published 15 July, 2026

DOI: https://doi.org/10.1103/b4n8-4bg7

Abstract

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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Phys. Rev. E 114, 014127– Published 15 July, 2026
  • Received 25 March 2026
  • Accepted 25 June 2026
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