{
  "id": "1810.10766",
  "version": "v1",
  "published": "2018-10-25T08:35:24.000Z",
  "updated": "2018-10-25T08:35:24.000Z",
  "title": "The probability distribution of Brownian motion in periodic potentials",
  "authors": [
    "Matan Sivan",
    "Oded Farago"
  ],
  "comment": "6 pages, 2 figures, Accepted for publication in Phys. Rev. E",
  "categories": [
    "cond-mat.stat-mech",
    "cond-mat.soft"
  ],
  "abstract": "We calculate the probability distribution function (PDF) of an overdamped Brownian particle moving in a periodic potential energy landscape $U(x)$. The PDF is found by solving the corresponding Smoluchowski diffusion equation. We derive the solution for any periodic even function $U(x)$, and demonstrate that it is asymptotically (at large time $t$) correct up to terms decaying faster than $\\sim t^{-3/2}$. As part of the derivation, we also recover the Lifson-Jackson formula for the effective diffusion coefficient of the dynamics. The derived solution exhibits agreement with Langevin dynamics simulations when (i) the periodic length is much larger than the ballistic length of the dynamics, and (ii) when the potential barrier $\\Delta U=\\max(U(x))-\\min(U(x))$ is not much larger than the thermal energy $k_BT$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-25T08:35:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "brownian motion",
      "periodic potential energy landscape",
      "probability distribution function",
      "corresponding smoluchowski diffusion equation",
      "langevin dynamics simulations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}