{
  "id": "2201.05388",
  "version": "v1",
  "published": "2022-01-14T11:00:35.000Z",
  "updated": "2022-01-14T11:00:35.000Z",
  "title": "First-encounter time of two diffusing particles in two- and three-dimensional confinement",
  "authors": [
    "F. Le Vot",
    "S. B. Yuste",
    "E. Abad",
    "D. S. Grebenkov"
  ],
  "journal": "Phys. Rev. E 105, 044119 (2022)",
  "doi": "10.1103/PhysRevE.105.044119",
  "categories": [
    "cond-mat.stat-mech",
    "physics.chem-ph"
  ],
  "abstract": "The statistics of the first-encounter time of diffusing particles changes drastically when they are placed under confinement. In the present work, we make use of Monte Carlo simulations to study the behavior of a two-particle system in two- and three-dimensional domains with reflecting boundaries. Based on the outcome of the simulations, we give a comprehensive overview of the behavior of the survival probability $S(t)$ and the associated first-encounter time probability density $H(t)$ over a broad time range spanning several decades. In addition, we provide numerical estimates and empirical formulas for the mean first-encounter time $\\langle \\cal{T}\\rangle $, as well as for the decay time $T$ characterizing the monoexponential long-time decay of the survival probability. Based on the distance between the boundary and the center of mass of two particles, we obtain an empirical lower bound $t_B$ for the time at which $S(t)$ starts to significantly deviate from its counterpart for the no boundary case. Surprisingly, for small-sized particles, the dominant contribution to $T$ depends only on the total diffusivity $D=D_1+D_2$, in sharp contrast to the one-dimensional case. This contribution can be related to the Wiener sausage generated by a fictitious Brownian particle with diffusivity $D$. In two dimensions, the first subleading contribution to $T$ is found to depend weakly on the ratio $D_1/D_2$. We also investigate the slow-diffusion limit when $D_2 \\ll D_1$ and discuss the transition to the limit when one particle is a fixed target. Finally, we give some indications to anticipate when $T$ can be expected to be a good approximation for $\\langle \\cal{T}\\rangle$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-14T11:00:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "diffusing particles",
      "three-dimensional confinement",
      "survival probability",
      "associated first-encounter time probability density",
      "contribution"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "APS",
      "journal": "Phys. Rev. E"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}