{
  "id": "1810.04639",
  "version": "v1",
  "published": "2018-10-10T16:59:20.000Z",
  "updated": "2018-10-10T16:59:20.000Z",
  "title": "Current with \"wrong\" sign and phase transitions",
  "authors": [
    "Roberto Boccagna"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove that under certain conditions, phase separation is enough to sustain a regime in which current flows along the concentration gradient, a phenomenon which is known in the literature as \\textit{uphill diffusion}. The model we consider here is a version of that proposed in [G. B. Giacomin, J. L. Lebowitz, Phase segregation dynamics in particle system with long range interactions, Journal of Statistical Physics 87(1) (1997): 37-61], which is the continuous mesoscopic limit of a $1d$ discrete Ising chain with a Kac potential. The magnetization profile lies in the interval $\\left[-\\varepsilon^{-1},\\varepsilon^{-1}\\right]$, $\\varepsilon>0$, staying in contact at the boundaries with infinite reservoirs of fixed magnetization $\\pm\\mu$, $\\mu\\in(m^*\\left(\\beta\\right),1)$, where $m^*\\left(\\beta\\right)=\\sqrt{1-1/\\beta}$, $\\beta>1$ representing the inverse temperature. At last, an external field of Heaviside-type of intensity $\\kappa>0$ is introduced. According to the axiomatic non-equilibrium theory, we derive from the mesoscopic free energy functional the corresponding stationary equation and prove the existence of a solution, which is antisymmetric with respect to the origin and discontinuous in $x=0$, provided $\\varepsilon$ is small enough. When $\\mu$ is metastable, the current is positive and bounded from below by a positive constant independent of $\\kappa$, this meaning that both phase transition as well as external field contributes to uphill diffusion, which is a regime that actually survives when the external bias is removed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-10T16:59:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "phase transition",
      "mesoscopic free energy functional",
      "axiomatic non-equilibrium theory",
      "long range interactions",
      "phase segregation dynamics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}