{
  "id": "1907.01582",
  "version": "v1",
  "published": "2019-07-02T18:39:38.000Z",
  "updated": "2019-07-02T18:39:38.000Z",
  "title": "Minimum Power to Maintain a Nonequilibrium Distribution of a Markov Chain",
  "authors": [
    "Dmitri S. Pavlichin",
    "Yihui Quek",
    "Tsachy Weissman"
  ],
  "comment": "9 pages, 5 figures",
  "categories": [
    "cond-mat.stat-mech",
    "cs.IT",
    "math.IT",
    "physics.bio-ph"
  ],
  "abstract": "Biological systems use energy to maintain non-equilibrium distributions for long times, e.g. of chemical concentrations or protein conformations. What are the fundamental limits of the power used to \"hold\" a stochastic system in a desired distribution over states? We study the setting of an uncontrolled Markov chain $Q$ altered into a controlled chain $P$ having a desired stationary distribution. Thermodynamics considerations lead to an appropriately defined Kullback-Leibler (KL) divergence rate $D(P||Q)$ as the cost of control, a setting introduced by Todorov, corresponding to a Markov decision process with mean log loss action cost. The optimal controlled chain $P^*$ minimizes the KL divergence rate $D(\\cdot||Q)$ subject to a stationary distribution constraint, and the minimal KL divergence rate lower bounds the power used. While this optimization problem is familiar from the large deviations literature, we offer a novel interpretation as a minimum \"holding cost\" and compute the minimizer $P^*$ more explicitly than previously available. We state a version of our results for both discrete- and continuous-time Markov chains, and find nice expressions for the important case of a reversible uncontrolled chain $Q$, for a two-state chain, and for birth-and-death processes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-02T18:39:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "markov chain",
      "minimum power",
      "nonequilibrium distribution",
      "minimal kl divergence rate lower",
      "mean log loss action cost"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}