{
  "abstract": "The efficiency of a machine at the scale of thermal fluctuations is random, and the conventional ratio $-W/Q_h$ has no moment of any order: its input heat fluctuates through zero. We work instead with the exergetic ratio $η=W/(W+T_0S)$, which lies in $[0,1]$ pointwise for non-negative dissipation, and ask what the energy budget alone determines about its mean. With $α=T_0\\langle S\\rangle/W$ and $σ^2$ the relative variance of the dissipation, $1/(1+α)<\\langleη\\rangle\\leσ^2/(1+σ^2)+1/\\{(1+σ^2)[1+α(1+σ^2)]\\}$, both ends sharp, with no distributional assumption. The floor is Jensen's inequality: fluctuating dissipation raises the mean efficiency above its deterministic value, and the mean alone gives nothing more. The ceiling is attained by an intermittently reversible law, dissipating nothing in a fraction $σ^2/(1+σ^2)$ of realisations, a prediction testable on trajectories. A third moment lifts the floor. Fixed delivered work is not required: when it too fluctuates, the bounds hold with the moments taken on $T_0S/W$, and the thermodynamic uncertainty relation on the work current converts the ceiling into a precision-efficiency frontier, whose zero-variance member is the known bound on a motor's ratio-of-means efficiency, shown here to be unsafe for the mean of the fluctuating ratio. Inside the interval lies the maximum-entropy benchmark $α^{-1}e^{1/α}E_1(1/α)$. Finally the bounds are worked out for a motor with futile cycles, observed until a fixed number of steps is delivered. There the dissipation cannot fall below the reversible cost of that work, and this floor $b$ sharpens the ceiling to $q/(1+αb)+(1-q)/(1+αc)$, $q=σ^2/[σ^2+(1-b)^2]$, $c=1+σ^2/(1-b)$, removing 40-67 per cent of the width. It is saturated when slips are rare: the extremal law is an operating regime, not an idealisation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15443",
  "author_total": 1,
  "authors": [
    {
      "name": "Badr Farih"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/83wxcfx03yqas.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15443",
  "fulltext_url": null,
  "id": "83wxcfx03yqas",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15443",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "physics.data-an"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.15443",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/83wxcfx03yqas.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:11Z",
      "native_id": "2609.15443",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15443"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:physics.data-an",
      "subfield": "computational"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Sharp Bounds on the Mean Efficiency of a Fluctuating Machine",
  "title_text": "Sharp Bounds on the Mean Efficiency of a Fluctuating Machine",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:11Z",
  "url": "https://research.draconitic.com/paper/83wxcfx03yqas",
  "venue": "arXiv",
  "versions": []
}