Moment generating function bound from detailed fluctuation theorem

Abstract

A famous consequence of the detailed fluctuation theorem (FT), $p(\Sigma )/p(-\Sigma )=\exp (\Sigma )$, is the integral FT $⟨\exp (-\Sigma )⟩=1$ for a random variable $\Sigma$ and a distribution $p(\Sigma )$. When $\Sigma$ represents the entropy production in thermodynamics, the main outcome of the integral FT is the second law, $⟨\Sigma ⟩\ge 0$. However, a full description of the fluctuations of $\Sigma$ might require knowledge of the moment generating function (MGF), $G(\alpha ):=⟨\exp (\alpha \Sigma )⟩$. In the context of the detailed FT, we show the MGF is lower bounded in the form $G(\alpha )\ge B(\alpha ,⟨\Sigma ⟩)$ for a given mean $⟨\Sigma ⟩$. As applications, we verify that the bound is satisfied for the entropy produced in the heat exchange problem between two reservoirs mediated by a weakly coupled bosonic mode and a qubit swap engine.

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Cited by in corpus (2)

• Work Fluctuations in Ergotropic Heat Engines
• From the Fokker-Planck equation to a contact Hamiltonian system