A famous consequence of the detailed fluctuation theorem (FT), $p\left(\mathrm{\Sigma }\right)/p(-\mathrm{\Sigma })=exp\left(\mathrm{\Sigma }\right)$, is the integral FT $\langle exp(-\mathrm{\Sigma })\rangle =1$ for a random variable $\mathrm{\Sigma }$ and a distribution $p\left(\mathrm{\Sigma }\right)$. When $\mathrm{\Sigma }$ represents the entropy production in thermodynamics, the main outcome of the integral FT is the second law, $\langle \mathrm{\Sigma }\rangle \ge 0$. However, a full description of the fluctuations of $\mathrm{\Sigma }$ might require knowledge of the moment generating function (MGF), $G\left(\alpha \right):=\langle exp\left(\alpha \mathrm{\Sigma }\right)\rangle$. In the context of the detailed FT, we show the MGF is lower bounded in the form $G\left(\alpha \right)\ge B(\alpha ,\langle \mathrm{\Sigma }\rangle )$ for a given mean $\langle \mathrm{\Sigma }\rangle$. 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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• Entropy production
• Fluctuation theorems
• Quantum thermodynamics
• Stochastic thermodynamics