Bound for the moment generating function from the detailed fluctuation theorem.

Abstract

A famous consequence of the detailed fluctuation theorem (FT), p(Σ)/p(-Σ)=exp(Σ), is the integral FT 〈exp(-Σ)〉=1 for a random variable Σ and a distribution p(Σ). When Σ represents the entropy production in thermodynamics, the main outcome of the integral FT is the second law, 〈Σ〉≥0. However, a full description of the fluctuations of Σ might require knowledge of the moment generating function (MGF), G(α):=〈exp(αΣ)〉. In the context of the detailed FT, we show the MGF is lower bounded in the form G(α)≥B(α,〈Σ〉) for a given mean 〈Σ〉. 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.