Fluctuation relations without uniform large deviations

We study the fluctuations of a stochastic Maxwell-Lorentz particle model driven by an external field to determine the extent to which fluctuation relations are related to large deviations. Focusing on the total entropy production of this model in its steady state, we show that, although the probability density of this quantity globally satisfies (by definition) a fluctuation relation, its negative tail decays exponentially with time, whereas its positive tail decays slower than exponentially with time because of long collision-free trajectories. This provides an example of a physical system for which the fluctuation relation does not derive, as commonly thought, from a probability density decaying everywhere exponentially with time or, in other words, from a probability density having a uniform large deviation form.