Exact four-vector work distribution and covariant fluctuation theorems of work for a relativistic particle in an expanding piston

We investigate the non-equilibrium four-vector work in an expanding relativistic piston. We derive the exact work distribution in this pedagogical model and find that the joint distribution of four-vector work $(W^0,W^1)$ concentrates on the origin and some curves in the $(W^0,W^1)$ space, rather than being smoothly distributed. In the non-relativistic limit, our model consistently recovers the non-relativistic dynamics. We further demonstrate that the momentum component of four-vector work remains significant in both the Lorentz-relativistic and Galilean-relativistic frameworks. On top of the work distribution, we verify a family of covariant fluctuation theorems of work. In addition, we introduce a novel geometrical technique for analyzing the dynamics of relativistic collision processes, which can be straightforwardly extended to multi-dimensional piston models.