The paper by Jorge Kurchan explores the Fluctuation Theorem (FT) for systems undergoing Langevin dynamics, which are characterized by finite degrees of freedom. The FT concerns the distribution of entropy production over long time intervals and states that the ratio of probabilities of having a given entropy production $\sigma_i$ averaged over a large time interval to having $-\sigma_i$ is $e^{\sigma_i}$. This theorem is derived under the assumption of ergodicity, which is particularly straightforward in the context of Langevin dynamics due to the trivial ergodicity of these systems. Kurchan's work has three main objectives: 1. To derive the Gallavotti-Cohen (GC) fluctuation theorem for Langevin dynamics, which simplifies the proof by bypassing complex ergodic theory issues. 2. To study how the FT can be violated in systems with infinitely many degrees of freedom, isolating violations due to complexity from those due to the non-applicability of the chaotic hypothesis. 3. To explore the applicability of the FT to various driven systems, such as Burgers-KPZ, phase separation under shear, and turbulence. The paper begins by reviewing the Langevin and Kramers equations, discussing detailed balance and its modification in the presence of non-conservative forces. It then derives the FT from the modified detailed balance property and presents a limit theorem for entropy production. The FT is shown to reduce to the Green-Kubo formula in the purely conservative limit. In the Fokker-Planck case, where the Langevin equation lacks an inertial term, the paper similarly derives the FT and a nonlinear fluctuation-dissipation theorem. The author concludes by discussing possible violations of the FT in systems with infinitely many degrees of freedom, particularly those with slow dynamics, and suggests that the violation of the fluctuation-dissipation theorem may be a signature of complex dynamics in strongly driven infinite systems.The paper by Jorge Kurchan explores the Fluctuation Theorem (FT) for systems undergoing Langevin dynamics, which are characterized by finite degrees of freedom. The FT concerns the distribution of entropy production over long time intervals and states that the ratio of probabilities of having a given entropy production $\sigma_i$ averaged over a large time interval to having $-\sigma_i$ is $e^{\sigma_i}$. This theorem is derived under the assumption of ergodicity, which is particularly straightforward in the context of Langevin dynamics due to the trivial ergodicity of these systems. Kurchan's work has three main objectives: 1. To derive the Gallavotti-Cohen (GC) fluctuation theorem for Langevin dynamics, which simplifies the proof by bypassing complex ergodic theory issues. 2. To study how the FT can be violated in systems with infinitely many degrees of freedom, isolating violations due to complexity from those due to the non-applicability of the chaotic hypothesis. 3. To explore the applicability of the FT to various driven systems, such as Burgers-KPZ, phase separation under shear, and turbulence. The paper begins by reviewing the Langevin and Kramers equations, discussing detailed balance and its modification in the presence of non-conservative forces. It then derives the FT from the modified detailed balance property and presents a limit theorem for entropy production. The FT is shown to reduce to the Green-Kubo formula in the purely conservative limit. In the Fokker-Planck case, where the Langevin equation lacks an inertial term, the paper similarly derives the FT and a nonlinear fluctuation-dissipation theorem. The author concludes by discussing possible violations of the FT in systems with infinitely many degrees of freedom, particularly those with slow dynamics, and suggests that the violation of the fluctuation-dissipation theorem may be a signature of complex dynamics in strongly driven infinite systems.