Explore the fundamental principles of statistical systems operating outside thermal equilibrium in this comprehensive lecture from the VI Siembra-HoLAGrav Young Frontiers Meeting. Delve into the theoretical framework and mathematical foundations that govern non-equilibrium statistical mechanics, examining how systems behave when they deviate from thermal equilibrium states. Learn about the key concepts, methodologies, and analytical tools used to understand and describe the dynamics of statistical systems that are driven away from equilibrium conditions. Discover the applications and implications of non-equilibrium statistical mechanics in various physical contexts, including quantum field theory and gravitational systems. Gain insights into current research developments and theoretical advances in this specialized field of physics through detailed explanations and mathematical derivations presented by an expert from Queen Mary University of London.