Thermodynamic Hamiltonian and Entropy Production

The variational method holds considerable significance within mathematical and theoretical physics. Its importance stems from its capacity to characterise natural systems through physical quantities, irrespective of the chosen frame of reference. This characteristic makes it a powerful tool for understanding the behaviour of diverse physical phenomena. A global and statistical approach originating from the principles of non-equilibrium thermodynamics has been developed. This approach culminates in the principle of maximum entropy generation, specifically tailored for open systems. The principle itself arises as a direct consequence of applying the Lagrangian approach to open systems. The work focuses on a generalised method for deriving the thermodynamic Hamiltonian. This Hamiltonian is essential to the dynamical analysis of open systems, allowing for a detailed examination of their time evolution. The analysis suggests that irreversibility appears to be a fundamental process related to the evolution of states within open systems.