Lagrangian for Non-Isolated Systems instead of Entropy for Isolated Systems

The Second Law of Thermodynamics states that entropy increases in a spontaneous process in an isothermal and isolated system and characterizes the direction of evolution. Real systems are not isolated. Here we suggest the description of progress in non-isolated and influenced by external fields systems. One of these fields is temperature field. In this case only entropy is not enough, and we suggest using a new function Ls, which is analogous to the Lagrangian in classical mechanics. Instead of mechanical kinetic energy, Ls includes the product ST, and the system always evolves towards the increase of this modified Lagrangian Ls. The system reaches an equilibrium when the gradient of a total potential force is balanced by the gradients of entropic and thermal forces. For isolated systems the description is reduced to Second Law and Clausius inequality. It has several advantages in comparison to Onsager’s non-equilibrium thermodynamics. This approach does not need a gradient of chemical potential, and easily explains the basic aspects of Soret thermodiffusion and thermoelectric Peltier-Seebeck and Thomson (Lord Kelvin) phenomena in non-isothermal and non-isolated systems. Inside the black hole balance of gravitational and entropic forces may lead to a steady state or the black hole evaporation.