Lagrangian for Real Systems Instead of Entropy for Ideal Isolated Systems

The Second Law of Thermodynamics states that entropy S increases in a spontaneous process in an ideal isothermal and isolated system, which characterizes the direction of evolution. Real systems are not isolated. They are influenced by external forces and fields. One of these fields is the temperature field. Here we suggest the description of progress in non-isolated and influenced by external fields system. 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. As before, it includes total potential energy but instead of mechanical kinetic energy, Ls includes the product ST, and the system always evolves towards increasing this modified Lagrangian. It reaches an equilibrium when the gradient of a total potential force is balanced by both the gradients of entropic and thermal forces. For isolated systems the description is reduced to Second Law and Clausius inequality. Our approach does not need a gradient of chemical potential, and it has several advantages compared to Onsager’s non-equilibrium thermodynamics. It easily explains the basic aspects of diffusion, Dufour effect and Soret thermodiffusion. The combination of electric, thermal, and entropic forces explains thermoelectric phenomena in non-isothermal and non-isolated systems, including Peltier-Seebeck and Thomson (Lord Kelvin) effects. Gravitational and entropic forces together inside a black hole may lead to a steady state or the black hole evaporation. They are also involved in influenced by Sun atmospheric processes.