Heat can flow from cold to hot in Microcanonical Thermodynamics of finite systems and the microscopic origin of phase transitions

Thermodynamics allows the application of Statistical Mechanics to finite and even small systems. As surface effects cannot be scaled away, one has to be careful with the standard arguments of splitting a system into two or bringing two systems into thermal contact with energy or particle exchange: Not only the volume part of the entropy must be considered. The addition of any other macroscopic constraint like a dividing surface, or the enforcement of gradients of the energy/particle reduce the entropy. As will be shown here, when removing such constraint in regions of a negative heat capacity, the system may even relax under a flow of heat against the temperature slope. Thus Clausius formulation of the Second Law: "Heat always flows from hot to cold" can be violated. However, the Second Law is still satisfied and the total Boltzmann-entropy is rising. In the final chapter the general microscopic mechanism leading to the convexity of the microcanonical entropy at phase separation is discussed. This is explained for the liquid--gas and the solid--liquid transition.