  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.
