Microscopic meaning of grand potential resulting from combinatorial approach to a general system of particles

We present an alternative approach to the problem of interacting fluids, which we believe may provide important insights into microscopic mechanisms that lead to the occurrence of phase transitions. The approach exploits enumerative properties and combinatorial meaning of Bell polynomials. We derive the exact formula for the probability of a general system of N particles at temperature T to consist of k weakly coupled clusters of various sizes. We also show that the grand potential of the system may be considered the exponential generating function for the number of internal states (thermodynamic probability) of these clusters. The microscopic interpretation of the grand potential is surprising, especially if one recalls that until now only the thermodynamic meaning of this free energy was known. We also derive an approximated expression for the density of states.