Remarks on phase transitions

It is usually said that, phase transitions are associated with singular, non-analytic and discontinuous behaviors of physical functions such as the thermodynamic potential or other non-equilibrium ones. But this is so only for infinite systems, in which any small difference in energy density can be infinitely magnified due the infinity of volumn. In infinite systems, phase transitions are sharp and abrupt, and ergodicity is completely lost when certain state is selected under symmetry breaking. This means infinite life time of the selected state, and thus very sharp transition. For finite systems, which represent the reality, phase transitions are never as sharp as that in infinite systems, since in this case the life time, though long, but finite. Also, no genuine singularities exist. Only strong crossovers can be observed, provided sufficient resolution. A crucial feature of finite systems might be that, configurations with small energy density differences could have very strong mixing and fluctuations and may not be distinguishable for certain resolutions.