Summary

Statistical mechanics supplements the deterministic fundamental laws with probabilistic tools in order to obtain effective descriptions of equilibrium macroscopic systems consisting of a large number of particles. This gives rise to emergent or effective laws that are not apparent at the microscopic level. The more useful of these laws are those that show universality, that is are independent of the microscopic details of theory. This was emphasized already for the case of the ideal gas law. For equilibrium systems, the concept of entropy summarizes via the Second Law the tendency of isolated systems to move towards greater disorder. As we saw from the statistical definition of entropy, the Second Law itself is an emergent law valid for large systems: Indeed the Second Law suggests macroscopic irreversibility eventhough the microscopic dynamics is reversible. For systems that are not isolated but with some state-variables kept constant, the free-energies $F$ and $G$ determine the equilibrium states through extremal principles. More usually quoted examples of universal emergent laws are those that arise near the critical point of a second-order phase transition. Near the critical point the correlation length diverges so that the system becomes scale-invariant, meaning that its properties become insensitive to the microscopic structure and display some universality. Percolation is a geometric analog of thermal systems that shows behaviour similar to that of second order phase transitions. It should be noted that because of universality near the critical point, the very simple models (whether for thermal systems or percolation) give results for critical properties that are in quantitative agreement with experiments even though the real microscopic dynamics of the experimental systems might be much more involved. This will not always be the case for other complex systems that we will study, where usually the agreement between models and reality will be qualitative at best. Nevertheless the virtue of the models even in those cases is that they highlight the important features that enable one to obtain crucial insight that might otherwise be lost in a mass of detail if the system was represented and studied by more accurate equations.