Elements of statistical physics and phase transitions

As there will be many references to the relationships of statistical physics, it seems relevant to present a brief summary of the most important formulae. 

If a system with a discrete set of its possible states s has a total energy H(s) in one of them, then the probability of the system being in the state. s is expressed by the Gibbs canonical distribution 

For systems whose energy varies quasicontiiiuously (i.e. the distances between energy levels are small in comparison with kT) the probability of the system being in the state with the energy between H and H + dH is expressed as 

the calculation of statistical integral 6 reduces to the computation of the configurative integral, the latter being a very difficult problem, for the solution of w hich different approximate methods have been developed. 

In view of Equation 1, any thermodynamic quantity, which characterizes the state of a system, is expressed in the standard way 

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