# Finite size effects on phase transitions

We have just seen that phase transitions can occur only in the thermodynamic limit. However, this is only a mathematical concept and can never be really reached: real systems, regardless of how much big they are, are not infinite! We can therefore ask if what we have just seen is useful at all, namely if and how much real systems differ from their thermodynamic limit.

Consider the correlation length of a thermodynamic system: we know that it depends on the temperature of the system, and that it diverges in the neighbourhood of a critical point. Let us suppose that our system is finite, and call ${\displaystyle L}$ the length of its size; of course the correlation length ${\displaystyle \xi }$ of the system cannot be greater that ${\displaystyle L}$ itself and so for temperatures near enough to the critical one the behaviour of the system will differ from that of its thermodynamic limit. Let us see with a realistic example how big this difference is; suppose that:

${\displaystyle \xi \sim \xi _{0}t^{-2/3}}$
where ${\displaystyle t=(T-T_{c})/T_{c}}$ is the reduced temperature and ${\displaystyle \xi _{0}\approx 10}$ Å is the correlation length far from the critical point. This form for ${\displaystyle \xi }$ is realistic for fluids and magnets, and ${\displaystyle \xi _{0}}$ is an overestimate in many real cases. If we suppose ${\displaystyle L=1{\text{ cm}}}$, we have ${\displaystyle \xi =L}$ when ${\displaystyle t\approx 10^{-11}}$: this means that we should be able to measure temperatures with a precision of one part in ${\displaystyle 10^{11}}$ in order to detect deviations from the thermodynamic limit!

We therefore see that even if in principle real systems behave differently from their thermodynamic limits, these differences are extremely small and negligible in all reasonable experimental conditions. We are thus legitimated to use thermodynamic limits in statistical mechanics in order to study phase transitions of real macroscopic systems.