- #1
maverick_starstrider
- 1,119
- 7
It is often said that phase transitions only exist in the thermodynamic limit based on some proof like:
-A system has time-reversal symmetry thus its TOTAL free energy F(H)=F(-H) (H is the field) therefore the magnetization is
[tex]M(H)=\frac{\partial F(H)}{\partial H}=\frac{\partial F(-H)}{\partial(- H)}=-\frac{\partial F(-H)}{\partial H}=-M(H) [/tex]
Thefore, [tex]M(H)=-M(H)[/tex] therefore at [tex]M=0[/tex] we have
[tex]M(0)=-M(0)[/tex] therefore M(0)=0, thus no spontaneous magnetization. However, in the thermodynamic limit [tex]f(H)[/tex] (the free energy per site) may have a discontinuity at H=0 and therefore the above is not necessarily true, and so on thus true phase transitions only exist in the thermodynamics limit.
Here's where I get confused. Can we not just say that the total free energy F(H) for a finite system of N spins is really [tex]F(H)=N f(H)[/tex] and the same argument applies. Therefore, isn't it really just the case the the thermodynamic limit EXIST and not necessarily that the system be at it?
-A system has time-reversal symmetry thus its TOTAL free energy F(H)=F(-H) (H is the field) therefore the magnetization is
[tex]M(H)=\frac{\partial F(H)}{\partial H}=\frac{\partial F(-H)}{\partial(- H)}=-\frac{\partial F(-H)}{\partial H}=-M(H) [/tex]
Thefore, [tex]M(H)=-M(H)[/tex] therefore at [tex]M=0[/tex] we have
[tex]M(0)=-M(0)[/tex] therefore M(0)=0, thus no spontaneous magnetization. However, in the thermodynamic limit [tex]f(H)[/tex] (the free energy per site) may have a discontinuity at H=0 and therefore the above is not necessarily true, and so on thus true phase transitions only exist in the thermodynamics limit.
Here's where I get confused. Can we not just say that the total free energy F(H) for a finite system of N spins is really [tex]F(H)=N f(H)[/tex] and the same argument applies. Therefore, isn't it really just the case the the thermodynamic limit EXIST and not necessarily that the system be at it?