Can susceptibility be infinite even for small systems? Susceptibility can be espressed in terms of Gibbs free energy as:
$$\chi^{-1}= \frac{\partial^2g}{\partial m^2}$$
Where $g$ is the intensive Gibbs free energy. So if the second derivative of $g$ with respect to $m$ is zero, even with $N$ ($N$ is the size of the system) finite we have a divergence of the susceptibility. Is it possible?
 A: I don't think so. Typically, the nonanalyticity of the free energy comes from the thermodynamic limit. In finite systems, the free energy is (almost) always a finite sum of analytic functions so there is no possibility of divergence of derivatives.
The divergence of the susceptibility is typically linked to a macroscopic change of order-parameter $m$ as a result of infinitesimally small source $h$. In a finite system, the response to a source would be always finite because the system can only have finite magnetisation.
Also, I think it should be $\frac{\partial^2 g}{\partial h^2}$?
A: In general, the critical behavior described in the thermodynamic limit is altered in finite systems. For me, the easiest example is the correlation length: in the thermodynamic limit it is infinite, yet in finite systems it is bounded by the size of the system. Correlations within the system cannot be bigger than the system itself!
In a magnetic system, the susceptibility diverges as a power law of the distance to the critical point in the thermodynamic limit ($\chi\sim(T-T_c)^{-\gamma}$ when $N\rightarrow \infty$). If $N<\infty$, this behavior is altered by a function that depends of the system size $\chi\sim(T-T_c)^{-\gamma} F(N)$. This function $F$ is called an "scaling function" and has to fulfill that it is finite when $N\rightarrow \infty$ and avoids divergences when $N<\infty$.
This is a brief summary of what is generally called "Theory of finite size effects". This is a good reference to read about this topic: Haye Hinrichsen - "Non-equilibrium critical phenomena and phase transitions into absorbing states".
