It is often confusing whether a susceptibility is the same as a response function, specially that often they are used interchangeably, in the context of statistical mechanics and thermodynamics. Very generally:

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**Response function:**

For response functions, typical examples would be thermal expansivity $\alpha,$ isothermal compressibility $\kappa\_T,$ specific heats $C\_v$, $C\_p,$ at least for these examples they seem all to be given by first derivatives of either a system parameter or a potential:

$$
\alpha = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)\_{P,N}, \, \kappa\_T = -\frac{1}{V} \left(\frac{\partial V}{\partial P}\right)\_{T,N}, \, C\_v = \left(\frac{\partial E}{\partial T}\right)\_{V,N}
$$ 

1. So can one define response functions as first derivatives (I guess talking of first derivatives already assumes linear responses) of a system's observables (e.g. $V$) and potentials (e.g. $E$) with respect to system parameters (e.g. $T,$ $P$) without loss of generality?

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**Susceptibilities:**

[Wikipedia definition:](http://en.wikipedia.org/wiki/Susceptibility)

> 
> In physics, the susceptibility of a material or substance describes
>  its response to an applied field. More general, a susceptibility is a
>  quantification for the change of an extensive property under variation
>  of an intensive property.
> 
> 
> 

Typical quantities we refer to as susceptibilities are magnetic and electric susceptibilities, describe the change of magnetization and polarisation with respect to changes of the magnetic field $h$ and electric field $E$ respectively. So one writes, for the magnetic susceptibility e.g.:

$$
\chi = \left(\frac{\partial M}{\partial h}\right)\_T
$$
But the magnetization itself seems to be a response function given by:
$$
M = \left(\frac{\partial F}{\partial h}\right)\_T
$$
Where $F$ is the Helmholtz free energy. Combining the two expression we can write the susceptibility as the second derivative of $F$:
$$
\chi = \left(\frac{\partial^2 F}{\partial h^2}\right)\_T
$$

2. The above in mind, was it correct to call the magnetization a *response function?* As it would be well in line with the given definition of response functions in first part.
3. From the final expression of $\chi,$ can one conclude that susceptibilities are usually given by second order derivatives of thermodynamic potentials with respect to a system parameter or an external field?

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4. Closing remark: All of this seems to rather point at the fact that response functions and susceptibilities cannot actually be used interchangeably. Anyhow, I really hope someone can resolve such confusions by giving more consistent or complete definitions of response functions and susceptibilities.