# Response functions

Response functions are quantities that express how a system reacts when some external parameters are changed. Some important response functions for a $PVT$ system are:

• The isobaric thermal expansion coefficient:

$\alpha _{P}={\frac {1}{V}}{\frac {\partial V}{\partial T}}_{|P,N}$ • The isothermal and adiabatic compressibilities:

$K_{T}=-{\frac {1}{V}}{\frac {\partial V}{\partial P}}_{|T,N}=-{\frac {1}{V}}{\frac {\partial ^{2}G}{\partial P^{2}}}_{|T,N}$ $K_{S}=-{\frac {1}{V}}{\frac {\partial V}{\partial P}}_{|S,N}=-{\frac {1}{V}}{\frac {\partial ^{2}H}{\partial P^{2}}}_{|S,N}$ (the minus sign is needed to make them positive)

• The specific heats at constant volume and pressure:

$C_{V}={\frac {\delta Q}{\partial T}}_{|V,N}=T{\frac {\partial S}{\partial T}}_{|V,N}=-T{\frac {\partial ^{2}F}{\partial T^{2}}}_{|V,N}$ $C_{P}={\frac {\delta Q}{\partial T}}_{|P,N}=T{\frac {\partial S}{\partial T}}_{|P,N}=-T{\frac {\partial ^{2}G}{\partial T^{2}}}_{|P,N}$ • For a magnetic system, a very important response function is the isothermal magnetic susceptibility:

$\chi _{T}={\frac {\partial M}{\partial H}}_{|T}=-{\frac {\partial ^{2}G}{\partial H^{2}}}_{|T}$ where of course $H$ is the external field. To be more precise, this should be a tensor instead of a scalar:
${\chi _{T}}_{\alpha \beta }={\frac {\partial M_{\alpha }}{\partial H_{\beta }}}_{|T}$ As we can see, response functions are related to the second derivatives of thermodynamic potentials.

With the response functions and Maxwell relations we can express many quantities otherwise hard to guess. For example, suppose we want to know what

${\frac {\partial S}{\partial V}}_{|T,N}$ is. Using Maxwell relations, we have:
${\frac {\partial S}{\partial V}}_{|T,N}={\frac {\partial P}{\partial T}}_{|V,N}$ and using the fact that:
${\frac {\partial P}{\partial T}}_{|V,N}{\frac {\partial V}{\partial P}}_{|T,N}{\frac {\partial T}{\partial V}}_{|P,N}=-1$ (which comes from mathematical analysis), then:
${\frac {\partial P}{\partial T}}_{|V,N}=-{\frac {1}{{\frac {\partial V}{\partial P}}_{|T,N}{\frac {\partial T}{\partial V}}_{|P,N}}}=-{\frac {{\frac {\partial V}{\partial T}}_{|P,N}}{{\frac {\partial V}{\partial P}}_{|T,N}}}={\frac {\alpha _{P}}{K_{T}}}$ Now, response functions must obey some simple yet important inequalities, which come from the thermal or mechanical stability of the system. For example, $C_{P}=\delta Q/\partial T_{|P}\geq 0$ and $C_{V}=\delta Q/\partial T_{|V}\geq 0$ , since giving heat to a system will increase its temperature. Similarly $K_{T}$ , $K_{S}\geq 0$ since an increase in pressure always decreases the volume.

With Maxwell's relations we can also obtain some equations that will be useful in the future. Let's start considering a system with a fixed number of particles (namely $dN=0$ ) and such that $S$ is explicitly expressed in terms of $T$ and $V$ . Then:

$dS={\frac {\partial S}{\partial T}}_{|V}dT+{\frac {\partial S}{\partial V}}_{|T}dV$ Dividing by $dT$ both sides keeping the pressure constant, and then multiplying by $T$ :
$T{\frac {\partial S}{\partial T}}_{|P}-T{\frac {\partial S}{\partial T}}_{|V}=T{\frac {\partial S}{\partial V}}_{|T}{\frac {\partial V}{\partial T}}_{|P}\quad \Rightarrow \quad C_{P}-C_{V}=T{\frac {\partial S}{\partial V}}_{|T}{\frac {\partial V}{\partial T}}_{|P}$ Now, using the Maxwell relation $\partial S/\partial V_{|T}=\partial P/\partial T_{|V}$ and , namely:
${\frac {\partial P}{\partial T}}_{|V}=-{\frac {\partial P}{\partial V}}_{|T}{\frac {\partial V}{\partial T}}_{|P}$ we get:
$C_{P}-C_{V}=-T{\frac {\partial P}{\partial V}}_{|T}\left({\frac {\partial V}{\partial T}}_{|P}\right)^{2}={\frac {TV}{K_{T}}}\alpha _{P}^{2}$ Similarly, for magnetic systems we have:
$C_{H}-C_{M}={\frac {T}{\chi _{T}}}\left({\frac {\partial M}{\partial T}}\right)^{2}$ Since all these quantities are positive, we also see that:
$C_{P}\geq C_{V}\quad \qquad C_{H}\geq C_{M}$ 1. In fact, let us call $x_{1}$ , $x_{2}$ and $x_{3}$ three variables, and define:
$x_{1}=f_{1}(x_{2},x_{3})\qquad x_{2}=f_{2}(x_{1},x_{3})\qquad x_{3}=f_{3}(x_{1},x_{2})$ Now, we have that $x_{3}=f_{3}(x_{1},f_{2}(x_{1},x_{3}))$ and so deriving with respect to $x_{3}$ and $x_{1}$ we get:
{\begin{aligned}1={\frac {\partial f_{3}}{\partial x_{2}}}{\frac {\partial f_{2}}{\partial x_{3}}}\quad \Rightarrow \quad {\frac {\partial f_{3}}{\partial x_{2}}}={\frac {1}{\frac {\partial f_{2}}{\partial x_{3}}}}\\{}\\0={\frac {\partial f_{3}}{\partial x_{1}}}+{\frac {\partial f_{3}}{\partial x_{2}}}{\frac {\partial f_{2}}{\partial x_{1}}}\quad \Rightarrow \quad {\frac {\partial f_{3}}{\partial x_{1}}}=-{\frac {\partial f_{3}}{\partial x_{2}}}{\frac {\partial f_{2}}{\partial x_{1}}}\end{aligned}} Therefore we have indeed:
${\frac {\partial f_{3}}{\partial x_{1}}}{\frac {\partial f_{1}}{\partial x_{2}}}{\frac {\partial f_{2}}{\partial x_{3}}}=-1$ This way we can find relations similar to ${\textstyle {\frac {\partial P}{\partial T}}_{|V,N}{\frac {\partial V}{\partial P}}_{|T,N}{\frac {\partial T}{\partial V}}_{|P,N}=-1}$ , for example:
${\frac {\partial S}{\partial V}}_{|E,N}{\frac {\partial V}{\partial E}}_{|S,N}{\frac {\partial E}{\partial S}}_{|V,N}=-1\quad \qquad {\frac {\partial S}{\partial N}}_{|E,V}{\frac {\partial N}{\partial E}}_{|S,V}{\frac {\partial E}{\partial S}}_{|N,V}=-1$ 2. There are a few cases where the opposite occurs, but are rather exotic. The most notable ones are black holes.