# Response functions

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

• The isobaric thermal expansion coefficient:

${\displaystyle \alpha _{P}={\frac {1}{V}}{\frac {\partial V}{\partial T}}_{|P,N}}$

• The isothermal and adiabatic compressibilities:

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

• The specific heats at constant volume and pressure:

${\displaystyle C_{V}={\frac {\delta Q}{\partial T}}_{|V,N}=T{\frac {\partial S}{\partial T}}_{|V,N}=-T{\frac {\partial ^{2}F}{\partial T^{2}}}_{|V,N}}$
${\displaystyle C_{P}={\frac {\delta Q}{\partial T}}_{|P,N}=T{\frac {\partial S}{\partial T}}_{|P,N}=-T{\frac {\partial ^{2}G}{\partial T^{2}}}_{|P,N}}$

• For a magnetic system, a very important response function is the isothermal magnetic susceptibility:

${\displaystyle \chi _{T}={\frac {\partial M}{\partial H}}_{|T}=-{\frac {\partial ^{2}G}{\partial H^{2}}}_{|T}}$
where of course ${\displaystyle H}$ is the external field. To be more precise, this should be a tensor instead of a scalar:
${\displaystyle {\chi _{T}}_{\alpha \beta }={\frac {\partial M_{\alpha }}{\partial H_{\beta }}}_{|T}}$

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

${\displaystyle {\frac {\partial S}{\partial V}}_{|T,N}}$
is. Using Maxwell relations, we have:
${\displaystyle {\frac {\partial S}{\partial V}}_{|T,N}={\frac {\partial P}{\partial T}}_{|V,N}}$
and using the fact that:
${\displaystyle {\frac {\partial P}{\partial T}}_{|V,N}{\frac {\partial V}{\partial P}}_{|T,N}{\frac {\partial T}{\partial V}}_{|P,N}=-1}$
(which comes from mathematical analysis[1]), then:
${\displaystyle {\frac {\partial P}{\partial T}}_{|V,N}=-{\frac {1}{{\frac {\partial V}{\partial P}}_{|T,N}{\frac {\partial T}{\partial V}}_{|P,N}}}=-{\frac {{\frac {\partial V}{\partial T}}_{|P,N}}{{\frac {\partial V}{\partial P}}_{|T,N}}}={\frac {\alpha _{P}}{K_{T}}}}$

Now, response functions must obey some simple yet important inequalities, which come from the thermal or mechanical stability of the system. For example, ${\displaystyle C_{P}=\delta Q/\partial T_{|P}\geq 0}$ and ${\displaystyle C_{V}=\delta Q/\partial T_{|V}\geq 0}$, since giving heat to a system will increase its temperature[2]. Similarly ${\displaystyle K_{T}}$, ${\displaystyle K_{S}\geq 0}$ 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 ${\displaystyle dN=0}$) and such that ${\displaystyle S}$ is explicitly expressed in terms of ${\displaystyle T}$ and ${\displaystyle V}$. Then:

${\displaystyle dS={\frac {\partial S}{\partial T}}_{|V}dT+{\frac {\partial S}{\partial V}}_{|T}dV}$
Dividing by ${\displaystyle dT}$ both sides keeping the pressure constant, and then multiplying by ${\displaystyle T}$:
${\displaystyle T{\frac {\partial S}{\partial T}}_{|P}-T{\frac {\partial S}{\partial T}}_{|V}=T{\frac {\partial S}{\partial V}}_{|T}{\frac {\partial V}{\partial T}}_{|P}\quad \Rightarrow \quad C_{P}-C_{V}=T{\frac {\partial S}{\partial V}}_{|T}{\frac {\partial V}{\partial T}}_{|P}}$
Now, using the Maxwell relation ${\displaystyle \partial S/\partial V_{|T}=\partial P/\partial T_{|V}}$ and , namely:
${\displaystyle {\frac {\partial P}{\partial T}}_{|V}=-{\frac {\partial P}{\partial V}}_{|T}{\frac {\partial V}{\partial T}}_{|P}}$
we get:
${\displaystyle C_{P}-C_{V}=-T{\frac {\partial P}{\partial V}}_{|T}\left({\frac {\partial V}{\partial T}}_{|P}\right)^{2}={\frac {TV}{K_{T}}}\alpha _{P}^{2}}$
Similarly, for magnetic systems we have:
${\displaystyle C_{H}-C_{M}={\frac {T}{\chi _{T}}}\left({\frac {\partial M}{\partial T}}\right)^{2}}$
Since all these quantities are positive, we also see that:
${\displaystyle C_{P}\geq C_{V}\quad \qquad C_{H}\geq C_{M}}$

1. In fact, let us call ${\displaystyle x_{1}}$, ${\displaystyle x_{2}}$ and ${\displaystyle x_{3}}$ three variables, and define:
${\displaystyle x_{1}=f_{1}(x_{2},x_{3})\qquad x_{2}=f_{2}(x_{1},x_{3})\qquad x_{3}=f_{3}(x_{1},x_{2})}$
Now, we have that ${\displaystyle x_{3}=f_{3}(x_{1},f_{2}(x_{1},x_{3}))}$ and so deriving with respect to ${\displaystyle x_{3}}$ and ${\displaystyle x_{1}}$ we get:
{\displaystyle {\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:
${\displaystyle {\frac {\partial f_{3}}{\partial x_{1}}}{\frac {\partial f_{1}}{\partial x_{2}}}{\frac {\partial f_{2}}{\partial x_{3}}}=-1}$
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:
${\displaystyle {\frac {\partial S}{\partial V}}_{|E,N}{\frac {\partial V}{\partial E}}_{|S,N}{\frac {\partial E}{\partial S}}_{|V,N}=-1\quad \qquad {\frac {\partial S}{\partial N}}_{|E,V}{\frac {\partial N}{\partial E}}_{|S,V}{\frac {\partial E}{\partial S}}_{|N,V}=-1}$
2. There are a few cases where the opposite occurs, but are rather exotic. The most notable ones are black holes.