Response functions

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

  • The isobaric thermal expansion coefficient:

  • The isothermal and adiabatic compressibilities:

(the minus sign is needed to make them positive)

  • The specific heats at constant volume and pressure:

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

where of course is the external field. To be more precise, this should be a tensor instead of a scalar:

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

is. Using Maxwell relations, we have:
and using the fact that:
(which comes from mathematical analysis[1]), then:

Now, response functions must obey some simple yet important inequalities, which come from the thermal or mechanical stability of the system. For example, and , since giving heat to a system will increase its temperature[2]. Similarly , 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 ) and such that is explicitly expressed in terms of and . Then:

Dividing by both sides keeping the pressure constant, and then multiplying by :
Now, using the Maxwell relation and , namely:
we get:
Similarly, for magnetic systems we have:
Since all these quantities are positive, we also see that:

  1. In fact, let us call , and three variables, and define:
    Now, we have that and so deriving with respect to and we get:
    Therefore we have indeed:
    This way we can find relations similar to , for example:
  2. There are a few cases where the opposite occurs, but are rather exotic. The most notable ones are black holes.