Maxwell Relations are commonly known as a set of four partial differential equations between four thermodynamic quantities or potentials: pressure (P), volume (V), temperature (T), and entropy (S). So for example the expression (∂T/∂V) |S means the partial derivative of T with respect to V while keeping S constant.
- (∂T/∂V) |S = -(∂P/∂S) |V
- (∂P/∂T) |V = (∂S/∂V) |T
- -(∂S/∂P) |T = (∂V/∂T) |P
- (∂V/∂S) |P = (∂T/∂P) |S
In my diagram above, the expressions that are equal are on either side of the common leg of adjacent isosceles right triangles.
Thermodynamics and the Four Thermodynamic Potentials
Actually, just the left hand side of the equations, because it looks better that way. The right hand sides are just a click away.
E and B are the electric and magnetic fields, respectively. ∇ is the “del” operator.
The divergences ∇ · E and ∇ · B are the “fields emanating from the sources.”
The curls ∇ x E and ∇ x B are the “circulation of the fields.”
I am currently reading Tor Norretranders “The User Illusion”, and he opens the book with chapters on information theory and thermodynamics. His description of the thought experiment “Maxwell’s Demon” made me realize that this system has all the ingredients of a fourfold: the two chambers, the demon, the door between the two chambers that the demon operates, and the molecules that the demon allows to or prevents from passing through the door from one chamber to another. The only way between the two chambers is through the door. The molecules are in a gaseous state, with a distribution of velocities, so some are moving faster than others.
The major physical action that the demon can perform is to open or close the door. If the demon did this randomly, it would accomplish nothing, and the system would ultimately evolve the same as one without door or demon. Instead the demon is able to open and close the door so as to allow faster molecules to collect in one chamber and slower molecules to collect in the other. The demon doesn’t touch the molecules, but he is somehow able to tell whether a molecule is fast or slow so he can decide whether to open or close the door.
The purpose of the thought experiment is to suggest that this system could violate the second law of thermodynamics by increasing order. By letting faster molecules move to one chamber, the system would then be able to do work by, for example, powering an engine. Perhaps the temperature differential is also able to power the demon itself. We would then have a perpetual motion machine, one that could give us unlimited energy, as long as there was a surplus of energy after the needs of the demon.
However, the modern analysis of the thought experiment shows that entropy does indeed increase within this system because the demon would eventually have to forget information about the molecules, and his forgetting would increase entropy. The demon would have to be an irreversible process.
Tor Norretranders / The User Illusion: Cutting Consciousness Down to Size