Interpolation and free energies

(Sethna, “Entropy, Order Parameters, and Complexity”, ex. XXX)
© 2022, James P. Sethna, all rights reserved.

Here we provide the function w(A) satisfying log(w) = A w, implemented using the Lambert W function with branch k = -1. This is needed for part (f) of the exercise “Interpolation and free energies”.

The Lambert W function satisfies W(z) exp(W(z)) = z. Mathematica calls it ProductLog.

z = 10.
ProductLog[z] Exp[ProductLog[z]]

The Lambert W(z) function on branch k = -1 is real for -1/e < z < 0, so our w(A) function is real for 0<A<1/e.

w[A_] := -ProductLog[-1,-A]/A
Print[Log[w[0.3]], " = ",0.3 * w[0.3]]