Positive-feedback entropic chain ($x=\tanh [(T_c/T)x],T_c=Na{f}_0/k_B$)

A chain of $N$ up/down segments of length $a$ has extension $L=Nax$ and binary configurational entropy. An external force $f(L)=f_{0}L/a$ has potential $-f_0{L}^2/(2a)$, obtained by integrating the force. Its mean-field free-energy minima obey the displayed equation. Expansion gives $F-F(0)=N{k}_B(T-T_c)x^2/2+N{k}_{B}T{x}^4/12+\cdots$, so the ordered extension emerges continuously, $|x|\sim \sqrt{3(T_c-T)/T}$, rather than jumping. The equation matches zero-field mean-field theory of the Ising model; finite systems round the ideal singular behavior.