= Positive-feedback entropic chain
{title2=$x=\tanh[(T_c/T)x],\quad T_c=Naf_0/k_B$}
A chain of $N$ up/down segments of length $a$ has extension $L=Nax$ and binary <configurational entropy>. An external force $f(L)=f_0L/a$ has potential $-f_0L^2/(2a)$, obtained by integrating the force. Its mean-field free-energy minima obey the displayed equation. Expansion gives $F-F(0)=Nk_B(T-T_c)x^2/2+Nk_BTx^4/12+\cdots$, 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.
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