(*dynamical Landau free-energy for a KCM on a complete graph*)
F[ρ_,s_,c_]=c ρ (1-ρ) +(1-c) ρ^2-2 Exp[-s]Sqrt[c (1-c) ρ (1-ρ)]ρ

(*psi_K(s) is given by the minimum of F[ρ,s,c] for ρ on [0,1]*)
Manipulate[
cc=.6;
Plot[F[ρ,s,cc],{ρ,0,1},PlotStyle->{Thick,Darker@Purple},Frame->True,
PlotRange->{{0,1},{-0.02,.12}}],
{{s,0},-.1,.1}]

(*dynamical Landau free-energy for a KCM on a complete graph s conjugated to activity*)
F[ρ_,s_,c_]=c ρ (1-ρ) +(1-c) ρ^2-2 Sqrt[c (1-c) ρ (1-ρ)]ρ -s ρ

(*psi(s) is given by the minimum of F[ρ,s,c] for ρ on [0,1]*)
Manipulate[
cc=.6;
Plot[F[ρ,s,cc],{ρ,0,1},PlotStyle->{Thick,Darker@Purple},Frame->True,
PlotRange->{{0,1},{-0.02,.12}}],
{{s,0},-.1,.1}]