We consider the probability measure minimizing a free energy functional equal to sum of Coulomb interaction, confinement potential and an entropy term, which arises in statistical mechanics gases. In limit where inverse temperature $\beta$ tends $\infty$ term disappears measure, we call thermal equilibrium well-known can also be interpreted as solution classical obstacle problem. provide quanti...

Abstract We study planar polygonal curves with the variational methods. show a unified interpretation of discrete curvatures and Steiner-type formula by extracting notion curvature vector from first variation length functional. Moreover, we determine equilibrium for functional under area-constraint condition their stability.