Monte Carlo study of the frame, fluctuation and internal tensions of fluctuating membranes with fixed area

Three types of surface tensions can be defined for lipid membranes: the internal tension, $\sigma$, conjugated to the real membrane area in the Hamiltonian, the mechanical frame tension, $\tau$, conjugated to the projected area, and the "fluctuation tension", $r$, obtained from the fluctuation spectrum of the membrane height. We investigate these surface tensions by means of a Monge gauge lattice Monte Carlo simulation involving the exact, nonlinear, Helfrich Hamiltonian and a measure correction for excess entropy of the Monge gauge. Our results for the relation between $\sigma$ and $\tau$ agrees well with the theoretical prediction of [J.-B. Fournier and C. Barbetta, Phys. Rev. Lett., 2008, 100, 078103] based on a Gaussian approximation. This provides a valuable knowledge of~$\tau$ in the standard Gaussian models where the tension is controlled by $\sigma$. However, contrary to the conjecture in the above paper, we find that $r$ exhibits no significant difference from $\tau$ over more than five decades of tension. Our results appear to be valid in the thermodynamic limit and are robust to changing the ensemble in which the membrane area is controlled.