Inequivalence of statistical ensembles in single molecule measurements

We study the role of fluctuations in single molecule experimental measurements of force-extension $\left(f\text{−}\zeta \right)$ curves. We use the wormlike chain (WLC) model to bring out the connection between the Helmholtz ensemble characterized by the free energy $\left[F\left(\zeta \right)\right]$ and the Gibbs ensemble characterized by the free energy $\left[G\left(f\right)\right]$. We consider the rigid rod limit of the WLC model as an instructive special case to bring out the issue of ensemble inequivalence. We point out the need for taking into account the free energy of transition when one goes from one ensemble to another. We also comment on the “phase transition” noticed in an isometric setup for semiflexible polymers and propose a realization of its thermodynamic limit. We present general arguments which rule out nonmonotonic force-extension curves in some ensembles and note that these do not apply to the isometric ensemble.

Figure 1

Force-extension curve in the Helmholtz (1a) and Gibbs (1b) ensembles for $\beta =0$. We have set $L=1$.

Figure 2

Schematic experimental design for replica thermodynamic limit, shown above for $N=4$