Ensemble inequivalence in single-molecule experiments

In bulk systems the calculation of the main thermodynamic quantities leads to the same expectation values in the thermodynamic limit, regardless of the choice of the statistical ensemble. Single linear molecules can be still regarded as statistical systems, where the thermodynamic limit is represented by infinitely long chains. The question of equivalence between different ensembles is not at all obvious and has been addressed in the literature, with sometimes contradicting conclusions. We address this problem by studying the scaling properties of the ensemble difference for two different chain models as a function of the degree of polymerization. By characterizing the scaling behavior of the difference between the isotensional (Gibbs) and isometric (Helmholtz) ensembles in the transition from the low-stretching to the high-stretching regime, we show that ensemble equivalence cannot be reached for macroscopic chains in the low force regime, and we characterize the transition from the inequivalence to the equivalence regime.

Figure 1

(Color online) The sketch of analogies between conjugate chain ensembles and standard statistical mechanical ensembles.

Figure 2

(Color online) Definition of ensemble difference ${\Delta }^{\prime }$ between isotensional and isometric ensembles for a given value $F^{\ast }$ of the force applied in the isotensional ensemble.

Figure 3

(Color online) Sketch of the chains in the isometric and isotensional ensembles.

Figure 4

(Color online) The force-extension curve for the $r_{eq}=0$ case in the isometric (circles) and isotensional (triangles) ensembles. From left to right, $N=10$ (black) and $N=30$ (green). The dashed lines show the result of linear interpolation. Data have been offset along the $x$ axis for the sake of clarity.

Figure 5

(Color online) The force-extension curve for the $r_{eq}=d$ case in the isometric (circles) and isotensional (triangles) ensembles. From left to right, $N=10$ (black), $N=30$ (green), $N=60$ (red), and $N=90$ (blue). The solid lines represent the result of linear interpolation in the low-stretching regime. Data have been offset along the $x$ axis for the sake of clarity.

Figure 6

(Color online) Universality of the force-extension curves in the low-stretching regimes for the $r_{eq}=d$ case, reporting the rescaled force $\tilde{F}$ as a function of the rescaled end-to-end distance $\tilde{X}$ for the isometric (circles) and isotensional (triangles) ensembles. The white solid line on top of the isotensional sampled curve in the low-stretching regime is a fit to Eq. (8), while the horizontal dashed line represents an estimate for the low-stretching regime from dimensional analysis.

Figure 7

(Color online) Dependence of the ensemble difference $\Delta$ as a function of the force applied in the isotensional ensemble for different chain lengths, $N=15$ (circles), $N=25$ (squares), $N=60$ (upper triangles), and $N=170$ (lower triangles) in the $r_{eq}=0$ case.

Figure 8

(Color online) Logarithmic plot of the ensemble difference $\Delta$ as a function of the number of beads, $N-1$, for the $r_{eq}=0$ case. Different symbols correspond to different applied forces in the isotensional ensemble, going from the low-stretching regime (smaller slope) to the high-stretching regime (larger slope). Dashed lines correspond to the best fit to a power law. The applied forces ranged from 0.035 to $3.25{\text{d}}^{-1}$.

Figure 9

(Color online) Logarithmic plot of the ensemble difference $\Delta$ as a function of the number of beads, $N-1$, for the $r_{eq}=d$ case. Different symbols correspond to different applied forces in the isotensional ensemble, going from the low-stretching regime (smaller slope) to the high-stretching regime (larger slope). Dashed lines correspond to the best fit to a power law. The applied forces ranged from 0.035 to $0.7{\text{d}}^{-1}$.

Figure 10

(Color online) Dependence of the fitted scaling exponent $\alpha$ on the applied force for the $r_{eq}=0$ (squares) and $r_{eq}=d$ (circle) cases. The dashed lines are the result of a best fit to the error function.