Lee-Yang zeros and large-deviation statistics of a molecular zipper

The complex zeros of partition functions were originally investigated by Lee and Yang to explain the behavior of condensing gases. Since then, Lee-Yang zeros have become a powerful tool to describe phase transitions in interacting systems. Today, Lee-Yang zeros are no longer just a theoretical concept; they have been determined in recent experiments. In one approach, the Lee-Yang zeros are extracted from the high cumulants of thermodynamic observables at finite size. Here we employ this method to investigate a phase transition in a molecular zipper. From the energy fluctuations in small zippers, we can predict the temperature at which a phase transition occurs in the thermodynamic limit. Even when the system does not undergo a sharp transition, the Lee-Yang zeros carry important information about the large-deviation statistics and its symmetry properties. Our work suggests an interesting duality between fluctuations in small systems and their phase behavior in the thermodynamic limit. These predictions may be tested in future experiments.

Figure 1

Zipper model, Lee-Yang zeros, and large-deviation statistics. (a) The molecular zipper is a double-stranded macromolecule held together by $N$ links that can be either open (energy $\varepsilon )$ or closed (energy 0). At the transition temperature $T_c$, the system goes through a phase transition. (b) Lee-Yang zeros in the complex plane of the inverse temperature $\beta =1/k_{B}T$. (c) Large-deviation statistics of the energy per link for large $N$. Distributions are shown for three different temperatures $\beta =0.7{\beta }_c,{\beta }_c,1.3{\beta }_c$, with ${\beta }_c=1/k_B{T}_c$.

Figure 2

Free energy of the molecular zipper. (a) Free energy per link as a function of the temperature $T$ for different numbers of open links $n=0,1,...,N$. For $T<T_c$, the closed zipper has the lowest free energy. For $T>T_c$, the open zipper has the lowest free energy. The equilibrium free energy is shown with a thick black line. (b) Equilibrium free energy per link for different lengths of the zipper. In the thermodynamic limit, the equilibrium free energy per link becomes nonanalytic at $T=T_c$.

Figure 3

Average energy and fluctuations. (a) Average energy per link as a function of the temperature $T$. The various curves correspond to different lengths of the zipper. In the thermodynamic limit, the average energy per link develops a discontinuity at the transition temperature $T_c$. (b) Variance of the energy for zippers of different lengths. The energy fluctuations diverge [45] at the transition temperature $T_c$ in the thermodynamic limit.

Figure 4

Lee-Yang zeros for the zipper model. (a) Exact results for the leading Lee-Yang zeros in the complex plane of the inverse temperature for zippers of increasing size $N=1,2,...,30$. In the thermodynamic limit, the Lee-Yang zeros reach the inverse temperature on the real axis for which a phase transition occurs. The polar coordinates ${\rho }_{\mathrm{o}}$ and ${\varphi }_{\mathrm{o}}$ defined in Eq. (16) are indicated. (b) Extrapolation of the real part and the imaginary part of the Lee-Yang zeros in the thermodynamic limit.

Figure 5

Extraction of Lee-Yang zeros from the energy cumulants. (a) The Lee-Yang zeros are extracted from four consecutive cumulants up to order $n_{\max }$ for zippers of length $N=1,2,...,30$. The inverse temperature is $\beta =0.75{\beta }_c$. The inset illustrates how the accuracy of the method is improved by increasing the cumulant order. (b) Extrapolation of the real and imaginary parts of the Lee-Yang zeros in the thermodynamic limit. These results should be compared with the exact ones in Fig. 4.

Figure 6

Lee-Yang zeros for the broken zipper. (a) Leading Lee-Yang zeros in the complex plane of the inverse temperature. The zeros are extracted from the energy cumulants $\langle \langle U^n\rangle \rangle$ of order $n=9,10,11,12$ for zippers of length $N=1,2,...,30$ with $J=\varepsilon$ and $\beta =1.75(lng)/\varepsilon$. With increasing size, the Lee-Yang zeros move towards the convergence points marked with big red circles. (b) Extrapolation of the convergence points ${\beta }_c$ and ${\beta }_c^{*}$ in the thermodynamic limit. The zeros remain complex.

Figure 7

Large-deviation statistics for the broken zipper. The inverse temperature is (a) $\beta =0.7\mathrm{Re}\left[{\beta }_{\mathrm{c}}\right]$, (b) $\beta =\mathrm{Re}\left[{\beta }_{\mathrm{c}}\right]$, and (c) $\beta =1.3\mathrm{Re}\left[{\beta }_{\mathrm{c}}\right]$, where $\mathrm{Re}\left[{\beta }_{\mathrm{c}}\right]$ is the real part of the convergence points extracted in Fig. 6 with $J=\varepsilon$. The thick red lines are exact results based on Eq. (39) with $N={10}^3$, while the blue lines are the top of the ellipse described by Eq. (48). The fitting parameters $u_1,u_2$, and $\mathcal{N}$ depend weakly on the temperature. The lower parts of the ellipses are indicated with dashed lines.

Figure 8

Fluctuation relation. The solid lines show the left-hand side of Eq. (49) using the exact results from Fig. 7. The dashed lines correspond to the right-hand side of Eq. (49).

Figure 9

Product expansion of the partition function. The dashed lines show Eq. (A2) for $N=20$. We have truncated $n$ from $-n_{\max }$ to $+n_{\max }$. With increasing values of $n_{\max }$, the product expansion converges towards the exact partition function for the zipper model shown with a black line.