Nonlinear Structures and Thermodynamic Instabilities in a One-Dimensional Lattice System

The equilibrium states of the discrete Peyrard-Bishop Hamiltonian with one end fixed are computed exactly from the two-dimensional nonlinear Morse map. These exact nonlinear structures are interpreted as domain walls, interpolating between bound and unbound segments of the chain. Their free energy is calculated to leading order beyond the Gaussian approximation. Thermodynamic instabilities (e.g., DNA unzipping and/or thermal denaturation) can be understood in terms of domain wall formation.

Figure 1

The unstable manifold of the FP of the map (2) for $N=28$. Filled squares belong to stable equilibria, open circles to unstable ones. The horizontal line at $y=44$ demonstrates the multivaluedness of the manifold as a function of $y$ (4 stable and three unstable equilibria with that value of $y$; details in the inset). The vertical lines are drawn at $p_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and $p_{\mathrm{m}\mathrm{a}\mathrm{x}}$, the minimal and maximal asymptotic slopes of DWs.

Figure 2

The eight stable equilibria corresponding to $N=28$, $y_0=0$, $y_{N+1}=L=80$. Not shown are seven unstable equilibria enmeshed between the stable ones. Inset: total energies for both stable (filled squares) and unstable (open circles) equilibria. The continuous curve corresponds to a theoretical estimate which does not distinguish between stable and unstable equilibria (cf. text). Note that the typical energy difference between neighboring extrema—which does not shrink with increasing $L$—is about $0.15$.

Figure 3

Eigenvalue spectra of the Hessians $A^{(0)}$ (open squares) and $A^{(*)}$ (open circles) for $N=32$ and $L=100$. The DW’s spectrum consists of bands of optical and acoustic phonons, localized, respectively, in the bound and unbound portions of the chain, and a single local mode in the gap; both bands are well described (to order $\mathcal{O}(1/L)$) by the corresponding free phonon dispersion curves (dotted).

Figure 4

Lowest order anharmonic correction to the free energy difference $\Delta G$: The coefficient $\Delta c_2(p,L)$ for $L=100$, 140, 200, 300, 450 vs $p$; for each $L$ the difference $[C_2(p)-C_2^0]/L$ is numerically evaluated at all stable minima; the interpolates taken at $p=p^{*}$ are plotted against $1/L$ (inset); this procedure extrapolates to a value $\Delta c_2^{*}=-0.01784(5)$.

Figure 5

Phase diagram in the $f$-$T$ plane. The dotted line is obtained on the basis of the Gaussian approximation (predicted $T_c=1.291$). The solid line is Eq. (14) with $\Delta c_2^{*}=-0.01784$ (cf. Figure 4). Inset: detail of critical region, showing a predicted $T_c=1.215$; the triangle is the result of a finite-size scaling numerical TI calculation $T_c=1.227$ [1c].