Algebraic statistics of Poincar\'e recurrences in DNA molecule

Statistics of Poincar\'e recurrences is studied for the base-pair breathing dynamics of an all-atom DNA molecule in realistic aqueous environment with thousands of degrees of freedom. It is found that at least over five decades in time the decay of recurrences is described by an algebraic law with the Poincar\'e exponent close to $\beta=1.2$. This value is directly related to the correlation decay exponent $\nu = \beta -1$, which is close to $\nu\approx 0.15$ observed in the time resolved Stokes shift experiments. By applying the virial theorem we analyse the chaotic dynamics in polynomial potentials and demonstrate analytically that exponent $\beta=1.2$ is obtained assuming the dominance of dipole-dipole interactions in the relevant DNA dynamics. Molecular dynamics simulations also reveal the presence of strong low frequency noise with the exponent $\eta=1.6$. We trace parallels with the chaotic dynamics of symplectic maps with a few degrees of freedom characterized by the Poincar\'e exponent $\beta \sim 1.5$.