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We study the dynamics of an ensemble of non-interacting harmonic oscillators in a nonlinear dissipative environment described by the Nosé -Hoover model. Using numerical simulation we find the histogram for total energy, which agrees with the analysis of the Nosé -Hoover equations effected with the method of averaging. The histogram does not correspond to Gibbs' canonical distribution. We have found oscillations at frequency proportional to α/m, α the dissipative parameter of thermostat and m the characteristic mass of particle, about the stationary state corresponding to equilibrium. The oscillations could have an important bearing upon the analysis of simulating molecular dynamics in the Nosé -Hoover thermostat.

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We study the interplay of the tautomeric transitions amino/keto → imino/enol of DNA base pairs and the elastic properties of the DNA, by employing the numerical simulation of the nonlinear and nonlocal Schrödinger equation that describes the concerted tunneling of protons in the hydrogen bonds of base pairs. We show that the dynamics of tunneling is characterized by solitary waves for the tunneling amplitudes. The velocity of solitons is generally small, of the order 10-3–10-2 cm/sec . We also found a phenomenon similar to the freak wave of nonlinear theory; in the context of DNA, it means that the conformation of the base pairs and the proton states, for which a tautomeric transition is only of low probability, could move along the DNA molecule and focus on a smaller set of base pairs so that the rate of transition increases. This result may have a bearing on the phenomenon of spontaneous mutations. We suggest that the irradiation of DNA with electromagnetic waves at frequencies in the infrared region, corresponding to the proton tunneling, could cause mutations.