DNA viewed as an out-of-equilibrium structure

The complexity of the primary structure of human DNA is explored using methods from nonequilibrium statistical mechanics, dynamical systems theory and information theory. The use of chi-square tests shows that DNA cannot be described as a low order Markov chain of order up to $r=6$. Although detailed balance seems to hold at the level of purine-pyrimidine notation it fails when all four basepairs are considered, suggesting spatial asymmetry and irreversibility. Furthermore, the block entropy does not increase linearly with the block size, reflecting the long range nature of the correlations in the human genomic sequences. To probe locally the spatial structure of the chain we study the exit distances from a specific symbol, the distribution of recurrence distances and the Hurst exponent, all of which show power law tails and long range characteristics. These results suggest that human DNA can be viewed as a non-equilibrium structure maintained in its state through interactions with a constantly changing environment. Based solely on the exit distance distribution accounting for the nonequilibrium statistics and using the Monte Carlo rejection sampling method we construct a model DNA sequence. This method allows to keep all long range and short range statistical characteristics of the original sequence. The model sequence presents the same characteristic exponents as the natural DNA but fails to capture point-to-point details

Unitary Evolution on a Discrete Phase Space

We construct unitary evolution operators on a phase space with power of two discretization. These operators realize the metaplectic representation of the modular group SL(2,Z_{2^n}). It acts in a natural way on the coordinates of the non-commutative 2-torus, T_{2^n}^2$ and thus is relevant for non-commutative field theories as well as theories of quantum space-time. The class of operators may also be useful for the efficient realization of new quantum algorithms.Comment: 5 pages, contribution to Lattice 2005 (theoretical developments

Nonlinear dynamic systems in the geosciences

Geophysical phenomena are often characterized by complex, random-looking deviations of the relevant variables from their average values. Typical examples of such aperiodicity are the intermittent succession of Quaternary glaciations as revealed by the oxygen isotope record of deep-sea cores of the last 106 years or the pronounced spatial disorder characterizing geologic materials. A major task of the geoscientist is to reconstitute from this type of record the principal mechanisms responsible for the observed behavior. Traditional approaches attribute the complexity encountered in the record of a natural variable to external uncontrollable factors and to poorly known parameters whose presence tends to blur fundamental underlying regularities. Here, we consider that complexity might be an intrinsic property generated by the nonlinear character of the system's dynamics. We review bifurcations, chaos, and fractals, three important mechanisms leading to complex behavior in nonlinear dynamic systems, and stress the role of the theory of nonlinear dynamic systems as a major tool of interdisciplinary research in the geosciences. The general ideas are illustrated on the dynamics of Quaternary glaciations and the dynamics of tracer transport in a sediment

Nonlinear dynamic systems in the geosciences

Geophysical phenomena are often characterized by complex, random-looking deviations of the relevant variables from their average values. Typical examples of such aperiodicity are the intermittent succession of Quaternary glaciations as revealed by the oxygen isotope record of deep-sea cores of the last 106 years or the pronounced spatial disorder characterizing geologic materials. A major task of the geoscientist is to reconstitute from this type of record the principal mechanisms responsible for the observed behavior. Traditional approaches attribute the complexity encountered in the record of a natural variable to external uncontrollable factors and to poorly known parameters whose presence tends to blur fundamental underlying regularities. Here, we consider that complexity might be an intrinsic property generated by the nonlinear character of the system's dynamics. We review bifurcations, chaos, and fractals, three important mechanisms leading to complex behavior in nonlinear dynamic systems, and stress the role of the theory of nonlinear dynamic systems as a major tool of interdisciplinary research in the geosciences. The general ideas are illustrated on the dynamics of Quaternary glaciations and the dynamics of tracer transport in a sediment

Thermostating by Deterministic Scattering: Heat and Shear Flow

We apply a recently proposed novel thermostating mechanism to an interacting many-particle system where the bulk particles are moving according to Hamiltonian dynamics. At the boundaries the system is thermalized by deterministic and time-reversible scattering. We show how this scattering mechanism can be related to stochastic boundary conditions. We subsequently simulate nonequilibrium steady states associated to thermal conduction and shear flow for a hard disk fluid. The bulk behavior of the model is studied by comparing the transport coefficients obtained from computer simulations to theoretical results. Furthermore, thermodynamic entropy production and exponential phase-space contraction rates in the stationary nonequilibrium states are calculated showing that in general these quantities do not agree.Comment: 16 pages (revtex) with 9 figures (postscript

Thermostating by deterministic scattering: the periodic Lorentz gas

We present a novel mechanism for thermalizing a system of particles in equilibrium and nonequilibrium situations, based on specifically modeling energy transfer at the boundaries via a microscopic collision process. We apply our method to the periodic Lorentz gas, where a point particle moves diffusively through an ensemble of hard disks arranged on a triangular lattice. First, collision rules are defined for this system in thermal equilibrium. They determine the velocity of the moving particle such that the system is deterministic, time reversible, and microcanonical. These collision rules can systematically be adapted to the case where one associates arbitrarily many degrees of freedom to the disk, which here acts as a boundary. Subsequently, the system is investigated in nonequilibrium situations by applying an external field. We show that in the limit where the disk is endowed by infinitely many degrees of freedom it acts as a thermal reservoir yielding a well-defined nonequilibrium steady state. The characteristic properties of this state, as obtained from computer simulations, are finally compared to the ones of the so-called Gaussian thermostated driven Lorentz gas.Comment: 13 pages (revtex) with 10 figures (encapsulated postscript

Thermostatting by deterministic scattering

We present a mechanism for thermalizing a moving particle by microscopic deterministic scattering. As an example, we consider the periodic Lorentz gas. We modify the collision rules by including energy transfer between particle and scatterer such that the scatterer mimics a thermal reservoir with arbitrarily many degrees of freedom. The complete system is deterministic, time-reversible, and provides a microcanonical density in equilibrium. In the limit of the disk representing infinitely many degrees of freedom and by applying an electric field the system goes into a nonequilibrium steady state.Comment: 4 pages (revtex) with 4 figures (postscript