Non-integrability of the mixmaster universe

We comment on an analysis by Contopoulos et al. which demonstrates that the governing six-dimensional Einstein equations for the mixmaster space-time metric pass the ARS or reduced Painlev\'{e} test. We note that this is the case irrespective of the value, $I$, of the generating Hamiltonian which is a constant of motion. For $I < 0$ we find numerous closed orbits with two unstable eigenvalues strongly indicating that there cannot exist two additional first integrals apart from the Hamiltonian and thus that the system, at least for this case, is very likely not integrable. In addition, we present numerical evidence that the average Lyapunov exponent nevertheless vanishes. The model is thus a very interesting example of a Hamiltonian dynamical system, which is likely non-integrable yet passes the reduced Painlev\'{e} test.Comment: 11 pages LaTeX in J.Phys.A style (ioplppt.sty) + 6 PostScript figures compressed and uuencoded with uufiles. Revised version to appear in J Phys.

Microscopic expressions for the thermodynamic temperature

We show that arbitrary phase space vector fields can be used to generate phase functions whose ensemble averages give the thermodynamic temperature. We describe conditions for the validity of these functions in periodic boundary systems and the Molecular Dynamics (MD) ensemble, and test them with a short-ranged potential MD simulation.Comment: 21 pages, 2 figures, Revtex. Submitted to Phys. Rev.

A Geometric, Dynamical Approach to Thermodynamics

We present a geometric and dynamical approach to the micro-canonical ensemble of classical Hamiltonian systems. We generalize the arguments in \cite{Rugh} and show that the energy-derivative of a micro-canonical average is itself micro-canonically observable. In particular, temperature, specific heat and higher order derivatives of the entropy can be observed dynamically. We give perturbative, asymptotic formulas by which the canonical ensemble itself can be reconstructed from micro-canonical measurements only. In a purely micro-canonical approach we rederive formulas by Lebowitz et al \cite{LPV}, relating e.g. specific heat to fluctuations in the kinetic energy. We show that under natural assumptions on the fluctuations in the kinetic energy the micro-canonical temperature is asymptotically equivalent to the standard canonical definition using the kinetic energy.Comment: 7 pages, LaTeX, uses RevTex. New sections and examples using fluctuations in the kinetic energy adde

Fractal Scales in a Schwarzschild Atmosphere

Recently, Glass and Krisch have extended the Vaidya radiating metric to include both a radiation fluid and a string fluid [1999 Class. Quantum Grav. vol 16, 1175]. Mass diffusion in the extended Schwarzschild atmosphere was studied. The continuous solutions of classical diffusive transport are believed to describe the envelope of underlying fractal behavior. In this work we examine the classical picture at scales on which fractal behavior might be evident.Comment: to appear in Class. Quantum Gra

Controlling complex networks: How much energy is needed?

The outstanding problem of controlling complex networks is relevant to many areas of science and engineering, and has the potential to generate technological breakthroughs as well. We address the physically important issue of the energy required for achieving control by deriving and validating scaling laws for the lower and upper energy bounds. These bounds represent a reasonable estimate of the energy cost associated with control, and provide a step forward from the current research on controllability toward ultimate control of complex networked dynamical systems.Comment: 4 pages paper + 5 pages supplement. accepted for publication in Physical Review Letters; http://link.aps.org/doi/10.1103/PhysRevLett.108.21870

Efficient estimation of energy transfer efficiency in light-harvesting complexes

The fundamental physical mechanisms of energy transfer in photosynthetic complexes is not yet fully understood. In particular, the degree of efficiency or sensitivity of these systems for energy transfer is not known given their non-perturbative and non-Markovian interactions with proteins backbone and surrounding photonic and phononic environments. One major problem in studying light-harvesting complexes has been the lack of an efficient method for simulation of their dynamics in biological environments. To this end, here we revisit the second-order time-convolution (TC2) master equation and examine its reliability beyond extreme Markovian and perturbative limits. In particular, we present a derivation of TC2 without making the usual weak system-bath coupling assumption. Using this equation, we explore the long time behaviour of exciton dynamics of Fenna-Matthews-Olson (FMO) protein complex. Moreover, we introduce a constructive error analysis to estimate the accuracy of TC2 equation in calculating energy transfer efficiency, exhibiting reliable performance for environments with weak and intermediate memory and strength. Furthermore, we numerically show that energy transfer efficiency is optimal and robust for the FMO protein complex of green sulphur bacteria with respect to variations in reorganization energy and bath correlation time-scales.Comment: 16 pages, 9 figures, modified version, updated appendices and reference lis

Accelerating cycle expansions by dynamical conjugacy

Periodic orbit theory provides two important functions---the dynamical zeta function and the spectral determinant for the calculation of dynamical averages in a nonlinear system. Their cycle expansions converge rapidly when the system is uniformly hyperbolic but greatly slowed down in the presence of non-hyperbolicity. We find that the slow convergence can be associated with singularities in the natural measure. A properly designed coordinate transformation may remove these singularities and results in a dynamically conjugate system where fast convergence is restored. The technique is successfully demonstrated on several examples of one-dimensional maps and some remaining challenges are discussed

Hopf's last hope: spatiotemporal chaos in terms of unstable recurrent patterns

Spatiotemporally chaotic dynamics of a Kuramoto-Sivashinsky system is described by means of an infinite hierarchy of its unstable spatiotemporally periodic solutions. An intrinsic parametrization of the corresponding invariant set serves as accurate guide to the high-dimensional dynamics, and the periodic orbit theory yields several global averages characterizing the chaotic dynamics.Comment: Latex, ioplppt.sty and iopl10.sty, 18 pages, 11 PS-figures, compressed and encoded with uufiles, 170 k

Continued-fraction expansion of eigenvalues of generalized evolution operators in terms of periodic orbits

A new expansion scheme to evaluate the eigenvalues of the generalized evolution operator (Frobenius-Perron operator) $H_{q}$ relevant to the fluctuation spectrum and poles of the order-$q$ power spectrum is proposed. The ``partition function'' is computed in terms of unstable periodic orbits and then used in a finite pole approximation of the continued fraction expansion for the evolution operator. A solvable example is presented and the approximate and exact results are compared; good agreement is found.Comment: CYCLER Paper 93mar00

Microcanonical temperature for a classical field: application to Bose-Einstein condensation

We show that the projected Gross-Pitaevskii equation (PGPE) can be mapped exactly onto Hamilton's equations of motion for classical position and momentum variables. Making use of this mapping, we adapt techniques developed in statistical mechanics to calculate the temperature and chemical potential of a classical Bose field in the microcanonical ensemble. We apply the method to simulations of the PGPE, which can be used to represent the highly occupied modes of Bose condensed gases at finite temperature. The method is rigorous, valid beyond the realms of perturbation theory, and agrees with an earlier method of temperature measurement for the same system. Using this method we show that the critical temperature for condensation in a homogeneous Bose gas on a lattice with a UV cutoff increases with the interaction strength. We discuss how to determine the temperature shift for the Bose gas in the continuum limit using this type of calculation, and obtain a result in agreement with more sophisticated Monte Carlo simulations. We also consider the behaviour of the specific heat.Comment: v1: 9 pages, 5 figures, revtex 4. v2: additional text in response to referee's comments, now 11 pages, to appear in Phys. Rev.