arXiv subjects

Carol Griswold Hoover

Publications and source records attributed to Carol Griswold Hoover.

31 records · Page 2Linked to original sources

arXiv2017

Instantaneous Pairing of Lyapunov Exponents in Chaotic Hamiltonian Dynamics and the 2017 Ian Snook Prize

The time-averaged Lyapunov exponents support a mechanistic description of the chaos generated in and by nonlinear dynamical systems. The exponents are ordered from largest to smallest with the largest one describing the exponential growth rate of the (small) distance between two neighboring phase-space trajectories. Two exponents describe the rate for areas defined by three nearby trajectories. The sum of the first three exponents is the rate for volumes defined by four nearby trajectories, and so on. Lyapunov exponents for Hamiltonian systems are symmetric. The time-reversibility of the motion equations links the growth and decay rates together in pairs. This pairing provides a more detailed explanation than Liouville's for the conservation of phase volume in Hamiltonian mechanics. Although correct for long-time averages, the dependence of trajectories on their past is responsible for the observed lack of detailed pairing for the instantaneous "local" exponents, $\{\ λ_i(t) \ \}$ . The 2017 Ian Snook Prizes will be awarded to the author(s) of an accessible and pedagogical discussion of local Lyapunov instability in small systems. We desire that this discussion build on the two nonlinear models described here, a double pendulum with Hooke's-Law links and a periodic chain of Hooke's-Law particles tethered to their lattice sites. The latter system is the $ϕ^4$ model popularized by Aoki and Kusnezov. A four-particle version is small enough for comprehensive numerical work and large enough to illustrate ideas of general validity.

cond-mat.stat-mech↗

arXiv2017

Singly-Thermostated Ergodicity in Gibbs' Canonical Ensemble and the 2016 Ian Snook Prize Award

The 2016 Snook Prize has been awarded to Diego Tapias, Alessandro Bravetti, and David Sanders for their paper -- Ergodicity of One-Dimensional Systems Coupled to the Logistic Thermostat. They introduced a relatively stiff hyperbolic tangent thermostat force and successfully tested its ability to reproduce Gibbs' canonical distribution for the harmonic oscillator, the quartic oscillator, and the Mexican Hat potentials. Their work constitutes an effective response to the 2016 Ian Snook Prize Award goal -- Finding ergodic algorithms for Gibbs' canonical ensemble using a single thermostat variable. We confirm their work here and highlight an interesting feature of the Mexican Hat problem when it is solved with an adaptive integrator.

nlin.CD↗

arXiv2016

A Tutorial: Adaptive Runge-Kutta Integration for Stiff Systems : Comparing the Nosé and Nosé-Hoover Oscillator Dynamics

"Stiff" differential equations are commonplace in engineering and dynamical systems. To solve them we need flexible integrators that can deal with rapidly-changing righthand sides. This tutorial describes the application of "adaptive" [ variable timestep ] integrators to "stiff" mechanical problems encountered in modern applications of Gibbs' 1902 statistical mechanics. Linear harmonic oscillators subject to nonlinear thermal constraints can exhibit either stiff or smooth dynamics. Two closely-related examples, Nosé's 1984 dynamics and Nosé-Hoover 1985 dynamics, are both based on Hamiltonian mechanics, as was ultimately clarified by Dettmann and Morriss in 1996. Both these dynamics are consistent with Gibbs' canonical ensemble. Nosé's dynamics is "stiff" and can present severe numerical difficulties. Nosé-Hoover dynamics, though it follows exactly the same trajectory, is "smooth" and relatively trouble-free. Our tutorial emphasises the power of adaptive integrators to resolve stiff problems like the Nosé oscillator. The solutions obtained illustrate the power of computer graphics to enrich numerical solutions. Adaptive integration with computer graphics are basic to an understanding of dynamical systems and statistical mechanics. These tools lead naturally into the visualization of intricate fractal structures formed by chaos as well as elaborate knots tied by regular nonchaotic dynamics. This work was invited by the American Journal of Physics.

cond-mat.stat-mech↗

arXiv2016

Nonequilibrium Systems : Hard Disks and Harmonic Oscillators Near and Far From Equilibrium

We relate progress in statistical mechanics, both at and far from equilibrium, to advances in the theory of dynamical systems. We consider computer simulations of time-reversible deterministic chaos in small systems with three- and four-dimensional phase spaces. These models provide us with a basis for understanding equilibration and thermodynamic irreversibility in terms of Lyapunov instability, fractal distributions, and thermal constraints

cond-mat.stat-mech↗

arXiv2016

Yokohama to Ruby Valley : Around the World in 80 Years. II

We two had year-long research leaves in Japan, working together fulltime with several Japanese plus Tony De Groot back in Livermore and Harald Posch in Vienna. We summarize a few of the high spots from that very productive year ( 1989-1990 ), followed by an additional fifteen years' work in Livermore, with extensive travel. Next came our retirement in Nevada in 2005, which has turned out to be a long-term working vacation. Carol narrates this part of our history together.

physics.hist-ph↗

arXiv2016

Singly-Thermostated Ergodicity in Gibbs' Canonical Ensemble and the 2016 Ian Snook Prize

For a harmonic oscillator, Nosé's single-thermostat approach to simulating Gibbs' canonical ensemble with dynamics samples only a small fraction of the phase space. Nosé's approach has been improved in a series of three steps: [ 1 ] several two-thermostat sets of motion equations have been found which cover the complete phase space in an ergodic fashion, [ 2 ] sets of single-thermostat motion equations, exerting "weak control" over both forces and momenta, have been shown to be ergodic, and [ 3 ] sets of single-thermostat motion equations exerting weak control over two velocity moments provide ergodic phase-space sampling for the oscillator and for the rigid pendulum, but not for the quartic oscillator or for the Mexican Hat potential. The missing fourth step, motion equations providing ergodic sampling for anharmonic potentials requires a further advance. The 2016 Ian Snook Prize will be awarded to the author(s) of the most interesting original submission addressing the problem of finding ergodic algorithms for Gibbs' canonical ensemble using a single thermostat.

cond-mat.stat-mech↗

arXiv2015

The Equivalence of Dissipation from Gibbs' Entropy Production with Phase-Volume Loss in Ergodic Heat-Conducting Oscillators

Gibbs' thermodynamic entropy is given by the logarithm of the phase volume, which itself responds to heat transfer to and from thermal reservoirs. We compare the thermodynamic dissipation described by phase-volume loss with heat-transfer entropy production. Their equivalence is documented for computer simulations of the response of an ergodic harmonic oscillator to thermostated temperature gradients. In the simulations one or two thermostat variables control the kinetic energy or the kinetic energy and its fluctuation. All of the motion equations are time-reversible. We consider both strong and weak control variables. In every case the time-averaged dissipative loss of phase-space volume coincides with the entropy produced by heat transfer. Linear response theory nicely reproduces the small-gradient results obtained by computer simulation. The thermostats considered here are ergodic and provide simple dynamical models, some of them with as few as three ordinary differential equations, while remaining capable of reproducing Gibbs' canonical phase-space distribution and precisely consistent with irreversible thermodynamics.

cond-mat.stat-mech↗

arXiv2015

Ergodicity of a Singly-Thermostated Harmonic Oscillator

Although Nose's thermostated mechanics is formally consistent with Gibbs' canonical ensemble, the thermostated Nose-Hoover ( harmonic ) oscillator, with its mean kinetic temperature controlled, is far from ergodic. Much of its phase space is occupied by regular conservative tori. Oscillator ergodicity has previously been achieved by controlling two oscillator moments with two thermostat variables. Here we use computerized searches in conjunction with visualization to find singly-thermostated motion equations for the oscillator which are consistent with Gibbs' canonical distribution. These models are the simplest able to bridge the gap between Gibbs' statistical ensembles and Newtonian single-particle dynamics.

cond-mat.stat-mech↗

arXiv2015

Time-Reversible Ergodic Maps and the 2015 Ian Snook Prize

The time reversibility characteristic of Hamiltonian mechanics has long been extended to nonHamiltonian dynamical systems modeling nonequilibrium steady states with feedback-based thermostats and ergostats. Typical solutions are multifractal attractor-repellor phase-space pairs with reversed momenta and unchanged coordinates, $(q,p)\longleftrightarrow (q,-p)$ . Weak control of the temperature, $\propto p^2$ and its fluctuation, resulting in ergodicity, has recently been achieved in a three-dimensional time-reversible model of a heat-conducting harmonic oscillator. Two-dimensional cross sections of such nonequilibrium flows can be generated with time-reversible dissipative maps yielding æsthetically interesting attractor-repellor pairs. We challenge the reader to find and explore such time-reversible dissipative maps. This challenge is the 2015 Snook-Prize Problem.

cond-mat.stat-mech↗

arXiv2015

Comparison of Very Smooth Cell-Model Trajectories Using Five Symplectic and Two Runge-Kutta Integrators

Time-reversible symplectic methods, which are precisely compatible with Liouville's phase-volume-conservation theorem, are often recommended for computational simulations of Hamiltonian mechanics. Lack of energy drift is an apparent advantage of such methods. But all numerical methods are susceptible to Lyapunov instability, which severely limits the maximum time for which chaotic solutions can be "accurate". The "advantages" of higher-order methods are lost rapidly for typical chaotic Hamiltonians. We illustrate these difficulties for a useful reproducible test case, the two-dimensional one-particle cell model with specially smooth forces. This Hamiltonian problem is chaotic and occurs on a three-dimensional constant-energy shell, the minimum dimension for chaos. We benchmark the problem with quadruple-precision trajectories using the fourth-order Candy-Rozmus, fifth-order Runge-Kutta, and eighth-order Schlier-Seiter-Teloy integrators. We compare the last, most-accurate particle trajectories to those from six double-precision algorithms, four symplectic and two Runge-Kutta.

nlin.CD↗

arXiv2015

Ergodicity of the Martyna-Klein-Tuckerman Thermostat and the 2014 Snook Prize

Nosé and Hoover's 1984 work showed that although Nosé and Nosé-Hoover dynamics were both consistent with Gibbs' canonical distribution neither dynamics, when applied to the harmonic oscillator, provided Gibbs' Gaussian distribution. Further investigations indicated that two independent thermostat variables are necessary, and often sufficient, to generate Gibbs' canonical distribution for an oscillator. Three successful time-reversible and deterministic sets of two-thermostat motion equations were developed in the 1990s. We analyze one of them here. It was developed by Martyna, Klein, and Tuckerman in 1992. Its ergodicity was called into question by Patra and Bhattacharya in 2014. This question became the subject of the 2014 Snook Prize. Here we summarize the previous work on this problem and elucidate new details of the chaotic dynamics in the neighborhood of the two fixed points. We apply six separate tests for ergodicity and conclude that the MKT equations are fully compatible with all of them, in consonance with our recent work with Clint Sprott and Puneet Patra.

cond-mat.stat-mech↗

arXiv2014

Ergodicity of a Time-Reversibly Thermostated Harmonic Oscillator and the 2014 Ian Snook Prize

Shuichi Nosé opened up a new world of atomistic simulation in 1984. He formulated a Hamiltonian tailored to generate Gibbs' canonical distribution dynamically. This clever idea bridged the gap between microcanonical molecular dynamics and canonical statistical mechanics. Until then the canonical distribution was explored with Monte Carlo sampling. Nosé's dynamical Hamiltonian bridge requires the "ergodic" support of a space-filling structure in order to reproduce the entire distribution. For sufficiently small systems, such as the harmonic oscillator, Nosé's dynamical approach failed to agree with Gibbs' sampling and instead showed a complex structure, partitioned into a chaotic sea, islands, and chains of islands, that is familiar textbook fare from investigations of Hamiltonian chaos. In trying to enhance small-system ergodicity several more complicated "thermostated" equations of motion were developed. All were consistent with the canonical Gaussian distribution for the oscillator coordinate and momentum. The ergodicity of the various approaches has undergone several investigations, with somewhat inconclusive ( contradictory ) results. Here we illustrate several ways to test ergodicity and challenge the reader to find even more convincing algorithms or an entirely new approach to this problem.

nlin.CD↗

arXiv2014

Heat Conduction, and the Lack Thereof, in Time-Reversible Dynamical Systems: Generalized Nosé-Hoover Oscillators with a Temperature Gradient

We use nonequilibrium molecular dynamics to analyze and illustrate the qualitative differences between the one-thermostat and two-thermostat versions of equilibrium and nonequilibrium (heat-conducting) harmonic oscillators. Conservative nonconducting regions can coexist with dissipative heat conducting regions in phase space with exactly the same imposed temperature field.

cond-mat.stat-mech↗