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Blog Archives
The φ4 Model, Chaos, Thermodynamics, and the 2018 SNOOK Prizes in Computational Statistical Mechanics
The one-dimensional φ4 Model generalizes a harmonic chain with nearest-neighbor Hooke’s-Law interactions by adding quartic potentials tethering each particle to its lattice site. In their studies of this model Kenichiro Aoki and Dimitri Kusnezov emphasized its most interesting feature: because ...
Ergodic Isoenergetic Molecular Dynamics for Microcanonical-Ensemble Averages
Considerable research has led to ergodic isothermal dynamics which can replicate Gibbs’ canonical distribution for simple (small) dynamical problems. Adding one or two thermostat forces to the Hamiltonian motion equations can give an ergodic isothermal dynamics to a harmonic oscillator, to a qu ...
On Local Lyapunov Exponents of Chaotic Hamiltonian Systems
Chaos in conservative systems, particularly in Hamiltonian systems, is different from chaos in dissipative systems. For example, not only the eigenvalues of the symmetric Jacobian, but also the global Lyapunov exponents of Hamiltonian systems occur in pairs (λ, −λ). In this article, we even s ...
Instantaneous Pairing of Lyapunov Exponents in Chaotic Hamiltonian Dynamics and the 2017 Ian Snook Prizes
The time-averaged Lyapunov exponents, {λi}, 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 desc ...
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 ...
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 Liver ...
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 ...
Time-Reversible Ergodic Maps and the 2015 Ian Snook Prizes
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 r ...
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 susce ...
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 canon ...