Blog Archives

Mixing Rates of Ergodic Algorithms

Sprott Julien C.

In response to the 2024 Snook Prize Problem, this paper compares the mixing rates of six simple numerical algorithms that produce an ergodic Gaussian distribution of position and momentum for a one-dimensional harmonic oscillator. A hundred thousand initial conditions spread uniformly over the co ...

2024 Snook Prize Problem: Ergodic Algorithms’ Mixing Rates

Hoover William G., Hoover Carol G.

In 1984 Shuichi Nosé invented an isothermal mechanics designed to generate Gibbs’ canonical distribution for the coordinates {q} and momenta {p} of classical N-body systems [1, 2]. His approach introduced an additional timescaling variable s that could speed up or slow down the {q, p} motion i ...

Ergodicity of One-dimensional Oscillators with a Signum Thermostat

Sprott Julien C.

Gibbs’canonicalensembledescribestheexponentialequilibriumdistributionf(q,p,T)∝e−H(q,p)/kT foran ergodic Hamiltonian system interacting with a ‘heat bath’ at temperature T . The simplest deterministic heat ba ...

Ergodic Isoenergetic Molecular Dynamics for Microcanonical-Ensemble Averages

Hoover Wm.G. , Hoover C.G.

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 ...

Ergodicity of One-dimensional Systems Coupled to the Logistic Thermostat

Tapias Diego, Bravetti Alessandro, Sanders David

We analyze the ergodicity of three one-dimensional Hamiltonian systems, with harmonic, quartic and Mexican-hat potentials, coupled to the logistic thermostat. As criteria for ergodicity we employ: the independence of the Lyapunov spectrum with respect to initial conditions; the absence of visual ...

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

Hoover Wm.G. , Hoover C.G.

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 ...

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

Hoover Wm.G. , Hoover C.G.

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

Hoover Wm.G. , Hoover C.G.

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 ...

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

Hoover Wm.G. , Hoover C.G.

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 t ...

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

Hoover Wm.G. , Hoover C.G.

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 ...