December 11, 2006
One-variable thermostats are studied as a generalization of the Nos\'e--Hoover method which is aimed to achieve Gibbs' canonical distribution with conserving the time-reversibility. A condition for equations of motion for the system with the thermostats is derived in the form of a partial differential equation. Solutions of this equation construct a family of thermostats including the Nos\'e--Hoover method as the minimal solution. It is shown that the one-variable thermostat coupled with the one-dimensional harmonic oscillator loses its ergodicity with large enough relaxation time. The present result suggests that multi-variable thermostats are required to assure the ergodicity and to work as heatbath.
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July 28, 2006
Ergodicity of the systems with Nos\'e-Hoover thermostat are studied. The dynamics of the heatbath variables are investigated and they can be periodic when the system has quick oscillation. The periodic behaviour of them causes the system to lose its ergodicity. The kinetic-moments method is also studied, and the heatbath variables in this method are found to be chaotic. The chaotic behaviour makes the whole system ergodic.
August 18, 2020
A simple proof and detailed analysis on the non-ergodicity for multidimensional harmonic oscillator systems with Nose-Hoover type thermostat are given. The origin of the nonergodicity is symmetries in the multidimensional target physical system, and is differ from that in the Nose-Hoover thermostat with the 1-dimensional harmonic oscillator. A new simple deterministic method to recover the ergodicity is also presented. An individual thermostat variable is attached to each deg...
September 25, 2019
Thermostats are dynamic equations used to model thermodynamic variables in molecular dynamics. The applicability of thermostats is based on the ergodic hypothesis. The most commonly used thermostats are designed according to the Nos\'e-Hoover scheme, although it is known that it often violates ergodicity. Here, following a method from our recent study \citep{SamoletovVasiev2017}, we have extended the classic Nos\'e-Hoover scheme with an additional temperature control tool. Ho...
December 17, 2008
The numerical integration of the Nose-Hoover dynamics gives a deterministic method that is used to sample the canonical Gibbs measure. The Nose-Hoover dynamics extends the physical Hamiltonian dynamics by the addition of a "thermostat" variable, that is coupled nonlinearly with the physical variables. The accuracy of the method depends on the dynamics being ergodic. Numerical experiments have been published earlier that are consistent with non-ergodicity of the dynamics for s...
November 7, 2005
The Nose-Hoover thermostat is a deterministic dynamical system designed for computing phase space integrals for the canonical Gibbs distribution. Newton's equations are modified by coupling an additional reservoir variable to the physical variables. The correct sampling of the phase space according to the Gibbs measure is dependent on the Nose-Hoover dynamics being ergodic. Hoover presented numerical experiments that show the Nose-Hoover dynamics to be non-ergodic when applie...
January 8, 2014
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.
April 28, 2015
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 sin...
January 29, 2004
We study the dynamics of an ensemble of non-interacting harmonic oscillators in a nonlinear dissipative environment described by the Nos\'e - Hoover model. Using numerical simulation we find the histogram for total energy, which agrees with the analysis of the Nos\'e - Hoover equations effected with the method of averaging. The histogram does not correspond to Gibbs' canonical distribution. We have found oscillations at frequency proportional to $\sqrt{\alpha/m}$, $\alpha$ th...
February 3, 2010
In this paper we formulate Bulgac-Kusnezov constant temperature dynamics in phase space by means of non-Hamiltonian brackets. Two generalized versions of the dynamics are similarly defined: one where the Bulgac-Kusnezov demons are globally controlled by means of a single additional Nos\'e variable, and another where each demon is coupled to an independent Nos\'e-Hoover thermostat. Numerically stable and efficient measure-preserving time-reversible algorithms are derived in a ...
January 16, 2015
The relative stability and ergodicity of deterministic time-reversible thermostats, both singly and in coupled pairs, are assessed through their Lyapunov spectra. Five types of thermostat are coupled to one another through a single Hooke's-Law harmonic spring. The resulting dynamics shows that three specific thermostat types, Hoover-Holian, Ju-Bulgac, and Martyna-Klein-Tuckerman, have very similar Lyapunov spectra in their equilibrium four-dimensional phase spaces and when co...