Non-ergodicity Of The Nosé-hoover Harmonic Oscillator Coupled To The Logistic Thermostat

Nosé-Hoover dynamical system is an important model in thermodynamics.When the reservoir quality is appropriate,the model has better ergodicity and can accurately describe Gibbs phenomenon in thermodynamics.With the increase of reservoir quality,the Nosé-Hoover system shows non-ergodicity and the model deviates from the hypothesis.Therefore,the study of ergodicity of the Nosé-Hoover system is of great significance in thermodynamics.Based on the thermodynamics,this paper studied the dynamics of the contact Hamiltonian system around the coupled Logistic thermostat Nosé-Hoover harmonic oscillator model.The main content of this paper includes two parts.In the first part,we establish the relationship between the Nosé-Hoover system and the contact Hamiltonian system.With ignoring the time scale,the Nosé-Hoover system and the contact Hamiltonian system are equivalent.In the second part,by using Moser's small twist theorem,it is proved that there are a lot of invariant curves in the Nosé-Hoover harmonic oscillator coupled to the logistic thermostat system under special potential and general potential.These invariant curves divide the phase space into multiple invariant regions,so as to prove that the system is non-ergodic.Our numerical results show that: when the reservoir quality in the model is small,the numerical study shows that the system has good ergodicity;when the reservoir quality is large,the Nosé-Hoover system is no longer ergodic.The results of this study show that when the reservoir quality is large,the new improved Nosé-Hoover model by Bravetti and Tapias is also not ergodic,which is of great guiding significance for those engaged in thermodynamics research.