This note studies Hamiltonian systems which are thermostated using the Jellinek–Berry thermostat (J. Phys. Chem. 1988; Phys. Rev. A 1988). Jellinek & Jellinek and Berry propose an extension of Nosé’s thermostat (J. Chem. Phys. 1984). They introduce multiple functional parameters in order to achieve ergodicity of the thermostated dynamics. This family of Hamiltonian thermostats aims to simulate the canonical ensemble of a Hamiltonian H by coupling H to a 1-d heat reservoir with potential energy v(s) and kinetic energy <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3536_Article_IEq1.gif" Format="GIF" Height="40" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">$\dfrac{1}{2Q}(p_s/u(s))^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>Q</mi> </mrow> </mfrac> </mstyle> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>s</mi> </msub> <mo stretchy="false">/</mo> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. This note derives a normal form for the reservoir’s potential energy; investigates when the Jellinek–Berry thermostated system admits a Hoover reduction; and, demonstrates that a Jellinek–Berry thermostated periodic ideal gas is completely integrable and satisfies a KAM twist condition called Rüssmann non-degeneracy. This is used to deduce that a thermostated, collision-less, non-ideal gas (i.e. one with a smooth potential energy) at sufficiently high temperatures of the reservoir has a positive measure set of invariant tori–hence, the thermostated dynamics are non-ergodic.

Notes on the Jellinek-Berry Thermostated Ideal Gas

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2025年11月10日

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Leo T. Butler,
Alireza Sharifi

摘要

This note studies Hamiltonian systems which are thermostated using the Jellinek–Berry thermostat (J. Phys. Chem. 1988; Phys. Rev. A 1988). Jellinek & Jellinek and Berry propose an extension of Nosé’s thermostat (J. Chem. Phys. 1984). They introduce multiple functional parameters in order to achieve ergodicity of the thermostated dynamics. This family of Hamiltonian thermostats aims to simulate the canonical ensemble of a Hamiltonian H by coupling H to a 1-d heat reservoir with potential energy v(s) and kinetic energy $\dfrac{1}{2Q}(p_s/u(s))^2$ $\frac{1}{2 Q} {(p_{s} / u (s))}^{2}$ . This note derives a normal form for the reservoir’s potential energy; investigates when the Jellinek–Berry thermostated system admits a Hoover reduction; and, demonstrates that a Jellinek–Berry thermostated periodic ideal gas is completely integrable and satisfies a KAM twist condition called Rüssmann non-degeneracy. This is used to deduce that a thermostated, collision-less, non-ideal gas (i.e. one with a smooth potential energy) at sufficiently high temperatures of the reservoir has a positive measure set of invariant tori–hence, the thermostated dynamics are non-ergodic.