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General formalism for singly thermostated Hamiltonian dynamics.

John D Ramshaw¹

¹Department of Physics, Portland State University, Portland, Oregon 97207, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|December 15, 2015

Summary

A new framework unifies modified Hamiltonian systems that maintain equilibrium distributions using thermostat variables. This formalism enables calculating ensemble averages from single trajectories for diverse systems.

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Area of Science:

Statistical mechanics
Theoretical physics
Dynamical systems theory

Background:

Canonical equilibrium distributions are fundamental in statistical mechanics.
Ergodic Hamiltonian systems allow ensemble averages via time averages.
Recent discoveries by Hoover and colleagues introduced novel systems with these properties.

Purpose of the Study:

Develop a general formalism for constructing modified Hamiltonian dynamical systems.
Ensure these systems preserve a canonical equilibrium distribution.
Facilitate the discovery, construction, and classification of such systems.

Main Methods:

Introduce a time evolution equation for a single thermostat variable.
Incorporate canonical and generalized Hamiltonian systems.
Consider state spaces of arbitrary dimensionality (even or odd).

Main Results:

A unified framework is established for a wide class of modified Hamiltonian systems.
Canonical ensemble averages can be computed as dynamical time averages for ergodic systems.
The formalism encompasses both few- and many-particle systems.

Conclusions:

The developed formalism provides a unified approach to modified Hamiltonian systems.
It aids in understanding and discovering systems that preserve canonical distributions.
The framework is applicable to diverse systems, including those with odd-dimensional state spaces.