Non-Hamiltonian equations of motion with a conserved energy

In 1980 Andersen introduced the use of "extended system" as a means of exploring by molecular dynamics simulation the phase space of a physical model according to a desired ensemble distribution different from the standard microcanonical function. Following his original work on constant pressure-constant enthalpy a large number of different equations of motion, not directly derivable from a Hamiltonian, have been proposed in recent years, the most notable of which is the so-called Nosé-Hoover formulation for "canonical" molecular dynamics simulation. Using a generalization of the symplectic form of the Hamilton equations of motion we show here that there is a unique general structure that underlies most, if not all the equations of motion for "extended systems." We establish a unifying formalism that allows one to identify and separately control the conserved quantity, usually known as the "total energy" of the system, and the phase-space compressibility. Moreover, we define a standard procedure to construct conservative non-Hamiltonian flows that sample the phase space according to a chosen distribution function [Tuckerman et al., Europhys. Lett. 45, 149 (1999)]. To illustrate the formalism we derive new equations of motion for two example cases. First we modify the equations of motion of the Nosé-Hoover thermostat applied to a one-dimensional harmonic oscillator, and we show how to overcome the ergodicity problem and obtain a canonical sampling of phase space without making recourse to additional degrees of freedom. Finally we recast an idea recently put forward by Marchi and Ballone [J. Chem. Phys. 110, 3697 (1999)] and derive a dynamical scheme for sampling phase space with arbitrary statistical biases, showing as an explicit application a demixing transition in a simple Lennard-Jones binary mixture.