content_image/362229.jpg content_image/362230.jpg content_image/362231.jpg content_image/362232.jpg
## II. NON-HAMILTONIAN BRACKETS AND MEASURE-PRESERVING ALGO- RITHMS 

Consider an arbitrary system admitting a time-independent (extended) Hamiltonian ex- pressed in terms of the phase space coordinates x$_{i}$ , i = 1 , . . . , 2 N . In this case, the Hamil- tonian can be interpreted as the conserved energy of the system. 

Upon introducing an antisymmetric tensor field (generalized Poisson tensor [21, 33]) in phase space, B ( x ) = − B $^{T}$( x ), one can define non-Hamiltonian brackets [14–16] as 

<img src='content_image/362229.jpg'>

where a = a ( x ) and b = b ( x ) are two arbitrary phase space functions. The bracket defined in Eq. (1) is classified as non-Hamiltonian [14–16] since, in general, it does not obey the Jacobi relation, i.e., in general the Jacobiator J ̸ = 0, where [21] 

<img src='content_image/362230.jpg'>

with c = c ( x ) arbitrary phase space function (in addition to the functions a and b , previously introduced). If J ̸ = 0, the tensor B$_{ij}$ is said to define an ‘almost-Poisson’ structure [34]. (Such systems have also been called ‘pseudo-Hamiltonian’ [33].) 

An energy-conserving and in general non-Hamiltonian phase space flow is then defined by the vector field 

<img src='content_image/362231.jpg'>

where conservation of H ( x ) follows directly from the antisymmetry of B$_{ij}$ . 

It has previously been shown how equilibrium statistical mechanics can be comprehen- sively formulated within this framework [16]. It is also possible to recast the above formalism and the corresponding statistical mechanics in the language of differential forms [17, 18]. If the matrix B is invertible (this is true for all the cases considered here), with inverse Ω$_{ij}$, we can define the 2-form [35] 

<img src='content_image/362232.jpg'>

The dynamics of Eq. (3) is then Hamiltonian if and only if the form (4) is closed , i.e., has zero exterior derivative, d Ω = 0 [35]. This condition is independent of the particular system of coordinates used to describe the dynamics. 

