Principles of non-Hamiltonian statistical mechanics

 

In the previous section, the concept of an ensemble was introduced, and the specific example of the microcanonical ensemble, as the ensemble of systems having was discussed. In general, an ensemble is defined by its phase space distribution function , which may possibly depend explicitly on time. The phase space distribution function must satisfy the Liouville equation, which for systems governed by Hamiltonian dynamics is

 

where is the 2dN-dimensional gradient on the phase space. The Liouville equation results from the requirement that the rate of change of the number of ensemble members in an arbitrary phase space volume is equal to the flux of members through the boundary of the volume. It can be seen that the Liouville equation is a statement of the conservation of f. Since f is a probability distribution function, the existence of a conservation law for f implies the existence of a conserved phase space measure, For Hamiltonian systems the invariant measure is, . It will shortly be shown below why this is an invariant measure for Hamiltonian systems. In summary, given an ensemble distribution function satisfying Eq. (4.1), the average of any observable can be defined by

 

Non-Hamiltonian equations of motion are typically used to generate ensembles other than the microcanonical, for describing systems subject to non-holonomic constraints, or for describing driven systems. The idea of generating ensembles dynamically began with the work of Andersen [24], who showed that by extending the phase space beyond the 2dN dimensions of the physical system, a dynamical scheme could be constructed to generate an isobaric distribution of the physical subsystem. Isothermal extensions followed [25, 26]. These original formulations of extended phase space dynamics were based on Hamiltonian systems, which possess certain undersirable features related to the definition of time. It was shown that these could be corrected by going over to a non-Hamiltonian formulation [27]. However, only recently has a consistent theoretical statistical framework underlying the use of general non-Hamiltonian systems been presented [20]. The theoretical underpinnings will be discussed below.

Consider the dynamical system,

 

which is assumed to be non-Hamiltonian (i.e., expressible in the form of Eqs. (3.4)). Here, is a generalized force, which may have an explicit time dependence. If the dynamical system is not Hamiltonian, then its phase space compressibility, defined to be

 

which vanishes for a Hamiltonian system (the incompressibility property), will generally be nonzero and the phase space measure is no longer invariant. In order to see this, one need only consider the Jacobian of the transformation from an initial phase space vector a time-evolved vector given by

 

where n is the dimension of the phase space. It can be shown (see, e.g. Ref. [20]) that satisfies the following evolution equation:

 

with initial condition . Equation (4.6) implies that J will only be 1 for all time if as it is for Hamiltonian systems. For non-Hamiltonian systems, the measure transforms according to

 

which demonstrates that if , .

A complete statistical theory of non-Hamiltonian systems has recently been presented in  [20]. The basic tenets of the theory are:

1.

There is an invariant measure that takes the form

where the metric factor is given by

 

and the function is related to the compressibility by

2.

There is a non-Hamiltonian generalization of the Liouville equation for a general ensemble distribution function , which takes the form

 

3.

In the absence of external driving and explicitly time-dependent forces, a non-Hamiltonian microcanonical ensemble can be defined based on the existence of the invariant measure. If the dynamical system in Eq. (4.3) has a set of M conserved quantities , satisfying then the microcanonical distribution function is given by

and the corresponding partition function is

 

The microcanonical distribution, , together with the metric, , satisfies the time-independent form of Eq. (4.11). This is a necessary but not sufficient condition to guarantee that a given is the correct equilibrium distribution function. For example, a distribution function , where M';SPMlt;M (i.e., a distribution constructed from a subset of the conservation laws satisfied by a system) also satisfies the generalized Liouville equation. However, all conservation laws are required if the correct microcanonical distribution is to be constructed.