(Published in Europhysics Letters 45, 149 (1999))
A consistent classical statistical mechanical theory of non-Hamiltonian dynamical systems is presented. It is shown that compressible phase space flows generate coordinate transformations with a nonunit Jacobian, leading to a metric on the phase space manifold which is nontrivial. Thus, the phase space of a non-Hamiltonian system should be regarded as a general curved Riemannian manifold. An invariant measure on the phase space manifold is then derived. It is further shown that a proper generalization of the Liouville equation must incorporate the metric determinant, and a geometric derivation of such a continuity equation is presented. The manifestations of the nontrivial nature of the phase space geometry on thermodynamic quantities is explored.
PACS numbers: 05.20.-y, 02.40.-k, 31.15.Qg
Classical statistical physics is traditionally based on a Hamiltonian
formulation of mechanics.
The phase space distribution function f(x,t) describing
an ensemble of systems with coordinates and momenta 
evolving according to Hamilton's equations of motion
satisfies the Liouville equation,
which is a statement of the conservation of f, df/dt=0. Accordingly, the average of an observable A(x) in such an ensemble is given by
where n is the dimension of the phase space.
The measure 
in the definition of the average is preserved by
Hamiltonian dynamics. That is, given a subset of systems with initial
conditions in a phase space volume element 
, the trajectories
obtained by solving Hamilton's equations of motion for
each initial condition 
will describe a volume element 
in phase space such that 
.
This is tantamount to the
incompressibility of phase space flow, which is a property satisfied
by Hamiltonian systems. It is also tantamount to the condition that
the Jacobian of the coordinate transformation specified by
is unity.
The implication of an invariant measure is that Eq. (2)
can be computed with respect to the phase space variables x at
any time t.
Despite their many simple and well known properties, Hamiltonian equations are not always the most useful in certain common problems in classical theoretical physics. Non-Hamiltonian equations of motion are often employed to describe the evolution of ``open'' systems (i.e., systems in contact with reservoirs) [, , ], driven and stressed systems [], and constrained systems [, ]. Since real systems should always be regarded, in some sense, as ``open'', both at the microscopic and macroscopic levels, it may be that non-Hamiltonian evolution equations may provide a more realistic description of their behavior [].
Despite their importance and frequent use, a fully consistent formulation of the statistical mechanics of non-Hamiltonian systems has never been presented. Moreover, a confusion in the literature on the thermodynamics of non-Hamiltonian systems exists, in spite of recent advances providing an impetus for a new formal development in non-Hamiltonian based statistical theory[, , ]. One of the signatures of non-Hamiltonian flow is that it can have a nonzero phase space compressibility, a property that distinguishes it from the Hamiltonian case, which is always incompressible. Since Hamiltonian flow preserve the measure of phase space, it is usually assumed that this space can be treated mathematically as a Euclidean manifold.
In this letter, the classical statistical mechanics of non-Hamiltonian systems will be revisited, and the fundamental concepts needed to develop a full statistical theory will be laid out. It will be shown that the assumption of a Euclidean phase space manifold cannot be made for non-Hamiltonian systems. Thus, in developing statistical mechanical concepts that refer to the geometry of the underlying space for a non-Hamiltonian system, one must treat the phase space as a general Riemannian manifold. It will, nevertheless, be shown that key concepts such as ``invariant measure'' and ``continuity,'' which describe Hamiltonian systems can be generalized to the non-Hamiltonian case by a proper treatment of the geometry of the phase space and that an invariant measure on the phase space manifold can be derived Following this, an equation of continuity, a ``generalized Liouville equation'' for the distribution function of an ensemble of systems evolving according to non-Hamiltonian dynamics, will be derived. The resulting equation will be seen to depend explicitly on the phase space metric and thus can be shown to reduce to the ordinary Liouville equation in the Hamiltonian limit. Finally the manifestation of these generalized concepts in the thermodynamics of non-Hamiltonian systems will be explored.
Consider a non-Hamiltonian dynamical system
for the evolution of the n coordinates 
with
initial values 
. The n coordinates
describe a point P of an
n-dimensional Riemannian manifold 
with metric G.
If we denote by 
the set of all n-tuples of real numbers,
then x belongs to this set.
The solutions 
of Eqs. (3) provide a coordinate transformation
on from the initial coordinates
to the coordinates at time t:
Since the generator of the coordinate transformations is the single parameter t, Eqs. (4) describe a one-parameter family of diffeomorphisms[].
Knowledge of the Jacobian of this transformation is essential for
an understanding of how the measure on transforms in time.
Using the well known relation for the determinant of a matrix, the
Jacobian can be expressed as
where the elements of the matrix 
are 
.
Eq. (5) will be used to derive an evolution
equation for 
.
Taking the time derivative of Eq. (5) gives
(using the Einstein summation convention on repeated indices)
where the quantity 
(implied summation over i) is
known as the phase space compressibility of the dynamical system.
Eq. (6) is subject to the obvious initial condition J(0)=1.
Note that for the inverse Jacobian, 
,
. Thus, it follows that the equation of motion for 
is
Eqs. (6) and (7) are not new but were known to
Liouville[] and appear elsewhere in the literature (see, for example,
Refs.[, ]). Eqs. (6) and (7) predict
a nonunit Jacobian for a compressible system. On the manifold , the
metric G has tensor components 
in the initial coordinate basis
and
components
in the
corresponding coordinate basis at time t. These representations of G are, in general,
different. The determinants of the metric in these two representations
are related by the Jacobian[]:
Since J (and ) are not unity, it is clear that the metric
determinant, in general, is also not unity.
This means that the phase space must be treated as a general Riemannian
manifold with arbitrary curvature, and the
volume n-form, which determines the volume element in an arbitrary coordinate system,
should be expressed as
, accounting
for a nontrivial metric. Moreover,
the integral
of any function 
over the space should be written as[]
The expression for the volume n-form, 
behaves
like a tensor under coordinate transformations, for which there is a Jacobian
related to the metric by Eq. (8) [].
Knowledge of the compressibility of a system allows an invariant measure for the
manifold to be derived.
Suppose that the compressibility 
contains no explicit
time dependence. Then Eq. (6) for the Jacobian can be easily
solved to yield
Since 
is a total time derivative, a variable w(x), related to
the compressibility by 
, can be introduced. Eq. (10)
can then be written as
The wedge product in the volume n-form transforms according to
Arranging Eq. (12 so that quantities at time t appear on the left and quantities at t=0 appear on the right leads to:
where the conventional volume element has been used.
Eq. (13)
shows that 
is an invariant volume form
on the manifold and, by extension, 
is an invariant measure.
Since the volume form is ,
the metric determinant can be identified as 
,
which satisfies the metric transformation rule 
.
The existence of an invariant measure means that there is an underlying fixed
manifold with possibly nontrivial curvature, for which the metric
is determined by the compressibility.
Eq. (8)
implies that 
satisfies the same differential equation as does
: 
.
Note that for Hamiltonian systems, Eq. (13) reduces to the well known
result 
and the familiar fixed Euclidean geometry of Hamiltonian phase space.
The existence of an invariant measure means that the phase space
average of some property, expressed in terms of an integral over the
manifold, can be related to the time average of the same property
over a trajectory generated by the non-Hamiltonian dynamics Eqs. (3)
under the usual assumption of ergodicity. Although the assumption that
contains no explicit time dependence was made, the above formalism
is also true for the case that contains explicit time dependence.
Consider next an arbitrary ensemble described by a distribution function
, i.e., f is a
function of n coordinates and time t.
A continuity equation for f can be derived by
considering the number of ensemble members N(t) in a volume
of the space, is given by 
.
The continuity condition is that the rate of change of the number of
ensemble members within is balanced by the flux of members
through the surface bounding , which is expressed mathematically as
where 
is the surface n-1 form, 
is the
unit normal one-form to the surface, and
the surface integral
has been converted to a volume integral via a generalization of the divergence
theorem to manifolds with nontrivial metrics using the Lie derivative 
along the vector 
.
Thus, Eq. (14) can be written as
Eq. (15) must hold independent of the choice of and
thus implies the local continuity condition:
Eq. (16) represents a continuity equation
for f on an arbitrary manifold but makes no reference to
a specific choice of coordinate basis. To project Eq. (16)
onto a coordinate basis, we first apply the Leibniz rule[],
to the
action of the Lie derivative on the product.
The action of the Lie derivative
on the scalar f and on the volume form 
is well
known:
where the component representation of the wedge product is
given by 
, the Levi-Civita tensor[, ].
The last line follows from the properties of the
[].
Combining the results of Eq. (17) with Eq. (16)
gives the general form for the continuity equation in an arbitrary coordinate basis
Since the volume n-form is not zero, the term in brackets must, therefore, vanish yielding
where 
.
Eq. (19) is a general form of the Liouville
equation, valid on a manifold with a nontrivial metric and hence
is valid for an ensemble in which the underlying dynamics is
compressible. Recall that satisfies
.
Substituting this into Eq. (19)
leads to the conservation condition for f, df/dt=0. Finally, combining
this with Eq. (13) leads to the general statement of
Liouville's theorem for a non-Hamiltonian system:
Note that for Hamiltonian dynamics, the compressibility
vanishes and 
. Then, Eq. (19) reduces to the
usual Liouville equation, Eq. (1).
Finally, we note that Eq. (19) is different from
the equation which has hitherto been assumed to be the generalized
form of the Liouville equation, 
.
Eq. (19) can be put in this form by
defining a new function 
. Such an identification is
not general!
The consequences of this identification
are explored in the following discussion.
The generalized Liouville equation and the invariant measure will be used to derive an important property satisfied by the Gibbs entropy of a system. Consider, first, a Hamiltonian system for which the Gibbs entropy S(t) is given by
where k is the Boltzmann constant. Although the standard integral notation
has been employed for simplicity, the connection between the volume
element and the contraction of the volume form in Eq. (9)
(with ) must be kept in mind.
By taking the time derivative of both sides of Eq. (20),
using the Hamiltonian form of Liouville's equation, and performing
a few integrations by parts, it can be shown that dS/dt=0.
Thus, for a Hamiltonian system, the
Gibbs entropy, which is a fine-grained quantity[], is constant in time.
The generalization of the Gibbs entropy for a non-Hamiltonian system can be shown to satisfy the same property, when the proper invariant measure is used. Recognizing that the phase space volume element now contains a metric determinant factor, the entropy can be expressed as[]
where the metric determinant is assumed to contain no explicit time dependence.
Again, the standard integral notation is connected to Eq. (9) through
a contraction of the volume n-form, but with correctly
determined by the compressibility.
Computing the time derivative of both sides gives
Performing two integrations by parts and expanding the spatial derivative, it follows that
since 
as can be seen from Eq. (8).
Thus, the fine-grained entropy for a non-Hamiltonian system shares the same property
as that of a Hamiltonian system, namely, that it remains constant in time.
Eq. (21) makes clear the ambiguity associated with the
redefinition . It is necessary to keep the metric
separate from the distribution function in order that the
entropy be definable in a consistent manner. Further
details on this issue can be found in Refs.[, ].
It is instructive to consider, next, a concrete example of a non-Hamiltonian system that is typically used in molecular dynamics calculations, that of the Gaussian isokinetic system[], for which the equations of motion take the form
where the dynamics is for a N particle system (unit mass) in d dimensions with
phase space vector 
and forces 
obtained from the negative gradient of a potential function 
.
The Lagrange multiplier, 
, ensures that the constraint of constant
total kinetic energy,
,
where T is the temperature, is satisfied.
is given explicitly by
.
Using this definition of , the compressibility of
Eqs. (24) can be easily computed and is
found to be
where N is assumed to be a large number. Thus,
, and the metric determinant becomes
,
leading to the well known result for the partition function generated
by Eqs. (24)[]:
where the fact that 
is the invariant measure
has been used. Eq. (26) shows that Gaussian isokinetic dynamics
generates a canonical distribution function in the position or configuration space
while restricting the momenta to remain on the constant kinetic energy surface.
Taking 
, it is straightforward to show that
the time-independent form of Eq. (19) is satisfied.
In conclusion, a formulation of classical statistical mechanics of non-Hamiltonian systems has been presented. No assumptions about the geometry of the underlying phase space have been made. Rather, the space has been treated as a general Riemannian manifold with a nontrivial metric. The connection between the dynamical Jacobian and the phase space compressibility has been reviewed, and it has been shown that the nonunit Jacobian determines the metric determinant on the space. Knowledge of the compressibility also allows an invariant measure on the manifold to be derived. Treating the continuity condition within the framework of a general manifold shows that the hitherto assumed form for the generalized Liouville equation is not correct but rather that the proper generalization is Eq. (19), which explicitly involves the metric determinant. When combined with the invariant measure, the generalized Liouville equation leads to a general statement of Liouville's theorem. Thus, statistical averages can be computed with respect to the coordinates x at any time t. All of the above mentioned results reduce to their Hamiltonian analogs when the compressibility vanishes. These results imply the existence of a smooth phase space distribution function even for non-Hamiltonian systems. Finally, it has been shown that the natural generalization of the fine-grained Gibbs entropy, accounting for the metric on the space, satisfies the relation dS/dt=0, exactly as it does for a closed Hamiltonian system. It would be interesting to explore the consequences of a coarse-graining procedure[] applied to Eq. (21) in relation to the entropy of a nonequilibrium state.
