1549380323-Statistical Mechanics Theory and Molecular Simulation

48 Classical mechanics

the entire system and determine the evolution of the physical subsystem. However, we
are rarely interested in all of the microscopic details of the bath. Wemight, therefore,
consider treating the effect of the bath in a more coarse-grained manner by replacing
its microscopic coordinates and momenta with a few simpler variables that couple to
the physical subsystem in a specified manner. In this case, a set ofequations of motion
describing the physical system plus the few additional variables used to represent the
action of the bath could be proposed which generally would not be Hamiltonian in
form because the true microscopic nature of the bath had been eliminated. For this
reason, non-Hamiltonian dynamical systems can be highly useful and it is instructive
to examine some of their characteristics.
We will restrict ourselves to dynamical systems of the generic form

̇x =ξ(x), (1.12.1)

where x is a phase space vector ofncomponents andξ(x) is a continuous, differentiable
n-dimensional vector function. A key signature of a non-Hamiltoniansystem is that it
can have a nonvanishing phase-space compressibility:

κ(x) =

∑n

i=1

∂ ̇xi

∂xi

=

∑n

i=1

∂ξi

∂xi

6 = 0. (1.12.2)

When eqn. (1.12.2) holds, many of the theorems about Hamiltonian systems no longer
apply. However, as will be shown in Chapter 2, some properties of Hamiltonian systems
can be generalized to non-Hamiltonian systems provided certain conditions are met. It
is important to note that when a Hamiltonian system is formulated in non-canonical
variables, the resulting system can also have a nonvanishing compressibility. Strictly
speaking, such systems are not truly non-Hamiltonian since a simple transformation
back to a canonical set of variables can eliminate the nonzero compressibility. How-
ever, throughout this book, we will group such cases in with our general discussion of
non-Hamiltonian systems and loosely refer to them as non-Hamiltonian because the
techniques we will develop for analyzing dynamical systems with nonzero compress-
ibility factors can be applied equally well to both types of systems.
A simple and familiar example of a non-Hamiltonian system is the case of the
damped forced harmonic oscillator described by an equation of motion of the form

mx ̈=−mω^2 x−ζx. ̇ (1.12.3)

This equation describes a harmonic oscillator subject to the action of a friction force
−ζx ̇, which could arise, for example, by the motion of the oscillator on a rough sur-
face. Obviously, such an equation cannot be derived from a Hamiltonian. Moreover,
the microscopic details of the rough surface are not treated explicitly but rather are
modeled grossly by the simple dissipative term in the equation of motionfor the phys-
ical subsystem described by the coordinatex. Writing the equation of motion as two
first order equations involving a phase space vector (x,p), we have

x ̇=

p

m