1549380323-Statistical Mechanics Theory and Molecular Simulation

Phase space 5

itself to an analytical solution could be introduced. Although often of limited util-
ity, important physical insights can sometimes be extracted from aclever model, and
it is usually possible to study the behavior of the model as external conditions are
varied, such as the number of particles, containing volume, applied pressure, and so
forth. Alternatively, one can consider a system, not of 10^23 particles, but of a much
smaller number, perhaps 10^2 –10^9 particles, depending on the nature of the system,
and solve the equations of motion numerically subject to initial conditions and the
boundary conditions of a containing volume. Fortunately, many macroscopic proper-
ties are well-converged with respect to system size for such small numbers of particles!
The rules of statistical mechanics are then used to analyze the numerical trajectories
thus generated. This is the essence of the technique known asmolecular dynamics. Al-
though the molecular dynamics approach is very powerful, a significant disadvantage
exists: in order to study the dependence on external conditions,a separate calculation
must be performed for every choice of these conditions, hence a very large number of
calculations is needed, for example, in order to map out a phase diagram. In addi-
tion, the “exact” forces between particles cannot be determinedand, hence, models
for these forces must be introduced. Usually, the more accuratethe model, the more
computationally intensive the numerical calculation, and the more limited the scope
of the calculation with respect to time and length scales and the properties that can
be studied. Often, time and length scales can be bridged by combiningmodels of dif-
ferent accuracy, including even continuum models commonly used in engineering, to
describe different aspects of a large, complex system, and devisingclever numerical
solvers for the resulting equations of motion. Numerical calculations (typically referred
to assimulations) have become an integral part of modern theoretical research,and
since many of these calculations rely on the laws of classical mechanics, it is impor-
tant that this subject be covered in some detail before advancingto a discussion of
the rules of statistical mechanics. The remainder of this chapter will, therefore, be
devoted to introducing the concepts from classical mechanics that will be needed for
our subsequent discussion of statistical mechanics.

1.3 Phase space: visualizing classical motion

Newton’s equations specify the complete set of particle positions{r 1 (t),...,rN(t)}and,
by differentiation, the particle velocities{v 1 (t),...,vN(t)}at any timet, given that
the positions and velocities are known at one particular instant in time. For reasons
that will be clear shortly, it is often preferable to work with the particle momenta,
{p 1 (t),...,pN(t)}, which, in Cartesian coordinates, are related to the velocities by

pi=mivi=mir ̇i. (1.3.1)

Note that, in terms of momenta, Newton’s second law can be writtenas

Fi=mai=mi

dvi

dt

=

dpi

dt

. (1.3.2)

Therefore, the classical dynamics of anN-particle system can be expressed by specify-
ing the full set of 6Nfunctions,{r 1 (t),...,rN(t),p 1 (t),...,pN(t)}. Equivalently, at any