Page 22 - STATISTICAL MECHANICS: From First Principles to Macroscopic Phenomena

8 1 The classical distribution function

action

δ L dt = 0 (1.1)

or equivalently by the Lagrangian equations of motion
∂L d ∂L
− = 0 (1.2)
∂q i dt ∂ ˙ q i
Alternatively one may define momenta

∂L
p i = (1.3)
∂ ˙ q i
and a Hamiltonian
N

H = p i ˙ q i − L (1.4)
i=1
Expressing H as a function of the momenta p i and the coordinates q i one then has
the equations of motion in the form
∂ H
= ˙ q i (1.5)
∂p i
∂ H
− = ˙ p i (1.6)
∂q i
In examples, we will often be concerned with a system of identical particles with
conservative pair interactions. Then it is convenient to use the various components
of the positions of the particles r 1 , r 2 ,... as the quantities q i , and the Hamiltonian
takes the form
2
H = p /2m + (1/2) V ( r k , r l ) (1.7)
k
k k =l
where the sums run over particle labels and p k =∇ ˙ H. Then the Hamiltonian
r k
equations reduce to simple forms of Newton’s equation of motion. It turns out,
however, that the more general formulation is quite useful at the fundamental level,
particularly in understanding Liouville’s theorem, which we will discuss later.
In keeping with the discussion in the Introduction, we wish to relate this mi-
croscopic description to quantities which are measured in experiment or which are
conveniently used to analyze the results of simulations in a very similar way. Gener-
ically we denote these observable quantities as φ(q i (t), p i (t)). It is also possible to
consider properties which depend on the microscopic coordinates at more than one
time. We will defer discussion of these until Part III. Generally, these quantities, for
example the pressure on the wall of a vessel containing the system, are not constant