Biomedical Engineering Reference
In-Depth Information
f(x)
a
b
x
Figure 9.2
Trapezium rule for a definite integral
9.1 An Early Paper
Metropolis and Ulam's (1949) keynote paper has an interesting abstract.
We shall present here the motivation and a general description of a method dealing with a class
of problems in mathematical physics. The method is, essentially, a statistical approach to the
study of differential equations, or more generally, of integro-differential equations that occur
in various branches of the natural sciences.
and there is much to be gained by studying the opening sentences...
Already in the nineteenth century a sharp distinction began to appear between two dif-
ferent mathematical methods for treating physical phenomena. Problems involving only a
few particles were studied in classical mechanics, through the study of systems of ordinary
differential equations. For the description of systems with very many particles, an entirely dif-
ferent technique was used, namely, the method of statistical mechanics. In this latter approach,
one does not concentrate on the individual particles but studies the properties of sets of
particles.
Straightforward thermodynamic quantities such as the pressure, the internal energy and
the Gibbs energy turn out to be impossible to calculate directly for a macroscopic system,
simply because of the large number of particles involved. In fact, it is not even sensible to
contemplate recording an initial starting point for 10
23
particles, let alone devising methods
for solving the equations of motion.
Suppose for the sake of argument that we have a systemof
N
simple atomic particles (such
as argon atoms) and we are interested in the total mutual potential energy (conventionally
written Φ rather than
U
). If the pair potential is
U
ij
(
R
) and the potentials are pair wise
additive, we have
N
−
1
N
U
ij
R
ij
Φ
=
i
=
1
j
=
i
+
1
If I denote the position vectors of the
N
particles measured at time
t
by
R
A
(
t
),
R
B
(
t
), ...,
R
N
(
t
), then these position vectors will depend on time and the instantaneous value of Φ
will depend on the values of the variables at that time. If we make enough measurements
