Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
5.1. FERMIONS IN A BOX. 97<br />
fα(z) =<br />
∞<br />
n=1<br />
z n<br />
n α (−1)n+1 , |z| < 1 (5.34)<br />
This defines the function for a disc around the origin. The values for arbitrary<br />
values of z are obtained by analytical continu<strong>at</strong>ion. For example:<br />
∞<br />
f0(z) = z n (−1) n+1 ∞<br />
= 1 − z n (−1) n = 1 − 1 z<br />
=<br />
1 + z 1 + z<br />
n=1<br />
n=0<br />
and this defines a function everywhere, except <strong>at</strong> the pole z = −1.<br />
Why is this important?<br />
(5.35)<br />
One might argue th<strong>at</strong> now we have computers available and we do not need<br />
all this analytical detail anymore. Since one use of power series is to be able<br />
to evalu<strong>at</strong>e functions over a reasonable domain of arguments, we can simply let<br />
computers take over and do the integr<strong>at</strong>ions for all values. This is not really<br />
correct, however. The analysis above tells us where we have poles and other<br />
singularities. These are places where computer calcul<strong>at</strong>ions will fail! Relying on<br />
computers also assumes th<strong>at</strong> there are no errors in the computer calcul<strong>at</strong>ions,<br />
and th<strong>at</strong> is not always true. Therefore, we need to be able to check our computer<br />
calcul<strong>at</strong>ions against the analytical solutions we can obtain via power series and<br />
other means. The analytical solutions can tell us th<strong>at</strong> near a limiting point<br />
or singularity an energy will behave like |T − T0| −0.3 , for example. This is<br />
very difficult to extract precisely from a computer calcul<strong>at</strong>ion. But, of course,<br />
comput<strong>at</strong>ional physics also has its place, because it can extend the analytical<br />
solutions to domains of the parameters where we cannot get analytical solutions.<br />
Simple example.<br />
As a simple example, study the differential equ<strong>at</strong>ion<br />
¨x + x = 0 (5.36)<br />
and try a power series of the form x(t) = ∞<br />
0 cnt n . This gives:<br />
∞<br />
cnn(n − 1)t n−2 ∞<br />
+ cnt n = 0 (5.37)<br />
0<br />
or after combining terms with equal powers of t, and defining c−2 = c−1 = 0 we<br />
get:<br />
∞<br />
0<br />
0<br />
(cn+2(n + 2)(n + 1) + cn)t n = 0 (5.38)