# Canonic bosons.py

### From Werner KRAUTH

This is the python2 program Canonic_bosons.py useful for the homework session of week 13 of my ICFP Lectures on statistical mechanics. Here, the density of state is 1,3,6,10,15, which corresponds to the three-dimensional isotropic harmonic trap. In the below program, we integrate in the complex plane from (-pi, lambda) to (pi, lambda) (neglecting the integral from (-pi, 0) to (-pi, lambda)), and always get the same result for the partition function. At T=1, for example, we get Z=17.373..., a result we can also obtain from the naive sum over the states, as implemented in Naive_bosons.py.

import math, cmath, numpy, pylab dos = [1, 3, 6, 10, 15] T = 1.0 beta = 1.0 / T Zint = complex(0.0, 0.0) dellambda = 0.01 oldl = complex(-math.pi, dellambda) complexi = complex(0.0, 1.0) for RL in numpy.arange(-math.pi, math.pi, 0.00001): newl = RL + dellambda * complexi integrand = cmath.exp(-complexi * 5 * newl) for E in range(5): integrand /= (1.0 - cmath.exp( - beta * E + complexi * newl)) ** dos[E] Zint += integrand * (newl - oldl) / (2.0 * math.pi) oldl = newl print Zint