Question

Consider an Ising model of 100 elementary dipoles. Suppose you wish to calculate the partition function for this system, using a computer that can compute one billion terms of the partition function per second. How long must you wait for the answer?

Short answer

The total time required is$4.13\times {10}^{13}\text{years}$

Step-by-step solution

Step 1. Given information

The equation for the partition function:

$Z=\sum _{s_i}{e}^{-\beta U}$

For $N$number of dipoles, each has two possibilities of alignments, the number of terms in this sum is $2^N$.

Given total number of dipoles in the system is } 100 \text {. Number of states in the system= $2^{100}$

The total number of partition function is,

$z={10}^9$

Step 2. The time required is 

$t=\frac{n}{z}$

$=\frac{2^{100}}{{10}^9}$

$=1.26\times {10}^{21}\mathrm{s}\left(\frac{1.0\mathrm{yr}}{365\text{days}}\right)\left(\frac{1.0\mathrm{day}}{24\mathrm{h}}\right)\left(\frac{1.0\mathrm{h}}{3600\mathrm{s}}\right)$

$=4.13\times {10}^{13}\text{years}$

Thus, the total time required is $=4.13\times {10}^{13}\text{years}$