Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
180 CHAPTER 8. MEAN FIELD THEORY: CRITICAL TEMPERATURE.<br />
but sometimes flips. A bar magnet will switch poles once in a while (but don’t<br />
wait for it to happen, folks). If we would take the thermodynamic limit first,<br />
the ground st<strong>at</strong>e of the system would always be disordered! This is the wrong<br />
picture of the ground st<strong>at</strong>e with zero magnetiz<strong>at</strong>ion. On the other hand, if we<br />
take a system <strong>at</strong> a small non-zero temper<strong>at</strong>ure, the thermodynamic limit then<br />
implies th<strong>at</strong> the equilibrium st<strong>at</strong>e of the system is a disordered st<strong>at</strong>e. Now the<br />
sp<strong>at</strong>ial and temporal averages of variables are the same. But <strong>at</strong> T = 0 they are<br />
not, and the point T = 0 is a singularity for the one dimensional Ising chain.<br />
Realistic systems have finite values of N and non-zero values of T . Keep<br />
this in mind, and w<strong>at</strong>ch out for sample size dependent behavior near phase<br />
transitions. Near a phase transition the correl<strong>at</strong>ion length becomes larger than<br />
the system size, just as in our example of the one dimensional Ising chain. Near<br />
a phase transition we have to look carefully <strong>at</strong> the thermodynamic limit and<br />
scale out finite size effects, using appropri<strong>at</strong>e scaling laws.<br />
The same argument can be applied in two dimensions. Assume th<strong>at</strong> the spins<br />
are arranged on a square of L × L sites. Consider all Bloch wall st<strong>at</strong>es which<br />
divide the sample into two parts, each of which is connected and contains <strong>at</strong><br />
least one surface spin. The average length of the Bloch wall LW is proportional<br />
to L, LW = cL. The way we defined the Bloch wall does not allow for fractal<br />
dimensions, because we have discrete building blocks, and finite steps along the<br />
wall.<br />
The number of possible walls depends on the number of neighbors of each<br />
site. In first approxim<strong>at</strong>ion it will depend on L in the form 2Lb LW . In this case b<br />
is the average number of choices we have for a wall to continue <strong>at</strong> a given point.<br />
For example, on a square l<strong>at</strong>tice we expect b to be between two and three. At<br />
each site the next point in the Bloch wall can either go right, straight, or left.<br />
This would mean three choices for each link in the wall. Since we have to avoid<br />
wall crossings, the actual value will be lower. The pre-factor 2L is due to the<br />
fact th<strong>at</strong> we have L possible starting points <strong>at</strong> a given side, and we can start <strong>at</strong><br />
two sides. The increase in energy due to the wall will be LW 2J, because each<br />
element of the wall breaks a bond. Hence we have<br />
∆G = 2JLW − kBT log(2[2Lb LW ]) + kBT log(2) (8.104)<br />
where we included a factor two because we can go spin up to down or down to<br />
up. This gives<br />
∆G = 2JLW − kBT log(b LW ) − kBT log(2[2L]) + kBT log(2) (8.105)<br />
In the limit L → ∞ the last two terms are not important and we see th<strong>at</strong><br />
∆G = LW (2J − kBT log(b)) (8.106)<br />
Therefore the ordered st<strong>at</strong>e is stable against the thermal introduction of Bloch<br />
walls if kBT log(b) < 2J. This gives an estim<strong>at</strong>e of