Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
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178 CHAPTER 8. MEAN FIELD THEORY: CRITICAL TEMPERATURE.<br />
h = 0 the difference in Gibbs free energy, ∆G, is therefore:<br />
∆G = 2J − T ∆S = 2J − kBT (log 2(N − 1) − log(2)) = 2J − kBT log(N − 1)<br />
(8.99)<br />
For every value of the temper<strong>at</strong>ure not equal to zero, this free energy difference<br />
becomes neg<strong>at</strong>ive in the thermodynamic limit N → ∞. Hence <strong>at</strong> a non-zero<br />
temper<strong>at</strong>ure the completely ordered st<strong>at</strong>e is not stable in the thermodynamic<br />
limit and the Ising chain should not show a phase transition in the traditional<br />
sense, even though <strong>at</strong> T = 0 the system is stable in the completely ordered<br />
st<strong>at</strong>e. The temper<strong>at</strong>ure T = 0 is a singularity, but for a real phase transition we<br />
require th<strong>at</strong> it must be possible to define values of the temper<strong>at</strong>ure below Tc.<br />
We will now look further into the physical explan<strong>at</strong>ion of this phenomenon.<br />
Consider a finite chain with all spins up <strong>at</strong> non-zero temper<strong>at</strong>ure. Each spin is<br />
able to fluctu<strong>at</strong>e, and spins th<strong>at</strong> are not <strong>at</strong> the end have to pay an energy 4J<br />
because they break two bonds by switching. The probability for this to happen<br />
is e −4Jβ . It is far more likely for a spin <strong>at</strong> the end to switch, since the energy<br />
cost is only 2J and hence the probability is e −2Jβ .<br />
How long does it take for such a defect to be introduced? Due to thermal<br />
effects (in real systems most likely rel<strong>at</strong>ed to phonons, but any small outside<br />
perturb<strong>at</strong>ion will do) the spin tries to change. We can represent these effects<br />
by an <strong>at</strong>tempt frequency A. The time scale typical for the introduction of such<br />
defects, tintro, then has to obey:<br />
1 = Ae −2Jβ tintro<br />
(8.100)<br />
and we say th<strong>at</strong> every tintro seconds a new defect is introduced.<br />
Wh<strong>at</strong> is the lifetime of such a defect? The easiest way is to think about it in<br />
terms of the motion of a Bloch wall. Such motion does NOT cost energy, since<br />
the energy is directly associ<strong>at</strong>ed with the two sites defining the Bloch wall, and<br />
no m<strong>at</strong>ter where it is the energy is the same 2J. Of course, <strong>at</strong> the end points<br />
the Bloch wall can disappear, with a gain of energy 2J.<br />
The motion of a Bloch wall can be represented by a random walk, since<br />
<strong>at</strong> each stage the probability of moving to the left is equal to the probability<br />
of moving to the right. We define thop to be the average time between hops.<br />
This is again a thermal effect, and the hopping time is the inverse of a different<br />
<strong>at</strong>tempt frequency, A ′ thop = 1, where there is no Boltzmann factor. We need<br />
some outside mechanism to cause these hops. They are most likely coupled to<br />
the same he<strong>at</strong> b<strong>at</strong>h which drives the spin flips, but the coupling strength is<br />
different because it is a different mechanism. Since the motion of the Bloch wall<br />
is a random walk, the average distance covered in l<strong>at</strong>tice spacings in a time t is<br />
t . The time it takes for a Bloch wall to move through the chain is therefore<br />
thop<br />
found by setting this expression equal to N, and we have<br />
tlife = thopN 2<br />
(8.101)