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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8.5. CRITICAL TEMPERATURE IN DIFFERENT DIMENSIONS. 179<br />
We can now distinguish two cases. If tlife ≫ tintro we find <strong>at</strong> each moment<br />
many Bloch walls in the system. This means th<strong>at</strong> the system looks like it is not<br />
ordered. On the other hand, if tlife ≪ tintro most of the time there are no Bloch<br />
walls in the system, and the chain is fluctu<strong>at</strong>ing back and forth between two<br />
completely ordered st<strong>at</strong>es, one with all spins up and one with all spins down.<br />
In both cases we have for the time average of a spin<br />
τ<br />
1<br />
〈σi〉 = lim σi(t)dt (8.102)<br />
τ→∞ τ 0<br />
which is zero in both case. Note th<strong>at</strong> we always need τ ≫ tlife, tintro.<br />
Wh<strong>at</strong> are values we expect for realistic systems. If we assume th<strong>at</strong> <strong>at</strong>tempts<br />
are caused by phonons,the <strong>at</strong>tempt frequencies for both cre<strong>at</strong>ion and hopping<br />
are of order 10 12 s −1 . This means th<strong>at</strong><br />
tintro<br />
tlife<br />
≈ A−1 e 2Jβ<br />
A −1 N 2<br />
(8.103)<br />
and with N = 10 8 as typical for a 1 cm chain and J = 1 meV as a typical spin<br />
energy, we have tintro ≈ tlife for T ≈ 1 K. But this temper<strong>at</strong>ure depends on the<br />
size of the system. At high temper<strong>at</strong>ures the shorter time is the introduction<br />
time. We always see many domain walls. This is the normal st<strong>at</strong>e of the system.<br />
If we go to lower temper<strong>at</strong>ures this picture should remain the same, but due<br />
to the finite length of the chain we see a switch in behavior. Below a certain<br />
temper<strong>at</strong>ure the chain <strong>at</strong> a given time is always uniformly magnetized, but<br />
over time it switches. The temper<strong>at</strong>ure <strong>at</strong> which this change of behavior takes<br />
place depends on the length of the chain. This is an example of a finite size<br />
effect. For large systems finite size effects take place <strong>at</strong> very low temper<strong>at</strong>ures,<br />
see example numbers above. For small samples this can be <strong>at</strong> much higher<br />
temper<strong>at</strong>ures!<br />
In the previous discussion we have three limits to worry about. In order<br />
for the averages to be the same, 〈σ〉time = 〈σ〉ensemble, we need τ → ∞. We<br />
are interested in low temper<strong>at</strong>ures, or the limit T → 0. Finally, we have the<br />
thermodynamic limit N → ∞. Since we need τ ≫ tlife = thopN 2 we see<br />
th<strong>at</strong> we need to take the limit τ → ∞ before the thermodynamic limit. Also,<br />
because τ ≫ tintro = A −1 e 2Jβ we need to take the limit τ → ∞ before the limit<br />
T → 0. Form our discussion it is clear th<strong>at</strong> limτ→∞〈σ〉τ = 0. Hence we have<br />
limT →0〈σ〉∞ = 0 and this remains zero in the TD limit. The one dimensional<br />
Ising chain is never ordered!<br />
Calcul<strong>at</strong>ions <strong>at</strong> T = 0 do not make sense. At T = 0 we have tintro = ∞ and<br />
we always have τ < tintro. Th<strong>at</strong> viol<strong>at</strong>es the ergodic theorem. So, even though<br />
people do calcul<strong>at</strong>ions <strong>at</strong> T = 0, one has to argue carefully if it makes sense.<br />
Th<strong>at</strong> is wh<strong>at</strong> we did for the system of Fermions, for example.<br />
After taking the limit τ → ∞, wh<strong>at</strong> does the system look like? If we now<br />
take T → 0 we find a system th<strong>at</strong> is most of the time completely ordered, but<br />
once in a while flips. The net magnetiz<strong>at</strong>ion is zero. Therefore, the real T = 0<br />
ground st<strong>at</strong>e of a finite system is a system th<strong>at</strong> is ordered most of the time,