Statistical Mechanics - Physics at Oregon State University

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178 CHAPTER 8. MEAN FIELD THEORY: CRITICAL TEMPERATURE. h = 0 the difference in Gibbs free energy, ∆G, is therefore: ∆G = 2J − T ∆S = 2J − kBT (log 2(N − 1) − log(2)) = 2J − kBT log(N − 1) (8.99) For every value of the temperature not equal to zero, this free energy difference becomes negative in the thermodynamic limit N → ∞. Hence at a non-zero temperature the completely ordered state is not stable in the thermodynamic limit and the Ising chain should not show a phase transition in the traditional sense, even though at T = 0 the system is stable in the completely ordered state. The temperature T = 0 is a singularity, but for a real phase transition we require that it must be possible to define values of the temperature below Tc. We will now look further into the physical explanation of this phenomenon. Consider a finite chain with all spins up at non-zero temperature. Each spin is able to fluctuate, and spins that are not at the end have to pay an energy 4J because they break two bonds by switching. The probability for this to happen is e −4Jβ . It is far more likely for a spin at the end to switch, since the energy cost is only 2J and hence the probability is e −2Jβ . How long does it take for such a defect to be introduced? Due to thermal effects (in real systems most likely related to phonons, but any small outside perturbation will do) the spin tries to change. We can represent these effects by an attempt frequency A. The time scale typical for the introduction of such defects, tintro, then has to obey: 1 = Ae −2Jβ tintro (8.100) and we say that every tintro seconds a new defect is introduced. What is the lifetime of such a defect? The easiest way is to think about it in terms of the motion of a Bloch wall. Such motion does NOT cost energy, since the energy is directly associated with the two sites defining the Bloch wall, and no matter where it is the energy is the same 2J. Of course, at the end points the Bloch wall can disappear, with a gain of energy 2J. The motion of a Bloch wall can be represented by a random walk, since at each stage the probability of moving to the left is equal to the probability of moving to the right. We define thop to be the average time between hops. This is again a thermal effect, and the hopping time is the inverse of a different attempt frequency, A ′ thop = 1, where there is no Boltzmann factor. We need some outside mechanism to cause these hops. They are most likely coupled to the same heat bath which drives the spin flips, but the coupling strength is different because it is a different mechanism. Since the motion of the Bloch wall is a random walk, the average distance covered in lattice spacings in a time t is t . The time it takes for a Bloch wall to move through the chain is therefore thop found by setting this expression equal to N, and we have tlife = thopN 2 (8.101)
8.5. CRITICAL TEMPERATURE IN DIFFERENT DIMENSIONS. 179 We can now distinguish two cases. If tlife ≫ tintro we find at each moment many Bloch walls in the system. This means that the system looks like it is not ordered. On the other hand, if tlife ≪ tintro most of the time there are no Bloch walls in the system, and the chain is fluctuating back and forth between two completely ordered states, one with all spins up and one with all spins down. In both cases we have for the time average of a spin τ 1 〈σi〉 = lim σi(t)dt (8.102) τ→∞ τ 0 which is zero in both case. Note that we always need τ ≫ tlife, tintro. What are values we expect for realistic systems. If we assume that attempts are caused by phonons,the attempt frequencies for both creation and hopping are of order 10 12 s −1 . This means that tintro tlife ≈ A−1 e 2Jβ A −1 N 2 (8.103) and with N = 10 8 as typical for a 1 cm chain and J = 1 meV as a typical spin energy, we have tintro ≈ tlife for T ≈ 1 K. But this temperature depends on the size of the system. At high temperatures the shorter time is the introduction time. We always see many domain walls. This is the normal state of the system. If we go to lower temperatures this picture should remain the same, but due to the finite length of the chain we see a switch in behavior. Below a certain temperature the chain at a given time is always uniformly magnetized, but over time it switches. The temperature at which this change of behavior takes place depends on the length of the chain. This is an example of a finite size effect. For large systems finite size effects take place at very low temperatures, see example numbers above. For small samples this can be at much higher temperatures! In the previous discussion we have three limits to worry about. In order for the averages to be the same, 〈σ〉time = 〈σ〉ensemble, we need τ → ∞. We are interested in low temperatures, or the limit T → 0. Finally, we have the thermodynamic limit N → ∞. Since we need τ ≫ tlife = thopN 2 we see that we need to take the limit τ → ∞ before the thermodynamic limit. Also, because τ ≫ tintro = A −1 e 2Jβ we need to take the limit τ → ∞ before the limit T → 0. Form our discussion it is clear that limτ→∞〈σ〉τ = 0. Hence we have limT →0〈σ〉∞ = 0 and this remains zero in the TD limit. The one dimensional Ising chain is never ordered! Calculations at T = 0 do not make sense. At T = 0 we have tintro = ∞ and we always have τ < tintro. That violates the ergodic theorem. So, even though people do calculations at T = 0, one has to argue carefully if it makes sense. That is what we did for the system of Fermions, for example. After taking the limit τ → ∞, what does the system look like? If we now take T → 0 we find a system that is most of the time completely ordered, but once in a while flips. The net magnetization is zero. Therefore, the real T = 0 ground state of a finite system is a system that is ordered most of the time,
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Statistical Mechanics Henri J.F. Ja
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IV CONTENTS 4.3 Gas of poly-atomic
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VI CONTENTS
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VIII INTRODUCTION Introduction. The
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X INTRODUCTION In chapter nine we d
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2 CHAPTER 1. FOUNDATION OF STATISTI
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120 CHAPTER 6. DENSITY MATRIX FORMA
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230 APPENDIX A. SOLUTIONS TO SELECT
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Statistical Mechanics - Physics at Oregon State University

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