Information Theory, Inference, and Learning ... - Inference Group

Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.404 31 — Ising ModelsN Mean energy and fluctuations Mean square magnetization16Energy0.50-0.5-1-1.5-21 10Temperature0.5Mean Square Magnetization10.80.60.40.201 10Temperature1Figure 31.3. Monte Carlosimulations of rectangular Isingmodels with J = 1. Mean energyand fluctuations in energy as afunction of temperature (left).Mean square magnetization as afunction of temperature (right).In the top row, N = 16, and thebottom, N = 100. For even largerN, see later figures.100Energy0-0.5-1-1.5Mean Square Magnetization0.80.60.40.2-21 10Temperature01 10TemperatureContrast with Schottky anomalyA peak in the heat capacity, as a function of temperature, occurs in any systemthat has a finite number of energy levels; a peak is not in itself evidence of aphase transition. Such peaks were viewed as anomalies in classical thermodynamics,since ‘normal’ systems with infinite numbers of energy levels (such asa particle in a box) have heat capacities that are either constant or increasingfunctions of temperature. In contrast, systems with a finite number of levelsproduced small blips in the heat capacity graph (figure 31.4).Let us refresh our memory of the simplest such system, a two-level systemwith states x = 0 (energy 0) and x = 1 (energy ɛ). The mean energy isexp(−βɛ)E(β) = ɛ1 + exp(−βɛ) = ɛ 11 + exp(βɛ)and the derivative with respect to β isSo the heat capacity is(31.22)dE/dβ = −ɛ 2 exp(βɛ)[1 + exp(βɛ)] 2 . (31.23)C = dE/dT = − dEdβ1k B T 2 =ɛ2k B T 2exp(βɛ)[1 + exp(βɛ)] 2 (31.24)and the fluctuations in energy are given by var(E) = Ck B T 2 = −dE/dβ,which was evaluated in (31.23). The heat capacity and fluctuations are plottedin figure 31.6. The take-home message at this point is that whilst Schottkyanomalies do have a peak in the heat capacity, there is no peak in theirfluctuations; the variance of the energy simply increases monotonically withtemperature to a value proportional to the number of independent spins. Thusit is a peak in the fluctuations that is interesting, rather than a peak in theheat capacity. The Ising model has such a peak in its fluctuations, as can beseen in the second row of figure 31.5.Figure 31.4. Schematic diagram toexplain the meaning of a Schottkyanomaly. The curve shows theheat capacity of two gases as afunction of temperature. Thelower curve shows a normal gaswhose heat capacity is anincreasing function oftemperature. The upper curve hasa small peak in the heat capacity,which is known as a Schottkyanomaly (at least in Cambridge).The peak is produced by the gashaving magnetic degrees offreedom with a finite number ofaccessible states.TRectangular Ising model with J = −1What do we expect to happen in the case J = −1? The ground states of aninfinite system are the two checkerboard patterns (figure 31.7), and they have