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.B.1: About phase transitions 603log ZN beta epsilonN log (2)(a)beta(c)var(E) N=24var(E) N=8betaFigure B.1. (a) Partition functionof toy system which shows a phasetransition for large N. The arrowmarks the point β c = log 2/ɛ. (b)The same, for larger N.(c) The variance of the energy ofthe system as a function of β fortwo system sizes. As N increasesthe variance has an increasinglysharp peak at the critical point β c .Contrast with figure B.2.log ZN beta epsilonN log (2)(b)betalog ZN beta epsilonN log (2)(a)beta(b)var(E) N=24var(E) N=8betaFigure B.2. The partition function(a) and energy-variance (b) of asystem consisting of Nindependent spins. The partitionfunction changes gradually fromone asymptote to the other,regardless of how large N is; thevariance of the energy does nothave a peak. The fluctuations arelargest at high temperature (smallβ) and scale linearly with systemsize N.The arrow marks the pointβ = ln 2(B.11)ɛat which these two asymptotes intersect. In the limit N → ∞, the graph ofln Z(β) becomes more and more sharply bent at this point (figure B.1b).The second derivative of ln Z, which describes the variance of the energyof the system, has a peak value, at β = ln 2/ɛ, roughly equal toN 2 ɛ 24 , (B.12)which corresponds to the system spending half of its time in the ground stateand half its time in the other states.At this critical point, the heat capacity of this system is thus proportionalto N 2 ; the heat capacity per spin is proportional to N, which, for infinite N, isinfinite, in contrast to the behaviour of systems away from phase transitions,whose capacity per atom is a finite number.For comparison, figure B.2 shows the partition function and energy-varianceof the ordinary independent-spin system.More generallyPhase transitions can be categorized into ‘first-order’ and ‘continuous’ transitions.In a first-order phase transition, there is a discontinuous change of one