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.410 31 — Ising ModelsRectangular:Triangular:W✻❄❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜W✻❄❜ ✟ ❜ ✟ ❜ ✟ ❜ ✟ ❜ ✟ ❜ ✟❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍ ❜ ✟ ❍❍ ❍ ❍ ❍ ❍ ❍Computational ideas:Only the largest eigenvalue is needed. There are several ways of getting thisquantity, for example, iterative multiplication of the matrix by an initial vector.Because the matrix is all positive we know that the principal eigenvectoris all positive too (Frobenius–Perron theorem), so a reasonable initial vector is(1, 1, . . . , 1). This iterative procedure may be faster than explicit computationof all eigenvalues. I computed them all anyway, which has the advantage thatwe can find the free energy of finite length strips – using equation (31.32) – aswell as infinite ones.Figure 31.14. Two long-thin-stripIsing models. A line between twospins indicates that they areneighbours. The strips have widthW and infinite length.Free Energy-1-2-3-4-5Ferromagnets of width 8TriangularRectangular-1-2-3-4-5-6Antiferromagnets of width 8TriangularRectangularFigure 31.15. Free energy per spinof long-thin-strip Ising models.Note the non-zero gradient atT = 0 in the case of the triangularantiferromagnet.-6-7-70 2 4 6 8 10Temperature-80 2 4 6 8 10TemperatureComments on graphs:For large temperatures all Ising models should show the same behaviour: thefree energy is entropy-dominated, and the entropy per spin is ln(2). The meanenergy per spin goes to zero. The free energy per spin should tend to −ln(2)/β.The free energies are shown in figure 31.15.One of the interesting properties we can obtain from the free energy isthe degeneracy of the ground state. As the temperature goes to zero, theBoltzmann distribution becomes concentrated in the ground state. If theground state is degenerate (i.e., there are multiple ground states with identicalEntropy0.70.60.50.40.30.20.1Triangular(-)RectangularTriangular(+)00 2 4 6 8 10TemperatureFigure 31.16. Entropies (in nats)of width 8 Ising systems as afunction of temperature, obtainedby differentiating the free energycurves in figure 31.15. Therectangular ferromagnet andantiferromagnet have identicalthermal properties. For thetriangular systems, the uppercurve (−) denotes theantiferromagnet and the lowercurve (+) the ferromagnet.