Footnotes

Chapter 1

1. An example of a physical model is a model of a plane used in wind tunnel tests to investigate its aerodynamic properties.

2. Properties of the real system which are initially assumed to exist (by analogy with properties of the model) may subsequently be observed as a result of advances in techniques of observation, e.g. electron microscopy.

3. Further consideration of this question is beyond the scope of this work but we may conjecture that light (as it is "in itself") is fundamental both to physical reality and to consciousness itself and so is unlikely to be fully comprehensible by discursive thought.

4. A fuller discussion of this point will be found at the end of this section.

5. The nature of such spin models is the subject of the following sections of this chapter.

6. The (pseudo-)random number generator functions in the model as the analogue of randomness in Nature.

7. Uhlenbeck and Goudsmit 1925, as cited by Domb 1974.

8. It may be noted that in fact bar magnets free to rotate and placed next to each do not align themselves with their north poles in the same direction, but rather in opposite directions.

9. In this report the following principle of italicisation is used: Where a mathematical expression occurs within a sentence it is italicised if and only if it consists of a single Roman character.

10. A more complicated model might allow energetic interactions between spins which are not nearest neighbours.

11. Renfrey Burnard Potts was Professor of Applied Mathematics, University of Adelaide, 1959-90.

12. A first-order phase transition is one in which a discontinuity occurs in a macroscopic quantity such as density, heat capacity, etc. as temperature or pressure is changed. It is the discontinuity that identifies the phase transition. In alloys and magnetic systems there are phase transitions that are second-order, meaning that the macroscopic properties do not change discontinuously at some particular temperature or pressure, but rather the rate of change of those quantities with respect to temperature or pressure does. Instead of jumping in value, a quantity suddenly goes from not changing to changing, or from changing at a certain rate to changing at a different rate. Examples are the magnetization of magnets and the crystal pattern of alloys, both of which are zero above a certain critical temperature and then start to increase below that temperature.

13. Prior to Onsager, Kramers and Wannier (1941) had made contributions to the solution of the problem.

* [Added 2001-02-17 CE] For information regarding the method of representing the lattice geometries studied in this work see Lattice Geometries.

14. The number of vertices to which a vertex in the lattice is directly connected by lines may be larger than the coordination number of the lattice. For example, it is possible to embed a honeycomb lattice (with coordination number of 3) in a square array of vertices, each of which is connected directly to four other vertices. In this case not all of the vertices are lattice sites (perhaps just half are).

15. In mathematics the term "lattice" is usually used in a more restricted sense, in which the binary relation is a partial ordering of a set satisfying certain conditions, and in which the set always contains a greatest and a least element with respect to this partial ordering.

16. In a spin model which is not a "nearest neighbour" spin model the interaction energy also depends on the distance between the spins (where "distance" means the length of the shortest connecting chain of spins).

17. In this report we shall often not distinguish notationally between a spin and its spin value in a given state, denoting both by terms such as Si.

18. This point is discussed in more detail in Section (iv) of Appendix 1.

19. This constancy at the macroscopic level does not imply constancy at the microscopic level, because at the level of atoms and electrons there is continual change. The spins of the electrons in a magnetic material are continually flipping, at a rate dependent on the temperature of the system, yet (if the system is in equilibrium) the overall magnetism remains constant (in the absence of outside influences).

20. For example, suppose a system consists of exactly two elements, each of which may have one of two energies, 0 or 1, then (assuming we can distinguish the two elements) there are four possible states of the system, one with energy 0, two with energy 1 and one with energy 2. Taking kB = 1 the partition function is Z = e0 + 2e-1/T + e-2/T. For T = 1, Z = 1.8711, and the probabilities of the system being in a certain state (at T = 1) are p(00) = 0.5344, p(01) = p(10) = 0.1966 and p(11) = 0.0723.

21. These two kinds of dynamics will be discussed further in the Sections 1.7 and 1.9.

22. A "plausibility argument" to show that this is so is given by Binder and Heermann (1998), pp.18-19.

23. Spins may be selected in various ways: (i) One may go through the lattice "typewriter fashion". (ii) One may select the spins randomly. (iii) One may divide the lattice into sublattices (similar to a checkerboard divided into squares) and select a spin from each sublattice in turn, going through the sublattices in typewriter fashion. (ii) is perhaps most realistic, but requires extra computing time to generate random numbers. In this study method (iii) is used in single spin flip dynamics.

24. The terms Si and Sj are here used to denote both the state of the spin system and the value of the spin under consideration.

25. Since the random spin value selection may produce the spin value already possessed by the spins of a virtual spin cluster not all virtual clusters are changed.

26. We can observe critical phenomena in physical reality, e.g. melting of solids, because the number of atoms involved is practically infinite.

27. It is interesting to note that work by Heermann and Stauffer (1980) suggests "that free edges approximate the infinite system as well as the more complicated periodic boundary conditions", at least in the case of the bond-diluted square lattice with lattice sizes from 10x10 to 240x240.

28. This is a favourite of FORTRAN programmers seeking to maximise execution speed.

29. Suppose n sizes L0, ..., Ln-1 are such that the reciprocals 1/L0, ..., 1/Ln-1 are equally spaced in the interval [0,1/L0]. Then for all 0 ≤ i ≤ n we have 1/Li - 1/Li+1 = 1/L0.1/n. Thus 1/L0 - 1/L1 = 1/(n.L0) so 1/L1 = (1/L0).(1-1/n), and by induction 1/Li = (1/L0).(1-i/n). So Ln-1 = n.L0 so L0 = Ln-1/n. Thus the choice of n and either L0 or Ln-1 determines all the other sizes (or approximate sizes if the factors of n.L0 do not allow for whole number solutions). Taking n = 6 and Ln-1 = 60 implies sizes 10, 12, 15, 20, 30 and 60, with the intervals between the reciprocals of the sizes being 1/60. Taking n = 6 and Ln-1 = 180 implies sizes 30, 36, 45, 60, 90 and 180, with the intervals between the reciprocals of the sizes being 1/180.

Chapter 2

1. The simulation software, with which all the simulation results in this work were obtained, was written by the author entirely ab initio except for the code for generating random numbers (which was obtained from Press et al. 1992 as stated in Appendix 2). The software was developed on the basis of published descriptions of spin models, cluster algorithms and the like, without the use of any pre-existing code either in C or in any other programming language (except for the random number generator).

2. ISM.EXE is provided on the accompanying computer disk. See Appendix 3, "Contents of the Disk".

3. This statement must be qualified in the case of the Wolff process, in which a sequence of clusters are created (and the spins in them flipped) but (in contrast to the Swendsen-Wang process) no subsequence of clusters forms a partition of the lattice. In this case one may group the clusters into subsequences whose combined number of spins is close to the number of spins flipped (on average) in one Swendsen-Wang sweep. For further details see Section 1.10.

4. See Section 6.3(iii).

5. Section 2.1 describes how these quantities were measured.

6. All tables referred to in Chapters 2 through 6 may be found in Appendix 5 unless otherwise stated.

7. Table 2.3.1 in Appendix 5 gives the data plotted in Figures 2.3.1 and 2.3.2.

Chapter 3

1. This sense of average is explained at the end of Section 3.2.

2. The correlation length is defined variously as the distance over which spins influence each other or the average size of a cluster of like-valued spins. It tends to infinity at the critical temperature.

3. Tc for this lattice type is 3.641 (see Section 3.3(b)).

4. See Appendix 3, (3)(i), regarding the input file for the simulation in the case of lattice size 20.

5. When measuring the Binder cumulant the errors become larger with larger temperature, as the magnetization tends toward zero.

6. See Appendix 3, (3)(ii), regarding the input file for the simulation in the case of lattice size 8.

7. It will be noted that these estimates do not all fall within the error limits of each other. E.g., the lower error limit of Adler (1983), i.e., 0.221660, is significantly above the upper error limit of Blöte and Kamieniarz (1994), i.e., 0.221653. On the other hand, the three most precise estimates are consistent with each other.

8. Computation time was 18.44 hours.

9. tanh(x) = (ex - e-x) / (ex + e-x)

10. On the 330 MHz PC on which all the simulations described in this work were performed this would require (assuming six temperatures) approximately 260 days of computation.

Chapter 4

1. See Appendix 3, 3(iii), regarding the input file for the simulations with lattice size 10.

2. A cluster of occupied sites is a set of occupied lattice sites such that between any two such sites there is a path consisting of open bonds.

3. Gaps in their deduction were finally filled only in 1981.

4. In an NxN square lattice there are, of course, N2 sites. If k of these are occupied, there are (N2)!/[(N2-k)!k!] possible site configurations. Thus in a 10x10 square lattice with site concentration 0.8 there are over 1020 possible site configurations.

5. In an NxN square lattice with free boundary conditions there are 2N(N-1) bonds. If k of these are open then there are [2N(N-1)]!/{[2N(N-1)-k]!k!} possible configurations of open bonds. In an NxN square lattice with N 3 and with periodic boundary conditions there are 2N2 bonds. If k of these are open then there are (2N2)!/[(2N2-k)!k!] possible configurations of open bonds.

Chapter 5

1. See Appendix 3, 5(i), for the input files which produced these results.

2. Okano et al. (1997, p.738) state: "It is known that the critical points [of the Potts model] locate at Jc = log(1+q)." Taking Tc = 1/Jc and q = 3 we obtain Tc = 1/ln(1+3) = 0.994973.

3. See Appendix 3, 3(iv), for the input data file used to obtain the data for this plot.

Chapter 6

1. Relaxation is the name given to the process whereby a system in an unstable state (e.g. a spin system at non-zero temperature with all spins oriented in one direction) attains an equilibrium state.

2. Re correlation length see footnote 2 in Section 3.1.

3. Discussed in Section 5.1.

4. Strictly speaking the symbol θ denotes only the new critical dynamic exponent, but sometimes here it is used to mean the slope of the log-log plot of magnetization against time for a particular lattice size and initial magnetization.

5. Okano et al.'s graphs (pp. 733 and 739) show the normed magnetization, M(t)/M(0). The simulation program calculates the normed magnetization values but they are not used in the analysis. Not norming the magnetization values does not change the slope of the graph or affect the estimates for θ arrived at here.

6. See Appendix 3, 5(ii), for the input file producing this least-squares fit.

7. In practice A(t) is obtained by averaging over a large number of samples.

8. The value of 0.191(1) obtained by Okano et al. for θ thus implies a value for z of 2.155(5).

9. Error bars are not shown because the standard deviation of the 2nd moment of the magnetization was not calculated.

10. And as usual an average is taken over many samples.

11. This is the result with Glauber dynamics. Metropolis dynamics gave 0.070(2).

12. For the square 3-state Potts model Schülke and Zheng (1995) obtained an estimate for θ of 0.0815(27). For the cubic 3-state Potts model Jaster et al. (1999) obtained an estimate for θ of 0.108(2).

13. APotts(t) can be less than 0, but is always ≥ -1/q.

14. The number of samples used is not stated.

15. With Glauber dynamics. Metropolis gave 0.841(5). The value of 0.075 obtained by Okano et al. for θ implies a value for z of 2.20.

16. Error bars are not shown because the standard deviation of the 2nd moment of the magnetization was not calculated.

17. See Appendix 3, (5)(iii), for the input file producing this least-squares fit.

18. The calculation of Q is quite sensitive to the values of the error bars, and an underestimate of the errors can easily produce a very low Q-value.

19. This is the same as the value calculated by Excel, as shown in Figure 6.4.7.

20. The linear relationship would be z = 2.07(2) + 0.042(8).q, implying a 5-state Potts z of 2.28(6).

Chapter 7

1. A universality class is defined by a set of specific values for critical exponents. Thus any two systems which belong to the same universality class exhibit the same critical behaviour, although the critical temperatures may differ.

2. Okano et al. used 80,000 samples for magnetizations 0.08 and 0.06, and 600,000 for 0.04 and 0.02.

3. Okano et al. do not state the number of samples used in the measurement of autocorrelation and the 2nd moment for the 3-state Potts model.

Chapter 8

* This chapter was added at the suggestion of the examiners during the viva voce examination, which took place 7th September 2000. The examiners were Dr. Geoff Rodgers, Brunel University (external examiner) and Dr Mirko Paskota, University of Derby (internal examiner).

1. The notions of site and bond percolation thresholds are explained in Sections 4.3 and 4.4 and in Appendix 6.

2. Binder (1997, p.518) describes a method for ascertaining the percolation threshold which is analogous to the method of ascertaining the critical temperature using the Binder cumulant (discussed in Section 3.1) in that the percolation threshold is given by the intersection of graphs of "spanning probability" against concentration for various lattice sizes.

3. The Metropolis algorithm is discussed in Section 1.7.

4. This cost function is unlike the Ising Hamiltonian in that it does not involve the concept of "nearest neighbour" spins.

Appendix 4

1. This function calls several auxiliary functions which are given on p.214 and pp.218-219 of Press et al. 1992.

Appendix 7

1. Some authors (e.g. Jain and Lage 1993) speak of a correlation length distinct from both the percolation correlation length and the thermal correlation length, and define it as the harmonic mean of these two.

2. See Appendix 6 for the definition of a spin cluster.

3. Actually Barber does not distinguish between these two types of correlation function, describing Γ(r,T) as "the pair correlation function" (with T > Tc).

4. Note that, since it is impossible for a lattice bond to be an open bond except where it connects two occupied sites, the bond concentration is defined as the number of open bonds divided, not by the number of lattice bonds (open or closed), but rather by the number of possible open bonds, which is the number of pairs of occupied nearest neighbour sites (assuming that a bond is possible only between nearest neighbour sites).

5. As defined in Appendix 6, a spin cluster is a set of spins, all of the same spin value, such that any two spins are connected by a chain of open bonds.

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