Modern classification of phase transitions –
discontinuous and continuous phase transitions
In nature there are two different kinds of phase transitions (Kadanoff, 2000).
The first kind, called discontinuous or first order phase transition, involves a discontinuous change in some intensive thermodynamic quantity. Two basic examples of the transition are:
evaporating water which is related to liquid → gas
transformation seen when one crosses an arbitrary point (T,p) ≠ (TC,pC) of the evaporation curve shown in the phase diagram (see figure). In this case, the discontinuity is observed in the density of water (liquid and vapour water, respectively) expressed as a function of temperature T or pressure p.
discontinuous jump of magnetization in a simple magnet observed below the Curie temperature TC, in the vicinity of vanishing external field |H|=0, when one continuously changes the field H from positive to negative values (see figure).
In both cases, at some critical temperature TC the discontinuities vanish, and the phase transition changes its character. At this special point, we say there is a continuous or second order phase transition.
At the moment let us explain that although, due to accompanying discontinuities, first order transitions may seem more dramatic in consequences than second order transitions, below we argue that due to the whole spectra of accompanying critical phenomena the latter are, in fact, more unpredictable and stormy.
In order to understand this unusual character of continuous phase transitions let us consider what happens when one moves along the discontinuous transition curve towards its critical point TC:
At the beginning let us concentrate on water, which for parameters (T,p) positioned
at the evaporation curve is a mixture of liquid water and vapour bubbles. Analysing sizes of these bubbles (i.e. their radius r or volume V) one would see that the correspond-ing probability distributions P(r) or P(V) characterizing the mixture, change from exponential (outside the critical point) to power-law (exactly at the critical point). The noticed scale-freeness, resulting from self-similarity of the system at
the critical point,has very serious consequences. Namely, due to dynamical character of the system, it implies that even small bubbles may grow to very large sizes. The last property exemplifies the most significant feature of continuous phase transitions which states: At critical points all fluctuations, regardless of
how minor, have to be taken into account, as they may have significant effects on the whole system. That is why physicists say that in the vicinity of critical points the considered
systems are
characterized by enormously large susceptibilities to external forces.
A similar analysis, such as the one described above, may be also performed for magnets. Thus, moving along the unique coexistence curve in the phase diagram of a simple magnet, and observing sizes n of magnetic domains, one would see that the corresponding distributions P(n) change their character from exponential to scale-free, analogously as it was described in the case of evaporating water.
Overview of critical phenomena in complex networks
Ising model
Ising model represents the best known, and the simplest model of a magnet. It is defined by:
the lattice or network on which the spins
si=±1 are positioned (e.g. two-dimensional square lattice or random network with a given degree distribution P(k)), and
by the interactions which favour the same alignment of the neighbouring spins.
At the critical temperature TC , and at zero applied magnetic field a magnet exhibits continuous phase transition, which corresponds to the emergence of the spontaneous magnetization M=N<s>. In regular lattices, the nonzero value of the average spin <s>, that occurs below TCrepresents order parameter characterising the considered phase transition.
In the context of Ising model defined on random uncorrelated networks with a given degree distribution P(k), it was shown (by mean field methods) (Bianconi 2002, Dorogovtsev 2002) that the critical temperature equals TC = <k2>/<k> , where <k> and <k2> stand for the first and the second moment of P(k), whereas the order parameter is given by the weighted spin <ks>/<k>, which reduces to <s> only for regular and classical random graphs.
The formulas describing the critical temperature, and the order parameter in complex networks have a very serious theoretical and practical consequences - especially in relation to scale-free networks P(k)~k-γ with the characteristic exponent 2<γ<3. The consequences are due to existence of hubs, i.e. highly connected nodes, which makes the critical temperature being a size-dependent parameter TC(N), and causes the overall network state (its ordering) being dependent on states of only a few nodes (Aleksiejuk et al.).
Percolation represents the simplest model of a disordered system
(Stauffer, 1992). In the most popular realization, it is defined on a square lattice, where each site / bond is randomly occupied with probability p, or empty with probability 1-p (see the attached applet). It is obvious that at low concentrations p, only small clusters of occupied sites / bonds exist. On the other hand, at large p values many paths consisting of occupied system entities, and spanning all over the system are present. The above suggests, that there exists the critical concentration pCwhich separates a phase of finite clusters (p< pC) from a phase where the giant / percolating cluster is present (p> pC).
In fact, percolation is an example of the continuous phase transition, which defines its own universality class, with well-established critical exponents characterizing behaviour of the considered system (including its order parameter - here given by the size of the giant cluster) in the vicinity of thepercolation thresholdpC.
In complex networks the phenomenon of percolation has been studied by several authors resulting in a number of important results of both theoretical and practical relevance. In particular, it was shown that the percolation threshold in uncorrelated random networks with arbitrary degree distribution P(k) is given by the simple formula <k2>=2<k> relating the first two moments of P(k) (Callaway et al. 2000, Newman et al. 2001, Fronczak et al. 2005). Moreover it was argued that the concept of percolation allows one to explore such important issues as: resilience of complex networks to random errors, as well as intentional attacks (Albert et al. 2000, Cohen et al. 2000a, Cohen et al. 2000b), and may very be very helpful in analysis of different transport phenomena taking place on the top of networks (Dorogovtsev et al. 2001).
Critical behaviour always arises from cooperative phenomena (with Bose-Einstein condensation being the only exception), which on their own turn result from the repeated interactions between “microscopic” elements of the considered systems. The above statement allows to expect that a number of different node-node interaction patterns must exist, which,
for a given level noise (i.e. temperature) should cause structural phase transitions in the considered networks (Palla et al. 2004).
A detailed reasoning behind this general thinking is the following: Since most of real networks exists for very specific functional reasons, their structure must reflect a certain optimisation process. The process can be viewed as an optimisation of a certain utility function (energy) characterizing the considered network given a well-established external conditions (temperature). Rephrasing what has been said above: In the case of reasonable utility function, the optimisation process (realized by rewiring of edges) must reveal existence of such external conditions (critical points TC), at which one can observe a complete reorganization in the network structure.
An example of such a reasonable network utility function has
been recently considered by (Biely & Thurner, 2005). The authors have assumed that individual nodes increase / decrease their utility by linking to nodes with higher / lower degree than their own. They have observed existence of the critical temperature TC, where total utility and n
etwork structure undergo radical changes. They have also shown, that in the vicinity of the critical point the considered networks are characterized by scale-free degree distributions, and exhibit a complex hierarchical topology expressed by nontrivial dependence of the local clustering coefficient c(k)~1/k.
Basing on the concepts of the Landau theory for continuous phase transitions, Golshev et al. have developed a phenomenological theory of critical phenomena in uncorrelated networks with an arbitrary distribution of connections P(k). In their paper, the authors have answered the basic questions related to the critical behaviour of complex networks, like: why critical phenomena in networks differ so much from those in regular lattices, and why all investigated models (including spin models, and percolation) demonstrate similar behaviour when the second moment of P(k) diverges. The authors have also shown that the critical behaviour in complex networks has a universal character, which is only determined by the structure of a network (i.e. its degree distribution P(k)), and the symmetry underlying a considered model.
