Über die Autorin bzw. den Autor

Ricard V. Sole is research professor and head of the Complex Systems Lab at Pompeu Fabra University and external professor at the Santa Fe Institute. He is the coauthor of "Signs of Life" (Basic) and "Self-Organization in Complex Ecosystems" (Princeton).

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"This ambitious book provides an elegant and much-needed synthesis to many of the ideas that have come to define the field of complex systems and their applications to nature and society. It makes an important contribution to the field, especially for researchers and students looking for an overview of the literature and entry points for research."--Luis Bettencourt, Los Alamos National Laboratory and the Santa Fe Institute

"This clear and easy-to-follow book is a valuable compilation of systems showing phase transition phenomena that have become more and more important in understanding natural and man-made complex systems. It is a useful addition to the already existing literature."--Stefan Thurner, Medical University of Vienna

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PHASE TRANSITIONS

PRINCETON UNIVERSITY PRESS

Contents

Preface..............................................xi1. Phase Changes.....................................12. Stability and Instability.........................253. Bifurcations and Catastrophes.....................374. Percolation.......................................535. Random Graphs.....................................636. Life Origins......................................707. Virus Dynamics....................................788. Cell Structure....................................919. Epidemic Spreading................................9910. Gene Networks....................................10911. Cancer Dynamics..................................12012. Ecological Shifts................................13413. Information and Traffic Jams.....................14814. Collective Intelligence..........................15715. Language.........................................16716. Social Collapse..................................179References...........................................189Index................................................213

Chapter One

1.1 Complexity

By tracking the history of life and society, we find much evidence of deep changes in life forms, ecosystems, and civilizations. Human history is marked by crucial events such as the discovery of the New World in 1492, which marked a large-scale transformation of earth's ecology, economics, and culture (Fernandez-Armesto 2009). Within the context of biological change, externally driven events such as asteroid impacts have also triggered ecosystem-scale changes that deeply modified the course of evolution. One might easily conclude from these examples that deep qualitative changes are always associated with unexpected, rare events. However, such intuition might be wrong. Take for example what happened around six thousand years ago in the north of Africa, where the largest desert on our planet is now located: the Sahara. At that time, this area was wet, covered in vegetation and rivers, and large mammals inhabited the region. Human settlements emerged and developed. There are multiple remains of that so-called Green Sahara, including fossil bones and river beds. The process of desertification was initially slow, when retreating rains changed the local climate. However, although such changes were gradual, at some point the ecosystem collapsed quickly. The green Sahara became a desert.

Transitions between alternate states have been described in the context of ecology (Scheffer 2009) and also in other types of systems, including social ones. Complex systems all display these types of phenomena (at least as potential scenarios). But transitions can also affect, sometimes dramatically, molecular patterns of gene activity within cells, behavioral patterns of collective exploration in ants, and the success or failure of cancer or epidemics to propagate (Solé et al. 1996; Solé and Goodwin 2001). When a given parameter is tuned and crosses a threshold, we observe change in a system's organization or dynamics. We will refer to these different patterns of organization as phases. The study of complexity is, to a large extent, a search for the principles pervading self-organized, emergent phenomena and defining its potential phases (Anderson 1972; Haken 1977; Nicolis and Prigogine 1977, 1989; Casti 1992a, b; Kauffman 1993; Cowan et al. 1994; Gell-Mann 1994; Coveney and Highfield 1995; Holland 1998; Solé and Goodwin 2001; Vicsek 2001; Mikhailov and Calenbuhr 2002; Morowitz 2002; Sornette 2004; Mitchell 2009). Such transition phenomena are collective by nature and result from interactions taking place among many interacting units. These can be proteins, neurons, species, or computers (to name just a few).

In physics, phase changes are often tied to changes between order and disorder as temperature is tuned (Stanley 1975; Binney et al. 1992; Chaikin and Lubensky 2001). Such phase transitions typically imply the existence of a change in the internal symmetry of the components and are defined among the three basic types of phases shown in figure 1.1. An example of such transition takes place between a fluid state, either liquid or gas, and a crystalline solid. The first phase deals with randomly arranged atoms, and all points inside the liquid or the gas display the same properties. In a regular (crystalline) solid, atoms are placed in the nodes of a regular lattice. In the gas phase (at high temperature) kinetic energy dominates the movement of particles and the resulting state is homogeneous and isotropic. All points are equivalent, the density is uniform, and there are essentially no correlations among molecules. In the liquid phase, although still homogeneous, short-distance interactions between molecules leads to short-range correlations and a higher density. Density is actually the fundamental difference distinguishing these two phases. Finally, the ordered arrangement observed at the solid phase is clearly different in terms of pure geometry. Molecules are now distributed in a highly regular way. Crystals are much less homogeneous than a liquid and thus exhibit less symmetry.

However, beyond the standard examples of thermodynamic transitions between these three phases, there is a whole universe. Matter, in particular, can be organized in multiple fashions, and this is specially true when dealing with so-called soft matter. But the existence of different qualitative forms of macroscopic organization can be observed in very different contexts. Many are far removed from the standard examples of physics and chemistry. And yet, some commonalities arise.

1.2 Phase Diagrams

Phase changes are well known in physics: boiling or freezing are just two examples of changes of phase where the nature of the basic components is not changed. The standard approach of thermodynamics explores these changes by defining (when possible) the so-called equation of state (Fermi 1953), which is a mathematical expression describing the existing relations among a set of state variables (i.e., variables defining the state of a system). For an ideal gas, when no interactions among molecules need be taken into account, the equation reads

pV = nRT (1.1)

where p, V, and T indicate pressure, volume, and temperature, respectively. Here R is the so-called gas constant (the same for all gases) and n the amount of substance (in number of moles). This equation is valid for a pure substance, and as we can see, it establishes a well-defined mathematical relation between p, V, T, and n. Given this equation, only three independent variables are at work (since the fourth is directly determined through the state equation). From this expression, we can plot, for example, pressure as a function of V and T:

p(V, T ) = nRT/V (1.2)

which describes a continuous surface, displayed in figure 1.2a. For a given amount of perfect gas n, each point on this surface defines the only possible states that can be observed. Using this picture, we can consider several special situations where a given variable, such as temperature, is fixed while the other two can be changed. Fixing a given temperature T₁, we can see, for example, that pressure decays with volume following an inverse law, that is, p(V; T₁) = nRT₁/V, which allows defining an isothermal process as the one moving on this line. The curve is called an isotherm. By using different temperatures, we can generate different isotherms (which are in fact cross sections of the previous surface). Similarly, we can fix the volume and define another set of curves, now given by p(T) = BT with B = nR/V. The important idea here is that all possible states are defined by the equation of state and that in this case all possible changes are continuous.

The previous equations are valid when we consider a very diluted gas at high temperature. However, in the real world, transitions between different macroscopic patterns of organization can emerge out of molecular interactions (figure 1.2b). Different phases are associated with different types of internal order and for example when temperature is lowered, systems become more ordered. Such increasing order results from a dominance of cohesion forces over thermal motion. Different combinations of temperature and pressure are compatible with different phases. An example is shown in figure 1.3, where the phase diagram for water is displayed involving liquid, solid, and gas phases. Two given phases are separated by a curve, and crossing one of these curves implies a sharp change in the properties of the system. We say that a first order transition occurs and, in this case, each phase involves a given type of organization though no coexistence between phases is allowed. The melting of a solid (such as ice) or the boiling of a liquid (such as water) is a daily example, where both solid and liquid are present at the melting point.

There is a special point in the previous diagram that appears when we follow the liquid-gas boundary curve and defines its limit. This so-called critical point describes a situation where there is in fact no distinction between the two phases. Moreover, we can see that the lack of a boundary beyond the critical point makes possible continuous movement from one phase to the other, provided that we follow the appropriate path (such as the one indicated in figure 1.3 with a dashed curve). The presence of this point has a crucial relevance in understanding the nature and dynamics of many natural and social phenomena.

1.3 Interactions Make a Difference

Phase transitions have been shown to occur in many different contexts (Chaikin and Lubensky 1995; Stanley et al. 2000; Solé and Goodwin 2001). These include physical, chemical, biological, and even social systems. In figure 1.4 we illustrate several examples, including those from molecular, behavioral, and cellular biology and cognitive studies. These are systems spanning many orders of magnitude in their spatial embodiment, and the nature of the transitions is very different in terms of its functional and evolutionary relevance, but all of them have been described by means of simple models.

Proteins exhibit two basic classes of spatial organization. Either they are folded in three-dimensional space or remain unfolded (as a more or less linear chain). The change from one state to the other (a–b) takes place under critical conditions (temperature or even the presence of other molecules). Our second example deals with a special and important group of molecules, so-called amphiphiles, which possess both hydrophilic and hydrophobic properties. Typically, there is a hydrophilic head part (see figure 1.4c) that gets in touch with water molecules, whereas the opposing side, the so-called hydrophobic tail, avoids interacting with water. As a consequence, sets of amphiphilic molecules will tend to form self-organized, spatial structures that minimize the energy of interaction (Evans and Wennerström 1999; Mouritsen 2005). By depending on the relative concentrations of water and amphiphiles, different stable configurations will be observed, defining well-defined phases (such as layers or vesicles).

Our third and fourth examples involve behavioral patterns of interactions among individuals within groups of animals, as these occur with a fish school (figure 1.4d) or, at a lower scale of organization, among cells in a cell culture (figure 1.4e). In the first case, interactions involve repulsion, attraction, the tendency to maintain the same movement as neighboring individuals or to remain isolated. This is easily modeled using computer simulations and mathematical models. By tuning the appropriate parameters, we can observe different types of collective motion and colony-level patterns (Vicsek et al. 1995; Toner and Tu 1998; Czirok et al. 1999; Camazine et al. 2001; Mikhailov and Calenbuhr 2002; Theraulaz et al. 2003; Schweitzer 2003; Sumpter 2006; Garnier et al. 2007; Gautrais et al. 2008). Growing bacterial cell populations in a Petri dish with variable concentrations of a critical nutrient also demonstrate that different forms of colony organization emerge once given thresholds are crossed (Ben Jacob 2003; Ben Jacob et al. 2004; Kawasaki et al. 1997; Matushita et al. 1998; Eiha et al. 2002; Wakano et al. 2003). The final case (figure 1.4f) involves cognitive responses to ambiguous figures (Attneave 1971) and a dynamical example of transitions among alternate states (Ditzinger and Haken 1989). Our brain detects one of the figures (the two faces) or the other (the vase), and both compete for the brain's attention, which alternates between the two "phases" (Kleinschmidt et al. 1998).

In this book we present several examples of phase transitions in very different systems, from genes and ecosystems to insect colonies or societies. Many other examples can be mentioned, including chemical instabilities (Nicolis et al. 1976; Feinn and Ortoleva 1977; Turner 1977; Nitzan 1978), growing surfaces (Barabási and Stanley 1995), brain dynamics (Fuchs et al. 1992; Jirsa et al. 1994; Kelso 1995; Haken 1996, 2002, 2006; Steyn-Ross and Steyn-Ross 2010), heart rate change (Kiyono et al. 2005), immunology (Perelson 1989; de Boer 1989; Tomé and Drugowich 1996; Perelson and Weisbuch 1997; Segel 1998), galaxy formation (Schulman and Seiden 1981, 1986) cosmological evolution (Guth 1999; Linde 1994), computation (Landauer 1961; Huberman and Hogg 1987; Langton 1990; Monasson et al. 1999; Moore and Mertens 2009), language acquisition (Corominas-Murtra et al. 2010), evolution of genetic codes (Tlusty 2007), politics and opinion formation (Lewenstein et al. 1992; Kacperski and Holyst 1996; Weidlich 2000; Schweitzer 2002, 2003; Buchanan 2007), game theory (Szabo and Hauert 2003; Helbing and Yu 2009; Helbing and Lozano 2010), and economic behavior (Krugman 1996; Arthur 1994, 1997; Ball, 2008; Haldane and May 2011) to cite just a few.

1.4 The Ising Model: From Micro to Macro

In the previous sections we used the term critical point to describe the presence of a very narrow transition domain separating two well-defined phases, which are characterized by distinct macroscopic properties that are ultimately linked to changes in the nature of microscopic interactions among the basic units. A critical phase transition is characterized by some order parameter f(µ) that depends on some external control parameter µ (such as temperature) that can be continuously varied. In critical transitions, f varies continuously at µ_c (where it takes a zero value) but the derivatives of f are discontinuous at criticality. For the so-called first-order transitions (such as the water-ice phase change) there is a discontinuous jump in f at the critical point.

Although it might seem very difficult to design a microscopic model able to provide insight into how phase transitions occur, it turns out that great insight has been achieved by using extremely simplified models of reality. In this section we introduce the most popular model of a phase transition: the Ising model (Brush 1967; Stanley 1975; Montroll 1981; Bruce and Wallace 1989; Goldenfeld 1992; Binney et al. 1993; Christensen and Moloney 2005). Although this is a model of a physical system, it has been used in other contexts, including those of membrane receptors (Duke and Bray 1999), financial markets (Bornhodlt and Wagner 2002; Sieczka and Holyst 2008), ecology (Katori et al. 1998; Schlicht and Iwasa 2004), and social systems (Stauffer 2008). Early proposed as a simple model of critical behavior, it was soon realized that it provides a powerful framework for understanding different phase transitions using a small amount of fundamental features.

The Ising model can be easily implemented using a computer simulation in two dimensions. We start from a square lattice involving L x L sites. Each site is occupied by a spin having just two possible states: -1 (down) and +1 (up). These states can be understood as microscopic magnets (iron atoms) pointing either north or south. The total magnetization M(T ) for a given temperature T is simply the sum M(T) = (1/N) [summation]^N_I = 1 S_i where N = L². Iron atoms have a natural tendency to align with their neighboring atoms in the same direction. If a "down" atom is surrounded by "up" neighbors, it will tend to adopt the same "up" state. The final state would be a lattice with only "up" or "down" units. This defines the ordered phase, where the magnetization either takes the value M = 1 or M = -1. The system tries to minimize the energy, the so-called Hamiltonian:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

where J > 0 is a coupling constant (the strength of the local interactions) and [summation]_{i, j} indicates sum over nearest neighbors.

The model is based on the following observation. If we heat a piece of iron (a so-called ferromagnet) to high temperature, then no magnetic attraction is observed. This is due to the fact that thermal perturbations disrupt atomic interactions by flipping single atoms irrespective of the state of their neighbors. If the applied temperature is high enough, then the atoms will acquire random configurations, and the global magnetization will be zero. This defines the so-called disordered phase. The problem thus involves a conflict between two tendencies: the first toward order, associated to the coupling between nearest atoms, and the second toward disorder, due to external noise.

(Continues...)

Excerpted from PHASE TRANSITIONSby Ricard V. Solé Copyright © 2011 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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