Challenges and trends in interacting particle systems - Semantic Scholar

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Two examples. A typical example of an interacting particle system is provided by the contact process. We consider individuals residing at the sites i of.
Challenges and trends in interacting particle systems Christian Maes1 Instituut voor Theoretische Fysica, K.U.Leuven Abstract: Some meditations are presented on the status of Interacting Particle Systems in the light of studies in complex systems. Probabilistic reasoning and a rigorous understanding of large scale dynamics contributes significantly in clarifying various complex issues. (Talk given in Warwick at the IMA conference on Mathematics in the Science of Complex Systems, 18–21 September 2006.)

1. Two examples A typical example of an interacting particle system is provided by the contact process. We consider individuals residing at the sites i of the one-dimensional lattice Z. Each individual can be in two possible states indicated by η(i) ∈ {0, 1}. They can be either healthy η(i) = 0 or infected η(i) = 1. There is a stochastic dynamics in which the state of the system is sequentially updated. It goes as follows. Infected individuals become healthy (they recover) at rate 1. If they are healthy they can get infected and that happens at a rate λ times the number of infected neighbors. In the simplest case we only consider i − 1 and i + 1 as the two neighbors of site i so that the updating rule at i ∈ Z becomes 1 −→ 0,

at rate 1

0 −→ 1,

at rate λ[η(i − 1) + η(i + 1)]

(1.1)

We have here a spatially extended Markov dynamics. The parameter λ ≥ 0 is obviously very important to understand if and when an initially localized infection would spread. There is a further dependence on the graph and on the architecture of connecting neighbors. It turns out that the situation is already very interesting in our onedimensional (nearest neighbor) case. What happens is that for λ > 0 small, all initial infection disappears; there is a unique and trivial stationary distribution concentrated on the state where all individuals are healthy. Yet, as λ grows larger than a critical value λc < +∞, there suddenly appears, first a small and then a larger density of infected individuals. There is thus a nontrivial stationary distribution for λ > λc , in which we see a positive density of infected individuals. More mathematical details and references can be found from the standard references [7, 3]. The reason why people have been interested 1

http://itf.fys.kuleuven.be/^ christ/ 1

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in that contact process is certainly because it can serve to illustrate some important tools and some interesting phenomena. Among the tools there are the coupling method, the use of duality and of monotonicity that are all related to intrinsically probabilistic concepts and that have much wider applications. For the phenomena, we can think of phase transitions and of collective behavior. The appearance of an “infected phase” is an emergent feature which is quite sensational: a small change in the parameter λ (that plays on the level of the individual) can imply the extinction versus the survival of an initial infection (that plays on the level of the society). It is a prototypical example of a complex system to which mathematics has greatly contributed. A second example is known as the kinetic or the stochastic Ising model. From the physics point of view it is more appropriate to use “spin language;” we put η(i) = ±1 (“up” and “down” spin) on the sites i ∈ Z2 of say the square lattice. (A one-dimensional lattice would here not be quite so interesting, see below.) The dynamics is called spin flip, because each transition involves the possible flipping of one spin. When the total configuration is η, the new configuration after a spin flip at the site i will be η i where η i (j) = η(j) when j 6= i and η i (i) = −η(i). Or, in each transition, only the spin at a single site can flip. The rate of such a transition is now slightly more complicated than for the contact process. One first introduces a so called energy function H(η), which must be such that the differences (∆i H)(η) ≡ H(η i ) − H(η) are well-defined for all η ∈ {+, −1}Z , i ∈ Z2 . That difference has the meaning of a relative energy, i.e., the energy difference after the spin at i has flipped. The spin flip rates c(i, η) are defined in terms of them. The rate c(i, η) gives the intensity of making the transition η → η i when the present configuration is η. In the stochastic Ising model one makes sure to satisfy 2

c(i, η) = exp[−β∆i H(η)] c(i, η i )

(1.2)

for some β ≥ 0. That is called the condition of detailed balance (or sometimes, of microscopic reversibility). The left hand-side give the ratio of rates for η → η i versus the reversed η i → η; it is given in terms of an energy difference. As a result, probability measures P for which, locally, P [η] ∝ exp[−βH], are all stationary. These distributions describe thermal equilibrium at inverse temperature β. They are of prime importance as models in statistical mechanics. The dynamics adds an interesting visualization. In fact, the spin flip dynamics provides a method to simulate these equilibrium spin models. A very

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well-known example is the Ising model where the energy H is a sum of nearest neighbor interactions of the form X η(i) η(j) (1.3) H(η) = − |i−j|=1

It has played a crucial role in the unravelling of critical and cooperative behavior. In two and higher dimensions, it shows a phase transition. To understand that effect we must define the equilibrium distribution P on finite volumes ΛN , say squares of side N centered around the origin. The probability to see a configuration η = (η(i), i ∈ ΛN ), is taken to be 1 P [η] = e−βHN (η) Z where Z = ZN (β) is a normalization and HN is the energy according to (1.3), but with the sum restricted to those sites i ∈ ΛN that are inside the square. There are also the sites j that are nearest neighbors of such i, and in fact they can fall outside ΛN . We must therefore impose boundary conditions, that is, tell what are the values of the sites η(j), j ∈ / ΛN . There are two extremal boundary conditions, the all plus and the all minus boundary conditions, with an obvious meaning. An important question is now whether the spin at the origin cares at all, i.e., whether its (marginal) distribution is at all (strongly) influenced by the boundary spins. More precisely, we consider the difference − δN ≡ hη(0)i+ N − hη(0)iN

of the expected spin at the origin between plus and minus boundary conditions. As the square grows with N ↑ +∞, the distance between the origin at the center and the boundaries increases. It is now very interesting that for large but finite β, the difference δN > 0 remains away from zero, uniformly in N ↑ +∞. That is so called long range order, or, the Ising model enjoys a “homeopathic” property: as the boundary recedes to infinity, it still has a non-zero influence, even at infinite distance so to speak. That long range order has much to do with spontaneous magnetization. One could say that the external field vanishes and yet, its influence remains felt and shows up in a strictly positive value for lim hη(0)i+ N > 0 N

The same (with a negative magnetization) is of course true for the minus boundary conditions. There is thus a spontaneous symmetry breaking; while the energy function (1.3) has a plus/minus symmetry, there appear different possible stationary distributions for the stochastic Ising model at large enough β. There is the distribution where the spins are mostly “up,” and there is the distribution where the spins are mostly “down.” Again, as in the contact process, the transition from a unique disordered phase to an ordered phase is sudden, for β > βc

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for some critical βc . The Ising model is therefore again a clear and interesting example of what we could mean by emergent behavior. 2. Standard aspects There are a number of standard features that go with interacting particle systems, as the ones we have been visiting in the previous section. We make a short list. Mostly we have to do with a fixed architecture. It forms the unchanging stage on which particles, spins, individuals or components do their thing. The architecture is mostly taken to be some regular lattice or graph, and it is not changing in time, nor does it have very different features from one place to another. Secondly, the components that make up the system are clearly distinguishable in their local or quasi-local interaction. The dynamical rules of updating are local, in the sense that whatever happens at a certain site or vertex is directly influenced only by its neighborhood as defined by the lattice or by the graph. As a consequence, if we look to the process as a space-time distribution, then we are able to find it in the exponential class, with a well-defined action. Thirdly, there has been a very strong influence from physics, and in particular, from equilibrium statistical mechanics. The inspiration and the constructions are often directly taken from physics problems. That includes the set-up `and the type of questions one asks. In particular, the condition of detailed balance, like in (1.2) is a kind of standard reference. Finally, and perhaps somewhat more vaguely, the attempt is more often to create complex and rich behavior from simple rules. There is the tendency to restrict the complexity of the updating; one is happy to find complex behavior emerging from more microscopic and conceptually more simple rules. Very complicated graphs, with involved state spaces and with difficult updating rules are very much avoided, even if only for simulation processes. 3. New features In more recent times, one can find studies on new types of interacting particle systems that somehow deviate from the standard set-up as described in the previous section. A first new aspect is that we see more interacting particle systems involving a variable or very complicated architecture. Moreover, sometimes the graph even takes part in the dynamics and provides feedback

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to the interacting components. The motivation and the inspiration appears to come from various sides. In biological processes, the tracks and the pathways are very much function of the activity. Also in computer science the grammar of a language enters as an important player. Today there is also a lot of interest on defining models on random graphs and on small worlds. In physics, random space-time in which and through which particles interact are making models of quantum gravity. A second new feature is that there is less hesitation to make the processes rather singular or effectively nonlocal. Again the motivation can come from the high connectivity in certain graphs, such as computer networks or in biological trees. Power law behavior is omnipresent and the analysis gets a very strong statistical flavor. Instead of measuring local or simple functions of the activity, one can be interested in global and/or sometimes very nonlocal events. As far as the physics influence goes, much more emphasis is put on outspoken nonequilibrium issues. Different constraints will replace the detailed balance condition, and in fact, these are sometimes inspired by economic or ecological realities. One notion that is more important in socio-economic models is that of Nash-equilibria and of stochastic stability. The interacting particles are agents having at their disposal certain strategies. Finally, more realistic and much more complicated updating rules are no longer avoided and it is more accepted that biological modelling in particular benefits from staying close enough to the true and diverse nature of the entities involved. In what follows, we will give some examples and more explanation to the above. We first present a somewhat more general connection with the study of complex systems. 4. Complexity The presence in more recent times of “complex systems” as a distinct scientific domain with its own goals and methods has brought some commotion in wider scientific circles. That does not only have science-political causes. A new direction has its initial successes and generates its journals, the creation of new centers and of new institutes with corresponding cash flows, and driving young and ambitious researchers. Yet, at the end of the day there is the unavoidable question: what is it about? What is the scientific subject and what themes are perhaps relevant for the larger scientific community or even for society?

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Clearly “complex systems” have been studied since the time people started to look at each other. It is fair to say however that a widely observed discipline of “complex systems” has started well in the years 1970 knowing its first great breakthrough in the 1980’s. That start has gone with a number of other developments. There was the ever growing power and availability of fast computers. That explosion in computer power, the establishment of computational centers and the spreading of reliable numerical algorithms have been very instrumental in the developing of “complex systems.” Methods of simulation and of visualization have opened, quite literally, new worlds for many researchers. A second also essential element has been the renewed interest in modern mechanics. Today we speak of the theory of dynamical systems. In the 1970’s important steps have been taken or have been remembered in the mathematics of dynamical systems. Sometimes we call it the theory of chaos or of nonlinear phenomena. Nonlinear systems are for many today almost synonymous with “complex systems.” A third tendency that often runs in parallel with the development of “complex systems” is that of interdisciplinarity. To be and to work in an interdisciplinary environment is thought to be methodologically useful and relevant. There is however more. Much of what drives “complex systems” comes from beyond one specific standard discipline, that is true, but there was the important discovery of universality in certain dynamical behavior. Some dynamical systems, networks etc. have shown features that remain true outside the strict initial context. One can classify behavior and universal features are the basis for very diverse realizations. Models also remained useful in domains as psychology, economy or in the social sciences. Natural scientists thought to discover new “markets” for their research. Finally, much of the development of “complex systems” was carried by statistical mechanics. It is mostly there that “complexity” has been cherished. An obvious reason is that statistical mechanics is a transfermechanics. It wants to bridge the gap between the complex microscopic world and the mesoscopic and macroscopic scales of description. An important notion is that of emergence, how, at a certain scale, new and unexpected behavior can arise. Phase transitions in physics are important examples. They have been mostly studied in equilibrium statistical mechanics. Much richer however is still the world of nonequilibrium phenomena, and it is there that the science of complexity found many connections. “Complex systems” indeed deals with phenomena that would be exceedingly complicated to solve exactly or for which no simple explicit or

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analytical solutions can be written out. Instead one tries a more qualitative analysis, making it somehow visible. If one considers systems with many, sometimes a huge number of interacting components, then specific methods are indeed needed. Not only does that require the appropriate tools; there opens a new world that could not have been guessed. It has always been one of the motivations of the study of interacting particle systems: the search for new phenomena. We have mentioned phase transitions and critical phenomena, but there are less traditional themes like the notion of self-organization. That opens yet a more sensational programme. It deals with the organization or creation of stable or robust characteristics. The system organizes itself apparently in complex patterns and structures, generically and without the exact tuning of some parameters. The self-organization manifests itself in spatial or temporal behavior but it could also become visible in the specific response to some action. The time-evolution of complex systems, that of an interacting particle system, can become rather unusual. Turbulence, that last great unsolved problem of classical physics, shows an exceptionally complex behavior. Dissipation goes hand in hand with the creation and the stabilization of schemes in which the system can “live” far from equilibrium. In all these problems, some leading principles (and also slogans) have entered. We have mentioned the ideas of universality and of emergence, of transfer between scales and of complications because of the number of components. Yet, there remains today the question for unifying schemes and theories. “Complex systems” will very much continue to benefit from clear mathematics and from rigorous analysis of “simple” complex systems. A new kind of science is not made by a great variety of examples but by providing unifying explanations and by developing the appropriate methods for dealing with nature. That is perhaps the greatest challenge — to help solve problems. 5. Examples 5.1. Nonlocal Markov processes. By now there is an enormous literature on the paradigm of self-organized criticality. Sandpiles are there the easiest models, [2]. There have been a continuous interest in simulating these models in all possible variants. That has been giving interesting results, but there is still a great deal of questions that require serious mathematical thought. Even some elementary questions are challenging us. Take a square V ⊂ Z2 and assign to each site a height hi = 1, 2, 3, 4. The dynamics of the standard abelian sandpile goes as follows. Choose a site i ∈ V ad random. Add one unit to its height sending hi 7→ h0i = hi + 1. There are two possibilities, Either h0i ∈ {1, 2, 3, 4}, and then we take h0i as the new height at site i. Or, h0i ≥ 5, in which case we start

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the toppling operation. By toppling we mean that the site gives one unit to each of its four neighbors and each neighbor repeats the same procedure: when the height is there above 4, it redistributes 4 units to its 4 neighbors. If such a neighbor is outside V , then that unit is lost for ever. All that toppling is continued until all heights at all sites of V are again in {1, 2, 3, 4}. One can easily perceive first questions. They ask for mathematical clarifications, such as whether the order of toppling matters, whether the toppling will ever stop, whether all height configurations are connected in one recurrent class etc. In fact, the answers are known and have given rise to some interesting algebra, see [4], which by itself has made the subject of sandpiles much more fascinating for all. From the description above, it should be clear that the addition of one unit to some site, can have (literally) a far reaching consequence, [9]. How far that perturbation reaches, depends on the density. It is as if the diffusion constant has a singularity as a function of the density (here height) — the unit gets transported through toppling via “high” sites. In other words, the range of interaction depends so to speak on the configuration. Or, thinking of the toppling as an instantaneous process, we have an interacting particle system which is highly nonlocal. In mathematical terms, it is not Feller (not leaving the set of continuous functions globally invariant). Such processes have unusual and, from a standard point of view perhaps horrifying properties. As an example of a simple non-Fellerian process, we can consider a Markov process on {0, 1}Z in which for each site 0 → 1 with rate 1. If all sites are at 1, they change (together) to all 0 at rate 1. As a consequence, when starting from a translation invariant distribution, the measure at time t becomes more and more equal to the delta-measure at all 1 (weak convergence), but that delta-measure at all 1 is not itself invariant. In fact, there is no invariant measure here. See [10] for further details and for some more examples of unusual “freezing” transitions. Sandpiles are not so pathological as in the previous example. In particular, we can characterize their invariant distribution (for finite volume). We have however no mathematical result which directly deals with the most important issue, that of establishing self-organized criticality. One needs to show there that the size of the avalanches (where the topplings occur) has a power-law statistics. Moreover, we need reliable algorithms to create also temporal criticality, see [1] for some attempts.

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5.2. Nonlinear Markov processes. All Markov processes, including interacting particle systems, have the following general feature: the evolution on the level of measures is linear. The set of probability measures is some simplex and is left invariant by the linear operation that defines the process. It is reflected in the linearity of the so called Master equation. That is of course very natural, and it reflects the idea that the measure at each time is made from drawing states according to the measure at the previous time, after which transitions are made with some probability (the updating rule). One can however imagine a much more complicated procedure, which gives almost no immediate results. To be specific, we take a simple set-up. Consider the state space K = {0, 1} for our stochastic process. Probability measures on K are completely determined by one number p ∈ [0, 1] (the probability of 1). Let us now assume that to each such p there is associated a transition (stochastic) matrix Tp . The stochastic process now runs as follows. We start from a given measure µ0 at time zero and we draw a state x ∈ K as our initial state x0 = x. The µ0 also determines a number p0 and hence a stochastic matrix Tp0 . That matrix contains the transition probabilities as its elements. In particular we can read the transition probability tp0 (y|x). We use it to draw x1 , the state at time 1. But we also get a new measure µ1 at time 1. That determines a number p1 and a matrix Tp1 . That will now be used to draw x3 given x2 . It is not hard to see that this strange process is not linear in the evolution of its measures; it is a nonlinear Markov process. Nonlinear Markov processes have actually appeared in recent examples of interacting particle systems. One recent source is in queueing systems,[12]. We give an example by A. Toom, [14] where the nonlinearity arises from a variable architecture. We are again with the same states as in the one-dimensional contact process, η ∈ {0, 1}Z . The first part of the updating is standard: the updating takes 1 → 0 at rate β, at each site of the lattice. Here comes however the unusual twist: when somewhere (η(i − 1), η(i)) = (0, 1), then the sites i − 1 and i are removed from the system at rate α. The previous sentence is easy to state informally but it is not so easy to incorporate it in a precise manner. Some mathematical precision is needed just to understand what is meant and what is going on. In particular, the very definition makes only some sense when doing it really on the infinite lattice but then we must still see for what initial measures. One can in fact start from all 1’s. Toom shows that for large enough β, there is convergence to all 0. For small enough β, the

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density of 0’s remains always small, and there is a real discontinuity in β (in contrast with the one-dimensional contact process). 5.3. Nonequilibrium issues. Since some years now there is a strong revival in attempts of constructing a nonequilibrium statistical mechanics. Obviously, the theory of interacting particle systems is a wonderful source for many effects and examples. There is the long tradition of stochastic lattice gases, e.g. to study hydrodynamic behavior, [13, 6]. There is the more recent addition of illustrating (more general) nonequilibrium fluctuation theory, [8]. There are however also new models that highlight some nonequilibrium effects, that use the context of interacting particle systems and still show some relevant departures. We have in mind the ratchet effect as e.g. realized in Brownian motors, [11]. Applications have been suggested to the theory of molecular motors. There are also analogues and suggested interpretations in game theory and in economy, where they are related to the so called Parrondo games. The set-up is at first very similar to that for stochastic lattice gases. The main difference is however that the potential landscape changes in time. One can imagine a random walker on a circle in some potential, that flashes between two values. That time-dependence breaks the condition of detailed balance and there can arise some net current. That current is not directly caused by some external force or by some boundary driving. It is therefore not immediately clear what would be the direction of that current. One speaks about fluctuations that drive the current. While many examples have been studied, it is fair to say that there does not really exist a unifying understanding of that ratchet effect, [5]. It again appears that specialized mathematical methods as those developed within the context of interacting particle systems will help to solve these questions. References [1] M. Baiesi and C. Maes, Realistic time correlations in sandpiles, Europhys. Lett. 75, 413–419 (2006). [2] P. Bak, K. Tang and K. Wiesenfeld: Self-Organized Criticality, Phys. Rev. A 38, 364–374 (1988). [3] Mu Fa Chen, From Markov Chains to Non-equilibrium Particle Systems, World Scientific, 2004. [4] D. Dhar: The Abelian Sandpiles and Related Models, Physica A 263, 4–25 (1999). [5] W. De Roeck and C. Maes, Symmetries of the ratchet current, arhive cond-mat/0610369. [6] C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems, Springer-Verlag, Berlin, 1999. [7] T.M. Liggett, Interacting Particle Systems, Springer-Verlag, 1985.

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[8] C. Maes, K. Netoˇcn´ y and B. Shergelashvili, An update on stochastic lattice gases: nonequilibrium issues. Lecture given at the Prague summer school on mathematical statistical mechanics, september 2006. [9] C. Maes, F Redig and E. Saada, Abelian Sandpile Models in Infinite Volume, Sankhya, the Indian Journal of Statistics 67, 634–661 (2005). [10] C. Maes, F Redig and E. Saada, Freezing transitions in non-Fellerian particle systems to appear in J. Stat. Phys. (2006), arXiv math.PR/0512525. [11] P. Reimann, Brownian Motors: Noisy Transport far from Equilibrium, Phys. Rep. 361, 57 (2002). [12] A. Rybko and S.Shlosman, Poisson Hypothesis for information networks (A study in non-linear Markov processes), math-ph/0303010, Moscow Mathematical Journal 5, 2005. [13] H. Spohn, Large Scale Dynamics of Interacting Particles, Springer Verlag, Heidelberg (1991). [14] A. Toom: Non-ergodicity in a 1-D particle process with variable length, J. Stat. Phys. 115, 895–924 (2004).