Semester Project:

Statistical Mechanics of Nonequilibrium Systems

Survey of Applications to Urban Systems

Maximilian Thess,Institute of Theoretical Physics, TU Berlin

June 27, 2012

Abstract

Cities constitute an exciting object of study from the perspective ofstatistical mechanics and nonlinear dynamics. First a number of applica-tions of statistical mechanics concepts in the simulation of urban systemsin the literature is surveyed. Subsequently numerical simulations of ancellular automaton reaction-diffusion urban growth model are carried outon the urban clusters of Europe in 2005.

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Contents

1 Introduction 3

2 Literature Review 32.1 Reaction diffusion systems . . . . . . . . . . . . . . . . . . . . . . 32.2 Urban Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Structure and Growth . . . . . . . . . . . . . . . . . . . . 62.2.2 Structure and Optimization . . . . . . . . . . . . . . . . . 7

2.3 Thoughts on a Connection to Statistical Mechanics . . . . . . . . 9

3 Numerical Simulations of a Cellular Automaton Model 93.1 Data and Setup of Simulation . . . . . . . . . . . . . . . . . . . . 103.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

A MATLAB Script for Cellular Automaton 14

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1 Introduction

Cities constitute an exciting object of study both from a physics as well as a so-cial science perspective. While this sounds obvious for the social sciences wherea number of disciplines such as urban planning, urban economics, human geog-raphy or history engage in the description and development of theories muchless is known about activities of physicists in that field. In the context of sta-tistical physics and nonlinear dynamics (more recently summarized under theumbrella term complex systems) urban systems have been investigated for quitesome time. Concepts and tools familiar to physicists such as self-organization,entropy, reaction-diffusion systems, phase-transitions, fractals and cellular au-tomata are used. The goal of this project is to illustrate through a number ofstudies the possible applications of those concepts and tools.1

In the literature review section a number of papers which methodologicallycan be classified into “reaction-diffusion systems” and “networks” are surveyed.In the simulation part a cellular automaton model of urban growth will beimplemented and tested on the current distribution of urban clusters in Europe.

2 Literature Review

2.1 Reaction diffusion systems

A short review of literature in statistical physics which has relevance to the studyof urban systems with an emphasis on observations and modelling approaches.

A reaction-diffusion system is a system of partial differential equationsof the type

∂tu = f(u) +D∇2u (1)

Discretizing time and space as one has to do for numerical simulations (e.g. byusing a first-order taylor approximations for the derivatives) one can connectthese systems to the concept of cellular automata. Cellular automata (CA)encompass a much broader class of systems than reaction-diffusion type differ-ential equations. In general a CA is a discrete model of a system composed ofa finite number of cells on a lattice. These cells can be in a finite number ofstates for which rules can be given which determine the change of state of acell as a function of the state of the cell’s neighbours. These systems are usedby a number of authors [4], [5], [13], [9] to account for some quite remarkableproperties of cities:

� When ordered according to their rank (biggest city first, second biggestcity second) they follow what is called Zipf’s law which relates rank r andsize N through a relationship of the form:

N ∝ r−α (2)

1For example “What is the ideal size of a city?” – A research question put forward byStefano Negri constitutes from a quantitative viewpoint of the physicist a probably nonlinearoptimization problem under constraints.

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Remarkably the exponent α ≈ 2 is stable over long periods of time as wellas across countries [9].

� When looking at the distribution of the size of urban clusters (see [9] fora detailed explanation) one similarly observes that larger clusters (size s)occur with smaller frequency P(s):

P (s) ∝ s−α (3)

� Starting from the center of mass of the population density distribution σof a city one observes that the density decreases as a function of distancewhich obeys the form:

σ(d; t) ∝ exp(−λ(t)d) (4)

The function λ(t) decreases over time which takes account of the fact thatolder cities are often less centralized.

� The morphology of the urban boundary has the shape of a fractal witha fractal dimension (there are several measures for this, see http://en.

wikipedia.org/wiki/Fractal_dimension#specific_definitions) be-tween 1.2 and 1.4 [5].

The above observations can be reproduced by a simple reaction-diffusioncellular automaton [5] which will be simulated numerically in the next chapter.The cellular automaton iterates on a two-dimensional field of population-densityu(x, y, t) ∈ R, {x, y} ∈ N with two free parameters p and α which determine

� a diffusion step in which α percent of each cells value are distributedamong its four nearest neighbors.

u(x, y, t+ 1/2) = (1− α)u(x, y, t) +α

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∑〈k,l〉

u(x, y, t) (5)

� a reaction step in which the cell increases its content by a factor of p−1

with probability p or is reset to zero with probability 1− p.

u(x, y, t+ 1) = p−1u(x, y, t+ 1/2) with probability p (6)

u(x, y, t+ 1) = 0 with probability 1-p (7)

Rigorous mathematical analysis of the model [4] can be performed for certainlimiting cases, for example the exponent of the power law relating the frequencyand density of cells can be determined in the stationary limit (α = 2).

In terms of interpretation p and α can be seen as mimicking the rate of urbansprawl (high α) and birth/migration rates (high p). While the older cities of

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Europe show low α and p the burgeoning metropolises of Asia can better becaptured by a high α, high p regime.

The model is quite remarkable since it allows the formation of clusters eventhough it does not contain any explicit process that leads to an aggregation ofthe content of cells, contrary it disperses the contents of cells (diffusion-step).

Figure 1: Comparison of the growth of Berlin beginning in 1875 until 1945 (left:real, right: model). From [3], note that this is not same CA model as mentionedabove.

Master-equations are also used in the context or reaction diffusion sys-tems, Schweitzer and Steinbrink [9] capture the evolution of the distribution ofcluster sizes which comprise the entire urban area P (n), n = {n1, n2, ..., nk, ..., nA}by two processes, the formation of new clusters and the growth of existingclusters. Growth of the existing cluster is captured by a symbolic reactionA

w1−−→ A+ 1 .The reaction rate w1 itself is a function of the total city cize Ntot =

∑i ni:

w1 = w(A + 1, t + 1|A, t) = f(Ntot). For an individual cluster with index k its

population can increase with rate nk: nkwk−−→ nk + 1. The probability of this

event is determined by the importance of the cluster:

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wk = w(nk + 1, t+ 1|nk, t) = γnkNtot

, γ = 1− f(Ntot) (8)

The factor γ can control the ration between new cluster formation and growthof existing clusters. This allows to write down a master-equation for the distri-bution of cluster sizes P (n, t):

∂P (n, t)

t= −

∑i

wk(nk, t)P (n, t) (9)

+∑i 6=j

wk(nk − 1, t)P (nnk→nk+1, t) (10)

+ w1(A, t)P (n, t) (11)

+ w(A− 1, t)P (nA→A−1, t) (12)

2.2 Urban Networks

Two recent studies in which networks are used to understand urban systems aresurveyed, emphasis is mostly put on observations not models.

Recently physicists have become interested in networks, in particular in theirstructure mostly in terms of statistical features (e.g. degree distributions) andhow their “macro”-structure can be related to processes governing the growthof the network on a microscopic scale. Among the growth-mechanisms the pref-erential attachment model by Barabasi-Albert [7] which reproduces the degreedistributions of scale free networks received quite some attention. In additionto growth mechanisms researchers are also interested in understanding how thestructure of networks relates to their function [7]. One question that arises ishow the optimization of certain cost-functions (which quantify the performanceof a network) give rise to different network topologies [10].

The study of urban road-, subway- and light rail networks differs from moststudies of complex networks since those networks are embedded in Euclideanspace while in the existing literature nodes mostly assumed to have no spatial re-lation to each other. The study of spatial networks [1] is a promising field withpotential applications in many areas ranging from the aforementioned trans-portation networks to electrical grids, the internet and also neural networks inwhich space-constraints play a role.

2.2.1 Structure and Growth

Along the lines of the structure ⇔ growth question a study by Roth and col-leagues [8] recently received quite some attention in the media [11]. Comparingthe subway networks of the worlds largest cities they claimed to have found signsof convergence in a number of structural properties. Especially they point outa ubiquitous core-periphery structure which they quantify through a structuralmeasure β (for details see [8]). Figure 2 shows this convergence.

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From a statistical mechanics standpoint this convergence is a convergenceof macro-observables. Unlike the case of urban growth phenomena in the pre-vious section there are of now no mechanistic models reproducing this macro-behaviour from microscopic interaction rules. Since those subway-systems haveemerged over a long period of time during which not only one planning au-thority was taking decisions it is not central planning but also self-organizationprocesses that play a role in their evolution.

Figure 2: Convergence of the ration β of core- to periphery-nodes in the worldssubway networks as a function of their size measured by N (number of nodes).From [8]

2.2.2 Structure and Optimization

A second example that relates the structure of a spatial network to an opti-mization process is a study of a slime mold by Tero and colleagues [12]. Theslime mold “Physarum polycephalum” is an organism that through millenia ofevolutional optimization had to struggle balancing two opposing goals. Thefirst and foremost was of course its foraging for food for which it is necessary tocover an area as large as possible. This happens under the constraint that onlya limited amount of biomass is available for the organism to grow. The grow-ing slime mold first covers a large area with its cell body, then within the cellbody a network forms which connects found food-sources through tubes. Thenetwork itself has to balance cost, transport efficiency and resilience to failureof its nodes. This inspired researchers to compare its optimization performanceto other systems that are optimized by humans.

Tero and colleauges confronted the slime mold with the problem that Tokyo’surban transportation planners face: connect the suburbs of Tokyo in a way suchthat travel times are minimized (transport efficiency) at low cost. Figure 3 showsboth the actual experiment as well as a model that reproduces some key features

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Figure 3: Left: model of the network evolution process, Right: growth processof the physarum polycephalum on distributed food sources. From [12]

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of the biological process. Introducing a number of structural measures they wereable to show that the performance of the slime mold does not quite match thatof the transportation planners. However the slime mold “solved the problem”through a microscopic, local processes of self-organization within its plasmod-ium while transportation planners have global knowledge of the distribution ofsuburbs and act as central planners.

The model they use to mimick the growth of the network consists of a rectan-gular grid on which food sources are distributed. In each time step two sourcesare selected at random and connected through a tube whose diameter growsas the flows increase, if tubes are “unused” for transport their diameter de-creases. Through these simple proceses a network similar to the one observedin experiments is built.

2.3 Thoughts on a Connection to Statistical Mechanics

In the previous sections the application of two modelling paradigms from physicsfor urban systems – networks and reaction-diffusion systems were discussed.How can this be connected to the theory of statistical mechanics of equilibriumand nonequilibrium systems?

The most obvious inspiration from statistical mechanics is the concept of“macro observables” that are created through “micro interaction”. Once castinto the same mathematical language that physicists are familiar with one canask a number of questions about such systems:

� How can master-equations be derived for cellular automata? What arethe steady state distributions?

� Is there convergence to a small set of possible patterns independent ofinitial conditions? For the urban growth model this means – do our citiesbecome more similiar?

� Are there bifurcations in the discrete dynamical systems? One possibilitywould be a shift from a circular urban form to an elongated urban clusteras the parameters are varied, or can such phenomena be ecluded on atheoretical basis? If there are such bifurcations, can we observe them inreality? Are there cities growing under similar external conditions butwith different shape?

� In the phase space of a cellular automaton are there attractors, e.g. limitcycles? Does the history of a city repeat itself?

3 Numerical Simulations of a Cellular Automa-ton Model

Even though Europe is one of the most urbanized continents in the world thetrend of urbanization still continues. Figure ?? shows this trend beginning in1950 to 2050.

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Figure 4: Urbanization trend in Europe. From [6]

3.1 Data and Setup of Simulation

Using the cellular automaton model developed in [5] we numerically explore howthe face of Europe will look in the future. Since the model does not have explicittimescales we unfortunately cannot relate the model-time to the year-count.

The cellular automaton is implemented in a MATLAB program which isstarted with the urban population distribution of Europe in 2005. The sourceof this data is the GPW project by Columbia University for the populationdensity [2] and Europes statistical agency EUROSTAT for the boundaries oflarger urban zones (LUZ).

The model parameters are chosen to be p = 0.1 and α = 0.5. In additionin each time-step each cell can receive an additional increase in population ofδm = 2 with probability pn = 0.02.

3.2 Simulation Results

Figure 5 and 6 show the initial distribution of the population as well as theresults after a number of iterations. The increase in brightness indicates anincreased population density in all cells, the black spots in the urban area arecells which have been set to zero according to the CA-rules. We see that clustershave increased in size as well as we would expect.

Another observation to be reproduced by the CA model was the power lawdistribution of the CA cells which constitute one urban cluster. Figure 7 showsthe cumulative distribution of the data, the power-law character over a broadrange of values is confirmed by straight lines in the double-logarithmic plot.

3.3 Discussion

A number of questions remain open and could be studied through further anal-ysis of the model either numerically or even analytically:

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Figure 5: Initial distribution and distribution after 50 timesteps for Europe.Source for population density [2].

Figure 6: Initial population distribution and distribution after 50 timesteps forParis. Source for population density [2].

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Figure 7: Cumulative distribution of the density of cells in the CA model.Straight lines in a double-logarithmic plot indicate a power-law of the formmentioned in section 1.

� What role do boundary conditions play for the long-term evolution of thecellular automaton?

� Is it possible to estimate the model-parameters p and α from actual dataso that we can assign a future year to the results of our simulations?

� What is the fractal dimension of the urban fringe? How does it changeover time? A box-count algorithm could be applied for example to check iffractal dimension described in [5] is conserved. Also the radial distributionof the population density could be studied.

� Are there any other regimes than growth in the parameter space of themodel? Do stable stady state formations of the cellular automaton modelexist?

Of course the model could also be extended to account for other factors, forexample the price of land which significantly influences the settlement of people.Transportation networks also allow cities to cover a much larger area in spaceand could be incorporated into the CA-model.

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References

[1] Marc Barthelemy. Spatial networks. arXiv:1010.0302v2, 2010.

[2] Columbia University; Center for International Earth Science Informa-tion Network (CIESIN) and Centro Internacional de Agricultura Tropi-cal (CIAT). Gridded population of the world, version 3 (gpwv3): Popu-lation density grid. Available at http: // sedac. ciesin. columbia. edu/gpw , 2005.

[3] Hernan A. Makse, Shlomo Havlin, and H. Eugene Stanley. Modelling urbangrowth patterns. Nature, 377, 1995.

[4] Susanna C Manrubia and Damian H Zanette. Intermittency model forurban development.

[5] Susanna C. Manrubia, Damin H. Zanette, and Ricard V. Sol. Transientdynamics and scaling phenomena in urban growth. Fractals, 7 No 1, 1999.

[6] United Nations. World urbanization prospects: The 2009 revision. 2009.

[7] M.E.J. Newman. The Structure and Function of Complex Networks. SIAMreview, 45(2):167–256, 2003.

[8] Camille Roth, Soong Moon Kang, Michael Batty, and Marc Barthelemy.Evolution of subway networks. arXiv:1105.5294v3.

[9] Frank Schweitzer and Jens Steinbink. Analysis and computer simulation ofurban cluster distributions. 2002.

[10] Sitabhra Sinha. Complexity vs. stability in small-world networks. PhysicaA: Statistical Mechanics and its Applications, 346(1-2):147–153, February2005.

[11] Der SPIEGEL. Unter der erde sind alle wieberlin. http: // www. spiegel. de/ wissenschaft/ mensch/

u-bahn-netze-in-grossen-staedten-aehneln-sich-sehr-a-839738.

html , 2012.

[12] A. Tero, S. Takagi, T. Saigusa, K. Ito, D. P. Bebber, M. D. Fricker, K. Yu-miki, R. Kobayashi, and T. Nakagaki. Rules for biologically inspired adap-tive network design. Science, 327(5964):439–442, January 2010.

[13] Damin H. Zanette and Susanna C. Manrubia. Role of intermittency inurban development: A model of large-scale city formation. Physical ReviewLetters, 79 Nb 3, 1997.

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A MATLAB Script for Cellular Automaton

1

2 L = 2e2;3 % area = rand(L); % alternative initial conditions4 % area = paris; % for Paris simulation5 area = extr s; % for all of Europe6 L = length(area);7 area b = area;8

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10 p = 0.1; % determines the reaction−process (birth/death)11 alpha = 0.5; % determines the diffusion process12

13 p n = 0.02; % add new population in new places14 q = 0.02; % parameter to tune the amount added in each step at ...

random15 ∆ m = 2;16

17 figure;18 colormap(hot);19 [L1, L2] = size(area);20 for time = 1 : 100021 area = area b;22 % option: change parameters in a stochastic manner in every ...

time−step23 % p = randn/2+0.5;24 % alpha = randn;25 %26 for x = 2 : L1−127 for y = 2 : L2−128 if randn < p n29 area b(x,y) = area b(x,y)+∆ m;30 end31 % diffusion at every step32 area b(x,y) = (1−alpha)*area(x,y)+ alpha/4* ...33 ( area(x−1,y) + area(x+1,y) + area(x,y−1) + ...

area(x,y+1));34 % to speed this up use convolution?35

36 % reaction increase every other step37 if mod(time,2) == 038 if randn < p39 area b(x,y) = (1−q)*1/p*area(x,y);40 else41 area b(x,y) = 0; % from 1999 MANRUBIA paper42 % area b(x,y) = q/(1−p)*area(x,y); % from ...

Manrubia 1997 paper43 end44 end45

46 end47 end48

49 imagesc(log(area b));

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50 pause(0.01);51

52 end

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