Phase transition in a model for territorial development

Transcription

1 Phase transition in a model for territorial development Alethea Barbaro Joint with Abdulaziz Alsenafi Supported by the NSF through Grant No. DMS Case Western Reserve University March 23, 27 A. Barbaro (CWRU) Model for Territorial Development March 23, 27 / 27

2 Introduction to Gangs Gangs are responsible for much of the violent crimes in the U.S. and worldwide One of the main concerns of a gang is its territory This territory is marked and defended Graffiti (tagging) is often used to claim or maintain territory Figure: Graffiti gang war in Los Angeles, California. Figure adopted from A. Barbaro (CWRU) Model for Territorial Development March 23, 27 2 / 27

3 Introduction to Gangs Gangs are responsible for much of the violent crimes in the U.S. and worldwide One of the main concerns of a gang is its territory This territory is marked and defended Graffiti (tagging) is often used to claim or maintain territory Figure: Graffiti gang war in Los Angeles, California. Figure adopted from A. Barbaro (CWRU) Model for Territorial Development March 23, 27 2 / 27

4 Introduction to Gangs Gangs are responsible for much of the violent crimes in the U.S. and worldwide One of the main concerns of a gang is its territory This territory is marked and defended Graffiti (tagging) is often used to claim or maintain territory Figure: Graffiti gang war in Los Angeles, California. Figure adopted from A. Barbaro (CWRU) Model for Territorial Development March 23, 27 2 / 27

5 Lattice Model Summary We consider two gangs, let s say red and blue Initially, agents are uniformly distributed over a toroidal lattice Agents have some probability of tagging the lattice site which they currently occupy Agents then are forced to move to one of the four neighboring sites ) The total number of agents in each gang is conserved A. Barbaro (CWRU) Model for Territorial Development March 23, 27 3 / 27

6 Lattice Model Summary We consider two gangs, let s say red and blue Initially, agents are uniformly distributed over a toroidal lattice Agents have some probability of tagging the lattice site which they currently occupy Agents then are forced to move to a neighboring site ) The total number of agents in each gang is conserved A. Barbaro (CWRU) Model for Territorial Development March 23, 27 4 / 27

7 Lattice Model Summary We consider two gangs, let s say red and blue Initially, agents are uniformly distributed over a toroidal lattice Agents have some probability of tagging the lattice site which they currently occupy Agents then are forced to move to a neighboring site ) The total number of agents in each gang is conserved A. Barbaro (CWRU) Model for Territorial Development March 23, 27 5 / 27

8 Our Discrete Model We base our model on the assumption that gangs claim territory by tagging Each gang preferentially avoids areas marked by the other gang Gang members do not interact directly Multiple agents may occupy the same site Agents interact only through the graffiti field A. Barbaro (CWRU) Model for Territorial Development March 23, 27 6 / 27

9 Our Discrete Model We base our model on the assumption that gangs claim territory by tagging Each gang preferentially avoids areas marked by the other gang Gang members do not interact directly Multiple agents may occupy the same site Agents interact only through the graffiti field A. Barbaro (CWRU) Model for Territorial Development March 23, 27 6 / 27

10 Discrete Dynamics The densitity of agents of gangs A and B at site (x, y) at time t are denoted by A (x, y, t) and B (x, y, t) The density of graffiti of gang A and B at site (x, y) at time t are A (x, y, t) and B (x, y, t) The probability of an agent from gang A to move to a neighbouring site is M A (x! x 2, y! y 2, t) := e P ( x,ỹ) (x,y ) B (x 2,y 2,t) e B ( x,ỹ,t) is parameter that symbolizes the avoidance of graffiti belonging to the other gang. A. Barbaro (CWRU) Model for Territorial Development March 23, 27 7 / 27

11 Discrete Dynamics The densitity of agents of gangs A and B at site (x, y) at time t are denoted by A (x, y, t) and B (x, y, t) The density of graffiti of gang A and B at site (x, y) at time t are A (x, y, t) and B (x, y, t) The probability of an agent from gang A to move to a neighbouring site is M A (x! x 2, y! y 2, t) := e P ( x,ỹ) (x,y ) B (x 2,y 2,t) e B ( x,ỹ,t) is parameter that symbolizes the avoidance of graffiti belonging to the other gang. A. Barbaro (CWRU) Model for Territorial Development March 23, 27 7 / 27

12 Discrete Dynamics The densitity of agents of gangs A and B at site (x, y) at time t are denoted by A (x, y, t) and B (x, y, t) The density of graffiti of gang A and B at site (x, y) at time t are A (x, y, t) and B (x, y, t) The probability of an agent from gang A to move to a neighbouring site is M A (x! x 2, y! y 2, t) := e P ( x,ỹ) (x,y ) B (x 2,y 2,t) e B ( x,ỹ,t) is parameter that symbolizes the avoidance of graffiti belonging to the other gang. A. Barbaro (CWRU) Model for Territorial Development March 23, 27 7 / 27

13 Discrete Equations According to our discrete model, employing this notation, we find: The expected density of gang A agents at site (x, y) 2 S at time t + t is A (x, y, t + t) = A (x, y, t)+ X A ( x, ỹ, t)m A ( x! x, ỹ! y, t) A (x, y, t) ( x,ỹ) (x,y) X ( x,ỹ) (x,y) M A (x! x, y! ỹ, t) The graffiti evolution is given by A (x, y, t + t) = A (x, y, t) t A (x, y, t)+ t A (x, y, t), A. Barbaro (CWRU) Model for Territorial Development March 23, 27 8 / 27

14 y y y y y y Simulations of Gang Dynamics: Well-Mixed Phase = 6 : Agent Density Agent Density Agent Density t = t = 5 t = 3 x Graffiti Territory t = x Graffiti Territory t = 5 x Graffiti Territory t = 3 x x x Figure: Temporal evolution of the densities for a well-mixed phase. A. Barbaro (CWRU) Model for Territorial Development March 23, 27 9 / 27

15 y y y y y y Segregated Phase: Agent Density = 2 5 : Agent Density Agent Density Agent Density t = t = t = x Agent Density t = 2 x Agent Density t = 2 x Agent Density t = 3 x x x Figure: Temporal evolution of the densities for a segregated phase. A. Barbaro (CWRU) Model for Territorial Development March 23, 27 / 27

16 y y y y y y Segregated Phase: Graffiti Field Graffiti Territory Graffiti Territory Graffiti Territory t = t = t = x Graffiti Territory t = 2 x Graffiti Territory t = 2 x Graffiti Territory t = 3 x x x Figure: Temporal evolution of the densities for a segregated phase. A. Barbaro (CWRU) Model for Territorial Development March 23, 27 / 27

17 Probability of Movement for Different Beta Values B = B = B = B = 5,.55 5,.5 5,.2 5, M left M right M up M down Table: Probabilities of an agent from gang A moving to a neighbouring site for different values. A. Barbaro (CWRU) Model for Territorial Development March 23, 27 2 / 27

18 Phase Transitions Phase transitions can be observed in many models for collective dynamics Different macroscopic behaviors depending on the parameter values Think of solid, liquid, and gas phases from physics Order parameters, Hamiltonians, and energies can be defined to help track the phase transition A. Barbaro (CWRU) Model for Territorial Development March 23, 27 3 / 27

19 Phase Transitions Phase transitions can be observed in many models for collective dynamics Different macroscopic behaviors depending on the parameter values Think of solid, liquid, and gas phases from physics Order parameters, Hamiltonians, and energies can be defined to help track the phase transition A. Barbaro (CWRU) Model for Territorial Development March 23, 27 3 / 27

20 Phase Transitions in Kinetic Models In the context of kinetic models, phase transitions come up frequently These models often exhibit collective dynamics in one parameter regime, but not in another The collective dynamics are then considered as a phase Consider flocking models: In one regime (e.g. high noise regime), there is very little order in the way the agents are moving In another regime (e.g. low noise), the agents align and move together in an organized way Polarity can be useful as an order parameter to track the phase A. Barbaro (CWRU) Model for Territorial Development March 23, 27 4 / 27

21 Phase Transitions in Kinetic Models In the context of kinetic models, phase transitions come up frequently These models often exhibit collective dynamics in one parameter regime, but not in another The collective dynamics are then considered as a phase Consider flocking models: In one regime (e.g. high noise regime), there is very little order in the way the agents are moving In another regime (e.g. low noise), the agents align and move together in an organized way Polarity can be useful as an order parameter to track the phase A. Barbaro (CWRU) Model for Territorial Development March 23, 27 4 / 27

22 Phase Transitions in Kinetic Models In the context of kinetic models, phase transitions come up frequently These models often exhibit collective dynamics in one parameter regime, but not in another The collective dynamics are then considered as a phase Consider flocking models: In one regime (e.g. high noise regime), there is very little order in the way the agents are moving In another regime (e.g. low noise), the agents align and move together in an organized way Polarity can be useful as an order parameter to track the phase A. Barbaro (CWRU) Model for Territorial Development March 23, 27 4 / 27

23 Phase Transitions in Kinetic Models In the context of kinetic models, phase transitions come up frequently These models often exhibit collective dynamics in one parameter regime, but not in another The collective dynamics are then considered as a phase Consider flocking models: In one regime (e.g. high noise regime), there is very little order in the way the agents are moving In another regime (e.g. low noise), the agents align and move together in an organized way Polarity can be useful as an order parameter to track the phase A. Barbaro (CWRU) Model for Territorial Development March 23, 27 4 / 27

24 Phase Transitions in Kinetic Models In the context of kinetic models, phase transitions come up frequently These models often exhibit collective dynamics in one parameter regime, but not in another The collective dynamics are then considered as a phase Consider flocking models: In one regime (e.g. high noise regime), there is very little order in the way the agents are moving In another regime (e.g. low noise), the agents align and move together in an organized way Polarity can be useful as an order parameter to track the phase A. Barbaro (CWRU) Model for Territorial Development March 23, 27 4 / 27

25 Phase Transitions in Kinetic Models In the context of kinetic models, phase transitions come up frequently These models often exhibit collective dynamics in one parameter regime, but not in another The collective dynamics are then considered as a phase Consider flocking models: In one regime (e.g. high noise regime), there is very little order in the way the agents are moving In another regime (e.g. low noise), the agents align and move together in an organized way Polarity can be useful as an order parameter to track the phase A. Barbaro (CWRU) Model for Territorial Development March 23, 27 4 / 27

26 Phase Transition in the Discrete Model We use an energy function to examine the system phases. The energy at time t is defined as E(t) = 2 X X ( A B )( A B ). 4 LN (x,y)2s ( x,ỹ) (x,y) Energy vs. Time.8 Energy β = β =. β =.65 β =.5 β = Time 4 The system has high energy in segregated phase. The system has low energy in well-mixed phase. A. Barbaro (CWRU) Model for Territorial Development March 23, 27 5 / 27

27 Phase Transitions for Different Mass Values Final Energy vs. β Mass = Final Energy vs. β Mass = Final Energy.6.4 Final Energy L =5 L =75 L = β -5.2 L =5 L =75 L = β -5 Figure: The energy at the final time step against of agents. for different lattice sizes and number A. Barbaro (CWRU) Model for Territorial Development March 23, 27 6 / 27

28 Phase Transitions for Different Ratio Values Final Energy vs. β γ/λ Ratio = /2 Final Energy vs. β γ/λ Ratio =.8.8 Final Energy.6.4 Final Energy L =5 L =75 L = β -5.2 L =5 L =75 L = β -5 Figure: The energy at the final time step against ratios. for different lattice sizes and A. Barbaro (CWRU) Model for Territorial Development March 23, 27 7 / 27

29 Deriving the Macroscopic Model from the Microscopic Recall the probability of movement from site x to site x 2 : M A (x! x 2, y! y 2, t) := e P ( x,ỹ) (x,y ) B (x 2,y 2,t) e B ( x,ỹ,t). We know the density of agents at site (x, y) at time t + t: A (x, y, t + t) = A (x, y, t)+ X A ( x, ỹ, t)m A ( x! x, ỹ! y, t) A (x, y, t) ( x,ỹ) (x,y) X ( x,ỹ) (x,y) M A (x! x, y! ỹ, t) We also have the density of the graffiti for each gang at site (x, y) at time t + t: A (x, y, t + t) = A (x, y, t) t A (x, y, t)+ t A (x, y, t) Our goal now is to find macroscopic equations which govern the evolution of these four quantities over time. A. Barbaro (CWRU) Model for Territorial Development March 23, 27 8 / 27

30 Deriving the Macroscopic Model from the Microscopic Recall the probability of movement from site x to site x 2 : M A (x! x 2, y! y 2, t) := e P ( x,ỹ) (x,y ) B (x 2,y 2,t) e B ( x,ỹ,t). We know the density of agents at site (x, y) at time t + t: A (x, y, t + t) = A (x, y, t)+ X A ( x, ỹ, t)m A ( x! x, ỹ! y, t) A (x, y, t) ( x,ỹ) (x,y) X ( x,ỹ) (x,y) M A (x! x, y! ỹ, t) We also have the density of the graffiti for each gang at site (x, y) at time t + t: A (x, y, t + t) = A (x, y, t) t A (x, y, t)+ t A (x, y, t) Our goal now is to find macroscopic equations which govern the evolution of these four quantities over time. A. Barbaro (CWRU) Model for Territorial Development March 23, 27 8 / 27

31 Deriving the Macroscopic Model from the Microscopic Recall the probability of movement from site x to site x 2 : M A (x! x 2, y! y 2, t) := e P ( x,ỹ) (x,y ) B (x 2,y 2,t) e B ( x,ỹ,t). We know the density of agents at site (x, y) at time t + t: A (x, y, t + t) = A (x, y, t)+ X A ( x, ỹ, t)m A ( x! x, ỹ! y, t) A (x, y, t) ( x,ỹ) (x,y) X ( x,ỹ) (x,y) M A (x! x, y! ỹ, t) We also have the density of the graffiti for each gang at site (x, y) at time t + t: A (x, y, t + t) = A (x, y, t) t A (x, y, t)+ t A (x, y, t) Our goal now is to find macroscopic equations which govern the evolution of these four quantities over time. A. Barbaro (CWRU) Model for Territorial Development March 23, 27 8 / 27

32 Deriving the Macroscopic Model from the Microscopic Recall the probability of movement from site x to site x 2 : M A (x! x 2, y! y 2, t) := e P ( x,ỹ) (x,y ) B (x 2,y 2,t) e B ( x,ỹ,t). We know the density of agents at site (x, y) at time t + t: A (x, y, t + t) = A (x, y, t)+ X A ( x, ỹ, t)m A ( x! x, ỹ! y, t) A (x, y, t) ( x,ỹ) (x,y) X ( x,ỹ) (x,y) M A (x! x, y! ỹ, t) We also have the density of the graffiti for each gang at site (x, y) at time t + t: A (x, y, t + t) = A (x, y, t) t A (x, y, t)+ t A (x, y, t) Our goal now is to find macroscopic equations which govern the evolution of these four quantities over time. A. Barbaro (CWRU) Model for Territorial Development March 23, 27 8 / 27

33 Deriving a Macroscopic Version of the Graffiti Density Evolution of the graffiti density: A (x, y, t + t) = A (x, y, t) t A (x, y, t)+ t A (x, y, t) Obtain a discrete derivative on the left-hand side: A (x, y, t + t) A (x, y, t) t = A (x, y, t)+ A (x, y, t) Simply taking t!, we have our two macroscopic (x, y, t) = A(x, y, t) A (x, y, (x, y, t) = B(x, y, t) B (x, y, t) A. Barbaro (CWRU) Model for Territorial Development March 23, 27 9 / 27

34 Deriving a Macroscopic Version of the Agent Density The derivation of the equations for the agent density are more complicated Considering the discrete equations: M A (x! x 2, y! y 2, t) := A (x, y, t + t) = A (x, y, t)+ A (x, y, t) X ( x,ỹ) (x,y) X ( x,ỹ) (x,y) e P ( x,ỹ) (x,y ) B (x 2,y 2,t) e B ( x,ỹ,t) A ( x, ỹ, t)m A ( x! x, ỹ! y, t) M A (x! x, y! ỹ, t) A. Barbaro (CWRU) Model for Territorial Development March 23, 27 2 / 27

35 Deriving a Macroscopic Version of the Agent Density We formally derive a macroscopic version Taylor expansion To circumnavigate the complication arising from the neighbors neighbors, we employ the discrete spatial Laplacian in two dimensions: 2 f (x, y, t) = 4 X l 2 f ( x, ỹ, t) 3 4f (x, y, t) 5 + O(l 2 ), ( x,ỹ) (x,y) We apply the discrete Laplacian several times to get rid of the dependence on the neighboring points. A. Barbaro (CWRU) Model for Territorial Development March 23, 27 2 / 27

36 Deriving a Macroscopic Version of the Agent Density We formally derive a macroscopic version Taylor expansion To circumnavigate the complication arising from the neighbors neighbors, we employ the discrete spatial Laplacian in two dimensions: 2 f (x, y, t) = 4 X l 2 f ( x, ỹ, t) 3 4f (x, y, t) 5 + O(l 2 ), ( x,ỹ) (x,y) We apply the discrete Laplacian several times to get rid of the dependence on the neighboring points. A. Barbaro (CWRU) Model for Territorial Development March 23, 27 2 / 27

37 Deriving a Macroscopic Version of the Agent Density We formally derive a macroscopic version Taylor expansion To circumnavigate the complication arising from the neighbors neighbors, we employ the discrete spatial Laplacian in two dimensions: 2 f (x, y, t) = 4 X l 2 f ( x, ỹ, t) 3 4f (x, y, t) 5 + O(l 2 ), ( x,ỹ) (x,y) We apply the discrete Laplacian several times to get rid of the dependence on the neighboring points. A. Barbaro (CWRU) Model for Territorial Development March 23, 27 2 / 27

38 Continuum Equations The full system of continuum equations (x, y, t) = A(x, y, t) A (x, y, (x, y, t) = B(x, y, t) B (x, y, r h (x, y, t) =D r A (x, y, t)+2 A (x, y, t)r B (x, y, t) B (x, y, t) = 4 r hr B (x, y, t)+2 B (x, y, t)r A (x, y, t) i i A. Barbaro (CWRU) Model for Territorial Development March 23, / 27

39 Linear Stability Analysis We linearize our model by considering a perturbation of the equilibrium solution A = A + A e t e ikx B = B + B e t e ikx A = A + A e t e ikx B = B + B e t e ikx, Solving for eigenvalue, we find the following four eigenvalues:,2 = q 4 + D k 2 ± D( + 4 q 4 + D k 2 ± 6 2 8D( 4 3,4 = 8 p A B ) k 2 + D 2 k 4 p A B ) k 2 + D 2 k 4. A. Barbaro (CWRU) Model for Territorial Development March 23, / 27

40 Critical for Continuum Model, 2, and 3 are always negative However, 4 is positive for 2( ) p, A B This defines a critical for the continuum system A. Barbaro (CWRU) Model for Territorial Development March 23, / 27

41 Comparing Discrete and Continuous Critical β * vs Mass Discrete Model Stability Analysis β * vs Ratio Discrete Model Stability Analysis β * 3 2 β * Mass 5 Figure: Critical γ / λ comparision. A. Barbaro (CWRU) Model for Territorial Development March 23, / 27

42 Future Work Numerical solution of the coupled system: how does it compare to the discrete model s evolution? Analysis of coupled system: energy, stability, coarsening rates, etc. Agent Density t = Agent Density t = Agent Density t = 2 y y y x x x A. Barbaro (CWRU) Model for Territorial Development March 23, / 27

43 Thank you for your attention! A. Barbaro (CWRU) Model for Territorial Development March 23, / 27