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1. Traffic jams(joint work with David Griffeath, Ryan Gantnerand related work with Levine, Ziv, Mukamel) To view the embedded objects in this Document, you will need the MCell player Available to download free from http://www.mirekw.com/ca/index.html

2. Traffic jams • TASEP has traffic jams caused by obstructions – not spontaneous

3. Traffic jams • TASEP has traffic jams caused by obstructions – not spontaneous • “Bus” model of O’Loan, Evans, Cates has convoys, due to non-conserved particles (riders). Convoys diffuse and merge.

4. Traffic jams • TASEP has traffic jams caused by obstructions – not spontaneous • “Bus” model of O’Loan, Evans, Cates has convoys, due to non-conserved particles (riders). Convoys diffuse and merge. • In 1992, Nagel and Schreckenberg invented a relatively simple particle model with realistic traffic jam behavior. It has been used to simulate actual traffic.

5. Nagel-Schreckenberg model (1992) • N-S model dynamics: • Discrete time, parallel updating • Particles are identical, but have “velocities” • Initial state is a random arrangement of cars • No passing, no collisions (exclusion) • Particles have finite acceleration (“slow to start”) • Particle velocities (jump size) limited by “head room”

6. Nagel-Schreckenberg model (1992) • N-S model dynamics: • Discrete time, parallel updating • Particles are identical, but have “velocities” • Initial state is a random arrangement of cars • No passing, no collisions (exclusion) • Particles have finite acceleration (“slow to start”) • Particle velocities (jump sizes) limited by “head room” • Numerical simulations show formation of spontaneous traffic jams in “cruise control limit” • Below critical density, free flow emerges • Above critical density, persistent jams form • Jam phase coexists with free flow phase • Proof?

7. A simpler traffic CA (TCA) • All particles identical, no obstructions, all quantities conserved, simple exclusion

8. A simpler traffic CA (TCA) • All particles identical, no obstructions, all quantities conserved, simple exclusion • Velocities replaced by jump probabilities, which are determined by positions of nearby particles

9. A simpler traffic CA (TCA) • All particles identical, no obstructions, all quantities conserved, simple exclusion • Velocities replaced by jump probabilities, which are determined by positions of nearby particles • Numerical simulations show wide variety of interesting behaviors, including: • Spontaneous jams that diffuse and merge • Varying parameter values leads to several qualitatively different jam phases • Rigorous results obtained for simple jam types

10. Jump probabilities When a is small, we get “slow to start” feature.

11. Jump probabilities Note that both of these rules are “particle-hole symmetric.

12. Jump probabilities It would seem natural to make g less than or equal to both a and b , but other choices can also be interesting

13. Jump probabilities Particle-hole symmetry requires g = d .

14. Basics for Traffic CA: • Look at process on infinite lattice, with random initial state, density r

15. Basics for Traffic CA: • Look at process on infinite lattice, with random initial state, density r • Primarily consider “cruise-control case” where d = 1 (deterministic motion in free flow phase)

16. Basics for Traffic CA: • Look at process on infinite lattice, with random initial state, density r • Primarily consider “cruise-control case” where d = 1 (deterministic motion in free flow phase) • If 0 < r < rc , then system goes into free flow, asymptotic throughput equals r (rigorous)

17. Basics for Traffic CA: • Look at process on infinite lattice, with random initial state, density r • Primarily consider “cruise-control case” where d = 1 (deterministic motion in free flow phase) • If 0 < r < rc , then system goes into free flow, asymptotic throughput equals r (rigorous) • If rc < r < rc , get mixture of free flow and jams, throughput is weighted average of throughput in jam phase and throughput in critical free flow

18. Basics for Traffic CA: • Look at process on infinite lattice, with random initial state, density r • Primarily consider “cruise-control case” where d = 1 (deterministic motion in free flow phase) • If 0 < r < rc , then system goes into free flow, asymptotic throughput equals r (rigorous) • If rc < r < rc , get mixture of free flow and jams, throughput is weighted average of throughputs in critical jam phase and critical free flow phase • If rc < r , system goes to a jam phase

19. Typical “fundamental diagram” for cruise control case when g small (q = asymptotic throughput)

20. Exactly solvable cases • TASEP: a = b = g = d

21. Exactly solvable cases • TASEP: a = b = g = d • Merging convoys if b = g = 1 (similar to “bus model”)

22. Exactly solvable cases • TASEP: a = b = g = d • Merging convoys if b = g = 1 (similar to “bus model”) • Solid fluctuating merging jams if b = d = 1, exactly computable throughput and critical density, equivalent to previous case (swapping g, d exchanges particles, holes)

23. Throughput case b = d = 1:

24. Exactly solvable cases • TASEP: a = b = g = d • Merging convoys if b = g = 1 (similar to “bus model”) • Solid fluctuating merging jams if b = d = 1, exactly computable throughput and critical density, equivalent to previous case (swapping g, d exchanges particles, holes) • Cruise-control casealso exactly solvable, but not typical, when a = 0 or b = 0

25. Interesting case: 0 < a , b < 1 and d = 1 • Fundamental diagram determined qualitatively correct by Gantner for g = 1, b and a small

26. Interesting case: 0 < a , b < 1 and d = 1 • Fundamental diagram determined qualitatively correct by Gantner for g = 1, b and a small • For this case, jams have simple interiors (“anti free flow”) with non-trivial boundaries

27. Interesting case: 0 < a , b < 1 and d = 1 • Fundamental diagram determined qualitatively correct by Gantner for g = 1, b and a small • For this case, jams have simple interiors (“anti free flow”) with non-trivial boundaries • Jams fluctuate and merge, leading to genuine phase separation

28. Interesting case: 0 < a , b < 1 and d = 1 • Fundamental diagram proved correct by Gantner for g = 1, b < .3 , and a small • For this case, jams have simple interiors (“anti free flow”) with non-trivial boundaries • Jams fluctuate and merge, leading to genuine phase separation • Proof requires obtaining control on jam boundaries, using comparison with finite Markov chain and branching annihilating random walk

29. Interesting case: 0 < a , b < 1 and d = 1 • Fundamental diagram proved correct by Gantner for g = 1, b < .3 , and a small • For this case, jams have simple interiors (“anti free flow”) with non-trivial boundaries • Jams fluctuate and merge, leading to genuine phase separation • Proof requires obtaining control on jam boundaries, using comparison with finite Markov chain and branching annihilating random walk • Proof probably also works for g = 0

30. More interesting case: 0 < a , b , g < 1 and d = 1(no anti-free flow) • Jams have complicated interiors, with stretches of free flow and congestion mixed together

31. More interesting case: 0 < a , b , g < 1 and d = 1(no anti-free flow) • Jams have complicated interiors, with stretches of free flow and congestion mixed together • Free flow behaves similar to so-called “chipping model”: blocks of free flow diffuse and merge, with fragmentation occurring at the edges

32. More interesting case: 0 < a , b , g < 1 and d = 1(no anti-free flow) • Jams have complicated interiors, with stretches of free flow and congestion mixed together • Free flow behaves similar to so-called “chipping model”: blocks of free flow diffuse and merge, with fragmentation occurring at the edges • True phase separation in TCA would correspond to formation of “condensate” in chipping model

33. More interesting case: 0 < a , b , g < 1 and d = 1(no anti-free flow) • Jams have complicated interiors, with stretches of free flow and congestion mixed together • Free flow behaves similar to so-called “chipping model”: blocks of free flow diffuse and merge, with fragmentation occurring at the edges • True phase separation in TCA would correspond to formation of “condensate” in chipping model • But Rajesh and Krishnamurthy (2002) claim chipping model does not form condensate when chipping is asymmetric (as it is in TCA)

34. A look at the chipping model

35. The version with one-sided chipping is most like the TCA

36. If chipping is symmetric, then it is known that particles cluster into a single large “aggregate”, surrounded by a “fluid” with finite density

37. If chipping is symmetric, then it is known that particles cluster into a single large “aggregate”, surrounded by a “fluid” with finite density • If chipping is not symmetric, numerical studies and heuristic arguments strongly suggest there is no aggregate; fluid can have arbitrarily high density

38. If chipping is symmetric, then it is known that particles cluster into a single large “aggregate”, surrounded by a “fluid” with finite density • If chipping is not symmetric, numerical studies and heuristic arguments strongly suggest there is no aggregate; fluid can have arbitrarily high density • Analogy with TCA suggests traffic jams and free flow always have finite length; this would mean no true phase separation

39. A closer look at the analogy suggests the following modified chipping model:

40. This “harmless” change introduces correlations between chipping and movement that make a big difference

41. This “harmless” change introduces correlations between chipping and movement that make a big difference • Large cluster becomes a “machine” that creates a fluid with particle density = 1

42. This “harmless” change introduces correlations between chipping and movement that make a big difference • Large cluster becomes a “machine” that creates a fluid with particle density = 1 • If initial particle density > 1, aggregate is formed, surrounded by density 1 fluid

43. This “harmless” change introduces correlations between chipping and movement that make a big difference • Large cluster becomes a “machine” that creates a fluid with particle density = 1 • If initial particle density > 1, aggregate is formed, surrounded by density 1 fluid • If initial particle density < 1, no aggregate

44. This “harmless” change introduces correlations between chipping and movement that make a big difference • Large cluster becomes a “machine” that creates a fluid with particle density = 1 • If initial particle density > 1, aggregate is formed, surrounded by density 1 fluid • If initial particle density < 1, no aggregate • Analogy with TCA suggests that both critical values exist and “typical fundamental diagram” is correct