A Boltzmann-type kinetic approach to the modeling of ...helper.ipam.ucla.edu/publications/traws1/traws1_13182.pdfThe spatially homogeneous model A Boltzmann-type kinetic approach to

The spatially homogeneous model

A Boltzmann-type kinetic approach to the modelingof vehicular traffic

Andrea Tosin

Istituto per le Applicazioni del Calcolo “M. Picone”Consiglio Nazionale delle Ricerche

Rome, Italy

Mathematical Foundations of TrafficUCLA, September 28-October 2, 2015

Andrea Tosin A Boltzmann-type kinetic approach to the modeling of vehicular traffic

Research outline

1 The spatially homogeneous modelContinuous velocity modelFrom continuous to discrete velocity modelFundamental diagrams and the phase transitionTraffic safety

2 The spatially non-homogeneous modelModel on a single roadModel on road networks

Research outline

1 The spatially homogeneous modelContinuous velocity modelFrom continuous to discrete velocity modelFundamental diagrams and the phase transitionTraffic safety

2 The spatially non-homogeneous modelModel on a single roadModel on road networks

The spatially homogeneous modelContinuous velocity modelFrom continuous to discrete velocity modelFundamental diagrams and the phase transition

Basics of kinetic modeling

Microscopic state of the vehicles: speed v ∈ [0, 1]Kinetic distribution function: f = f(t, v) s.t.

f(t, v) dv = fraction of vehicles with speed in [v, v + dv] at time t ≥ 0

The Boltzmann-type kinetic equation

∂tf = Q(f, f) :=

∫ 10

P(v∗ → v|v∗, ρ)f(t, v∗)f(t, v∗) dv∗ dv∗ − ρf (1)

P(v∗ → v|v∗, ρ) probability distribution of speed transitions due topairwise (binary) interactions among the vehicles:∫ 1

0

P(v∗ → v|v∗, ρ) dv = 1 ∀ v∗, v∗, ρ ∈ [0, 1] (2)

Mass conservation: ρ(t) :=∫ 1

0f(t, v) dv is constant, in fact from (1)-(2):

d

dt

∫ 10

f(t, v) dv =

∫ 10

Q(f, f) dv = 0

∂tf = Q(f, f) :=

∫ 10

0

P(v∗ → v|v∗, ρ) dv = 1 ∀ v∗, v∗, ρ ∈ [0, 1] (2)

d

dt

∫ 10

f(t, v) dv =

∫ 10

Q(f, f) dv = 0

∂tf = Q(f, f) :=

∫ 10

0

P(v∗ → v|v∗, ρ) dv = 1 ∀ v∗, v∗, ρ ∈ [0, 1] (2)

d

dt

∫ 10

f(t, v) dv =

∫ 10

Q(f, f) dv = 0

∂tf = Q(f, f) :=

∫ 10

0

P(v∗ → v|v∗, ρ) dv = 1 ∀ v∗, v∗, ρ ∈ [0, 1] (2)

d

dt

∫ 10

f(t, v) dv =

∫ 10

Q(f, f) dv = 0

∂tf = Q(f, f) :=

∫ 10

0

P(v∗ → v|v∗, ρ) dv = 1 ∀ v∗, v∗, ρ ∈ [0, 1] (2)

d

dt

∫ 10

f(t, v) dv =

∫ 10

Q(f, f) dv = 0

Qualitative properties

Let P(v∗ → ·|v∗, ρ) ∈P([0, 1]) for all v∗, v∗, ρ ∈ [0, 1]. We assume

W1(P(v∗ → ·|v∗, ρ), P(w∗ → ·|w∗, %)) ≤Lip(P) (|w∗ − v∗|+ |w∗ − v∗|+ |%− ρ|) ,

where W1 is the 1-Wasserstein metric for probability measures

Theorem (P. Freguglia, A. T., 2015 [4])

Fix ρ ∈ [0, 1] and f(0, v) =: f0(v) ∈M ρ+([0, 1]). There exists a uniquef ∈ C([0, +∞); M ρ+([0, 1])) which solves (1) in mild form:

f(t, v) = e−ρtf0(v)+

∫ t0eρ(s−t)

∫ 10

∫ 10P(v∗ → v|v∗, ρ)f(t, v∗)f(t, v∗) dv∗ dv∗ ds.

Given f01, f02 ∈M ρ+([0, 1]), the following continuous dependence estimate holds:

supt∈[0, T ]

W1(f1(t), f2(t)) ≤ e2 max{1,Lip(P)}TW1(f01, f02)

up to an arbitrarily large final time T < +∞.

Qualitative properties

Let P(v∗ → ·|v∗, ρ) ∈P([0, 1]) for all v∗, v∗, ρ ∈ [0, 1]. We assume

W1(P(v∗ → ·|v∗, ρ), P(w∗ → ·|w∗, %)) ≤Lip(P) (|w∗ − v∗|+ |w∗ − v∗|+ |%− ρ|) ,

where W1 is the 1-Wasserstein metric for probability measures

Fix ρ ∈ [0, 1] and f(0, v) =: f0(v) ∈M ρ+([0, 1]). There exists a uniquef ∈ C([0, +∞); M ρ+([0, 1])) which solves (1) in mild form:

f(t, v) = e−ρtf0(v)+

∫ t0eρ(s−t)

∫ 10

∫ 10P(v∗ → v|v∗, ρ)f(t, v∗)f(t, v∗) dv∗ dv∗ ds.

Given f01, f02 ∈M ρ+([0, 1]), the following continuous dependence estimate holds:

supt∈[0, T ]

W1(f1(t), f2(t)) ≤ e2 max{1,Lip(P)}TW1(f01, f02)

up to an arbitrarily large final time T < +∞.

Transition probabilities

We consider the following probability distribution of speed transitions:

P(v∗ → v|v∗, ρ) =

{(1− P )δv∗(v) + Pδmin{v∗+∆v, 1}(v) if v∗ ≤ v

∗

(1− P )δv∗(v) + Pδv∗(v) if v∗ > v∗

(3)where 0 < ∆v < 1 is given and

P = P (ρ) = 1− ργ (γ > 0)

is a probability of passing (cf. I. Prigogine, 1961)

The time-asymptotic solution of (1), with transition probabilities (3),concentrates only on speeds which are multiples of ∆v (G. Puppo, M.Semplice, A. T., G. Visconti, 2015 [6]) Picture

This suggests that, in order to study the macroscopic equilibria resultingfrom microscopic interactions, we can directly consider discrete velocitymodels

P(v∗ → v|v∗, ρ) =

∗

(1− P )δv∗(v) + Pδv∗(v) if v∗ > v∗

P = P (ρ) = 1− ργ (γ > 0)

P(v∗ → v|v∗, ρ) =

∗

(1− P )δv∗(v) + Pδv∗(v) if v∗ > v∗

P = P (ρ) = 1− ργ (γ > 0)

Discrete velocity model

Define a speed lattice

vj = (j − 1)∆v, j = 1, . . . , n, ∆v = 1n−1Assume that P is a discrete probability distribution over v:

P(v∗ → v|v∗, ρ) =n∑j=1

Pj(v∗, v∗, ρ)δvj (v) (4)

Fix ρ ∈ [0, 1] and take an initial condition of the form

f0(v) =n∑j=1

f0j δvj (v) with f0j ≥ 0,

n∑j=1

f0j = ρ (5)

The unique solution to (1) with transition probabilities (4) and initial condition (5) isf(t, v) =

∑nj=1 fj(t)δvj (v), where the fj ’s satisfy

dfj

dt=

n∑h=1

n∑k=1

Pjhk(ρ)fhfk − ρfj , fj(0) = f0j (6)

and Pjhk(ρ) := Pj(vh, vk, ρ).

P(v∗ → v|v∗, ρ) =n∑j=1

Pj(v∗, v∗, ρ)δvj (v) (4)

f0(v) =n∑j=1

n∑j=1

f0j = ρ (5)

dfj

dt=

n∑h=1

n∑k=1

P(v∗ → v|v∗, ρ) =n∑j=1

Pj(v∗, v∗, ρ)δvj (v) (4)

f0(v) =n∑j=1

n∑j=1

f0j = ρ (5)

dfj

dt=

n∑h=1

n∑k=1

P(v∗ → v|v∗, ρ) =n∑j=1

Pj(v∗, v∗, ρ)δvj (v) (4)

f0(v) =n∑j=1

n∑j=1

f0j = ρ (5)

dfj

dt=

n∑h=1

n∑k=1

Discrete transition probabilities

From (3) we deduce:

Pjhk(ρ) =

{(1− P )δjh + Pδj,min{h+1, n} if h ≤ k(1− P )δjk + Pδjh if h > k

(7)

which explicitly reads:

Pjhk(ρ) =

1− P if j = hP if j = h+ 1

0 otherwise

if h ≤ k, h < n

{1 if j = h

0 otherwiseif h = k = n

1− P if j = kP if j = h

0 otherwise

if h > k

We recall that P = P (ρ) = 1− ργ (γ > 0)

Asymptotic distributions

We study the asymptotic speed distributions f∞ = {f∞j }nj=1 resultingfrom (6)-(7): f∞j = lim

t→+∞fj(t)

The f∞j ’s form a one-parameter family, the parameter being the density ρwhich is conserved

Theorem (L. Fermo, A. T., 2014 [2])

For every n ≥ 2 and every ρ ∈ [0, 1] there exists a unique stable and attractiveequilibrium f∞ of (6), which satisfies:

f∞j ≥ 0 ∀ j = 1, . . . , n,n∑j=1

f∞j = ρ

In more detail, setting ρc :=(

12

) 1γ ,

for ρ < ρc there exists only one stable and attractive equilibriumfor ρ > ρc the previous equilibrium becomes unstable and a second stableand attractive one appears

ρc is a critical value for equilibria inducing a supercritical bifurcation

t→+∞fj(t)

f∞j ≥ 0 ∀ j = 1, . . . , n,n∑j=1

f∞j = ρ

12

) 1γ ,

t→+∞fj(t)

f∞j ≥ 0 ∀ j = 1, . . . , n,n∑j=1

f∞j = ρ

12

) 1γ ,

t→+∞fj(t)

f∞j ≥ 0 ∀ j = 1, . . . , n,n∑j=1

f∞j = ρ

12

) 1γ ,

t→+∞fj(t)

f∞j ≥ 0 ∀ j = 1, . . . , n,n∑j=1

f∞j = ρ

12

) 1γ ,

Fundamental and speed diagrams

We compute the macroscopic flux q and the mean speed u at equilibrium:

q(ρ) :=n∑j=1

vjf∞j (ρ), u(ρ) :=

q(ρ)

ρ

along with their standard deviations (dashed-red lines in the graphs below)

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1

Bifurcation phase transition, free (ρ < ρc) to congested flow (ρ > ρc)

Fundamental and speed diagrams

We compute the macroscopic flux q and the mean speed u at equilibrium:

q(ρ) :=n∑j=1

vjf∞j (ρ), u(ρ) :=

q(ρ)

ρ

along with their standard deviations (dashed-red lines in the graphs below)

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1

Bifurcation phase transition, free (ρ < ρc) to congested flow (ρ > ρc)

Traffic as a mixture of different vehicles

We consider two populations of vehicles, say cars (C) and trucks (T), withdifferent microscopic characteristics

Cars are shorter and faster while trucks are longer and slower

Speed grids:

vj = (j − 1)∆v, j = 1, . . . , np, p = C, T

∆v =1

nC − 1 , nT < nC

Characteristic lengths: `C = 1, `T > 1

Fraction of road occupancy:

s := ρC`C + ρT`T,

the admissible pairs of densities (ρC, ρT) ∈ [0, 1]2 being those s.t. s ≤ 1

Speed grids:

vj = (j − 1)∆v, j = 1, . . . , np, p = C, T

∆v =1

nC − 1 , nT < nC

s := ρC`C + ρT`T,

Speed grids:

vj = (j − 1)∆v, j = 1, . . . , np, p = C, T

∆v =1

nC − 1 , nT < nC

s := ρC`C + ρT`T,

Speed grids:

vj = (j − 1)∆v, j = 1, . . . , np, p = C, T

∆v =1

nC − 1 , nT < nC

s := ρC`C + ρT`T,

Speed grids:

vj = (j − 1)∆v, j = 1, . . . , np, p = C, T

∆v =1

nC − 1 , nT < nC

s := ρC`C + ρT`T,

Multi-population model

Model with self- and cross-interactions:

dfpjdt

=

np∑h,k=1

Pp,jhk (ρ)fphf

pk︸︷︷︸

self-interactions

+np∑h=1

nq∑k=1

Qpq,jhk (ρ)fphf

qk︸︷︷︸

cross-interactions

−(ρC+ρT)fpj (q := ¬p)

In the transition probabilities Pp,jhk , Qpq,jhk the density ρ is replaced by the

fraction of road occupancy s, i.e., the probability of passing is now

P = P (s) = 1− sγ (γ > 0)

Theorem (G. Puppo, M. Semplice, A. T., G. Visconti, 2015 [5])

If nC = nT and `C = `T = 1 the total kinetic distribution functionfj(t) := f

Cj (t) + f

Tj (t) solves the single-population model (6).

sc :=(

12

) 1γ is again a critical value for equilibria. For s = sc the

maximum flux is attained (G. Puppo, M. Semplice, A. T., G. Visconti,2015 [5]) Picture

dfpjdt

=

np∑h,k=1

Pp,jhk (ρ)fphf

pk︸︷︷︸

self-interactions

+np∑h=1

nq∑k=1

Qpq,jhk (ρ)fphf

qk︸︷︷︸

cross-interactions

P = P (s) = 1− sγ (γ > 0)

Cj (t) + f

sc :=(

12

dfpjdt

=

np∑h,k=1

Pp,jhk (ρ)fphf

pk︸︷︷︸

self-interactions

+np∑h=1

nq∑k=1

Qpq,jhk (ρ)fphf

qk︸︷︷︸

cross-interactions

P = P (s) = 1− sγ (γ > 0)

Cj (t) + f

sc :=(

12

dfpjdt

=

np∑h,k=1

Pp,jhk (ρ)fphf

pk︸︷︷︸

self-interactions

+np∑h=1

nq∑k=1

Qpq,jhk (ρ)fphf

qk︸︷︷︸

cross-interactions

P = P (s) = 1− sγ (γ > 0)

Cj (t) + f

sc :=(

12

References

[1] L. Fermo and A. Tosin.

A fully-discrete-state kinetic theory approach to modeling vehicular traffic.

SIAM J. Appl. Math., 73(4):1533–1556, 2013.

Fundamental diagrams for kinetic equations of traffic flow.

Discrete Contin. Dyn. Syst. Ser. S, 7(3):449–462, 2014.

A fully-discrete-state kinetic theory approach to traffic flow on road networks.

Math. Models Methods Appl. Sci., 25(3):423–461, 2015.

[4] P. Freguglia and A. Tosin.

Proposal of a risk model for vehicular traffic: A Boltzmann-like kinetic approach.

Preprint: arXiv:1506.05422, 2015.

[5] G. Puppo, M. Semplice, A. Tosin, and G. Visconti.

Fundamental diagrams in traffic flow: the case of heterogeneous kinetic models.

Commun. Math. Sci., 2015.

Accepted (Preprint: arXiv:1411.4988).

[6] G. Puppo, M. Semplice, A. Tosin, and G. Visconti.

Kinetic models for traffic flow resulting in a reduced space of microscopic velocities.

Preprint: arXiv:1507.08961, 2015.

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Figure: The asymptotic distribution function concentrates on multiples of ∆v Back

0 50 100 150 200 2500

1000

2000

3000

4000

5000

6000

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

density ρ / ρmax

flow

rate

: # v

ehic

les

/ lan

e / s

ec.

sensor data

Figure: Left: fundamental diagram from experimental data (Minnesota Departmentof Transportation, 2003). Right: fundamental diagram from the multi-population

model with γ = 0.5 Back

Appendix