SUMMER SCHOOL - KINETIC EQUATIONS LECTURE IV: HOMOGENIZATION OF KINETIC FLOWS IN NETWORKS. Shanghai, 2011.

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1 SUMMER SCHOOL - KINETIC EQUATIONS LECTURE IV: HOMOGENIZATION OF KINETIC FLOWS IN NETWORKS. Shanghai, C. Ringhofer ringhofer@asu.edu, math.la.asu.edu/ chris

2 OVERVIEW Quantum and classical description of many particle systems in confined geometries and periodic media. L1 Introduction. Transport pictures (Hamilton, Schrödinger, Heisenberg, Wigner). Thermodynamics and entropy. Two different spatial scales. Induce two different time scales. Macroscopic (simpler) description on the large scale. Retain the microscopic transport information on the small scale. L2 Semiclassical transport in narrow geometries. Free energy models. Treating complex geometries. Effective mass approximations in thin tubes. L3 Quantum transport in narrow geometries. Thin plates, narrow tubes. Sub-band modeling. L4 Homogenization in unstructured media and networks.

3 NON - PHYSICS APPLICATIONS OF KINETIC EQUATIONS Ensemble of objects (particles, agents), described by a state q. Two types of dynamics: 1 Continuous motion: 2 Random change: d dt q = v(q) q q new, dp[q new = q ] = P(q, q) dq Monte Carlo: Solve ODES for step 1 and randomly change the state for step 2. Multi - Agent Systems.

4 KINETIC EQUATIONS Kinetic density f (q, t) (probability density) Kinetic equation for the evolution t f + v(q) q f = P(q, q )ω(q )f (q, t) dq ω(q)f (q, t) ω: scattering frequency Gas dynamics: q = (x, p), v(x, p) = ( p m, xv(x)) T Advantage: coarse graining, large time scales, thermodynamics. Other applications: multi - agent models in economics, traffic flow, social science. Usually more complex interactions.

5 OUTLINE GOALS: 1 Continuous space models for flows on arbitrarily complex networks. 2 Replace network structure with interaction with a background in a kinetic equation. 3 Determine large time behavior by using averaging techniques. OUTLINE: 1 Motivation: Network examples. 2 Part I: Kinetic transport models for networks with enlarged state space. Large time averages. Diffusion. Transport coefficients. 3 Part II: Homogenization and network reorganization. Large networks. Large time scales. Topolgy. 4 Conclusions

6 GENERIC TRANSPORT MODEL FOR AGENTS IN A NETWORK 02 Graph with nodes x = z n, n = 1 : N and (possible) arcs d mn = z n z m. Wait: An agent (particle) arrives at z n at time t, and waits a random time τ to be processed. dp[τ = s] = u(s, n) ds. Choice: It then chooses the next node z k to go to. P[k = m] = A(m, n) (Kirchhoff matrix). Flight: It chooses a random travel time a. dp[a = s] = T(s, k, n) ds. Arrives at z k at time t + τ + a.

7 Example I (the easy case): Battiston - Stiglitz Network (Flow of products from raw material to customer. Flow of money and orders from to customer raw material.) Optimization: Göttlich, Herty, CR ( 08)

8 Example II (the hard case): Small World Network (Propagation of information (or disease) in a closed society) original arcs

9 Example III (the hard case): Small World Network (Air traffic networks, from D. Brockmann et. al. European J. Phys, )

10 THE PROBLEM WITH STANDARD KINETIC MODELS: 06 t f (x, q, t) + x (v(q)f ) = P(x, q, q )ω f dq ωf q : state, v(q): velocity Collisions (the changes of the state q) are instantaneous. Mean free path and mean collision frequency. Particles travel for an exponentially distributed time. q(t + t) = (1 r)q + rp, r {0, 1}, P[r = 1] = tω(q) dp[t = t q] = ω(q)e ω(q)t dt, Particles reach the next node only on average! Poissonparticlemoviehere dp dy [ x = y] = yω(q) ω(q) v(q) e v dq

11 ENLARGED STATE SPACE MODELS 09 Goal: Derive a kinetic equation in R d, such that the network structure enters only in the scattering cross sections, and the PDF stays concentrated on the nodes and arcs. State: (x, v, m, a, η, τ) m: node to travel to ( v v ) a: flight time ( v ) Clock variables: η: time elapsed since leaving the last node. τ: time elapsed since arrival at current node. Derive kinetic equation from a Monte Carlo model.

12 FREE FLIGHT PHASE: η < a x(t + t) = x(t) + tv(t) v(t + t) = v(t) m(t + t) = m(t) a(t + t) = a(t) η(t + t) = η(t) + t (flight clock) τ(t + t) = τ(t)

13 AT THE NEXT NODE: FOR η > a Monte Carlo: r: Decision variable whether to go to the next node m with a new a = a and a new v = v r {0, 1}, P[r = 1] = tω, P[r = 0] = 1 tω x(t + t) = x(t) v(t + t) = (1 r)v(t) + rv m(t + t) = (1 r)m(t) + rm a(t + t) = (1 r)a(t) + ra η(t + t) = (1 r)(η(t) + t) + r0 τ(t + t) = (1 r)(τ(t) + t) + r0

14 THE RANDOM PARTICLE MODEL q = (v, m, a, η, τ) m: node to travel to a: flight time η: time elapsed since we left the last node. τ: time elapsed since arrived at current node. x(t + t) = x(t) + H(a η) tv(t) + H(η a)0 v(t + t) = H(a η)v(t) + H(η a)[(1 r)v(t) + rv ] m(t + t) = H(a η)m(t) + H(η a)[(1 r)m(t) + rm ] a(t + t) = H(a η)a(t) + H(η a)[(1 r)a(t) + ra ] η(t+ t) = H(a η)(η(t)+ t)+h(η a)[(1 r)(η(t) + t) + r0] τ(t + t) = H(a η)τ(t) + H(η a)[(1 r)(τ(t) + t) + r0] Particle travels precisely a time a and a distance av!

15 PROBABILITIES OF THE NEW STATE: 11 q = (v, a, η, τ, m) dp[(v, a, η, m) new = (v, a, η, m )] x=zn = A(m, n)t(a, m, n)δ(η )δ(τ )δ(v z m zn a ) dv a m η Waiting time given by an imbedded Markov process for a general waiting time distribution u(n, τ) dτ. (Larsen 2010). P[r = 1] = tω(n, τ) = t τ u(n,τ) u(n,s) ds Particle travels precisely a time a and a distance av!

16 THE KINETIC EQUATION 13 Probability density: f (x, v, a, η, τ, m) t f + x [H(a η)vf ] + η f + H(η a) τ f = Q[f ] Q[f ](x) = δ(η) m A I (m, x)δ(v di m(x) ) a ω = H(η a)ω I (x, τ) A I (x), d I m(x), ω I : interpolants of A and the arc vectors: T(a, m, x)ω I f dv a η ω I f A I (m, z n ) = A(m, n), d m (z n ) = z m z n (???: How to construct the interpolants?)

17 CONCENTRATION: t f + x [H(a η)vf ] + η f + H(η a) τ f = Q[f ] Theorem If f is initially concentrated on the nodes f (x, v, a, η, τ, m, t = 0) = n δ(x z n)f i n(v, a, η, τ, m) then the density function f will stay concentrated on the nodes and the arcs for all time. On the kinetic level the choice of interpolants is irrelevant!, since Q is only evaluated at the nodes! particlemoviehere

18 LARGE TIME AVERAGES 15 Assume large graph and large time scale x x ε, t t ε t f + x [H(a η)vf ] = 1 ε C[f ] C[f ] x=zn = η f H(η a) τ f + + q = (v, a, η, τ, m) P(x, q, q )ω f dq ωf P(x, q, q ) x=zn = δ(η)δ(τ) m A(m, n)δ(v z m z n )T(a, m, n) a Generalization of Poupaud, Cercignani, Gamba, Levermore.

19 THE CHAPMAN - ENSKOG EXPANSION 16 : Derives a macroscopic equation for the spatio - temporal density ρ(x, t) = m f (x, v, a, η, τ, m, t) dvaητ Requires the computation of the kernel and the pseudo - inverse C + of C: P1 : C[f 0 ] = 0, f 0 = 1 P2 : C[f 1 ] = H(a η)vf 0, f 1 = 0 Result: Yields a macroscopic equation of the form t ρ(x, t) + x [Vρ εd x ρ] = 0

20 C can be factored in a relaxation operator and an invertible operator Transport coefficients computed exactly. C[f ] x=zn = P(x, q) ω f dq ωf η f H(η a) τ f C = R B R[f ] = P(x, q) f dq f, B[f ] = η f + H(η a) τ f + ωf C + = B 1 R + gives for for the kernel and the pseudo inverse: f 0 = B 1 P, C + = B 1 R +

21 TRANSPORT COEFFICIENTS: V = E[H(a η)vf 0 ], D = E[H(a η)vc + [H(a η)vf 0 ]] Mean velocity: The diffusion matrix D: Lemma D(x) is positive semidefinite V(x) = E[d(x)] E[a+τ](x) r T Ds = (r + s)t K(r + s) (r s) T K(r s), r, s R 2 8E[a + τ] K = cov[d] + var[a + τ] E[a + τ] 2 E[d]E[d]T E[d] = A I (m, x)d m (x), cov[d] = A I (m, x)d m (x)d m (x) T E[d]E[d] T

22 SUMMARY - PART I 19 : Kinetic model, for stochastic transport, restricted to an arbitrary network. Large time behavior (O( t ε ) time scale) given by asymptotics t ρ(x, t) + x [Vρ εd x ρ] = 0 Possibly even larger O( t ε 2 ) time scale if V = O(ε) (depends on arrangement of nodes). Requires the computation of the interpolants of the Kirchhoff matrix and the distance vectors A I (m, x), d I m(x). Not relevant on the kinetic microscopic level, since collisions only happen at the nodes. Necessary for computing large time averages.

23 PART II: HOMOGENIZATION AND REORGANIZATION OF UNSTRUCTURED NETWORKS 20 To compute on large scales the fine structured geometry of the network has to be approximated (interpolation, least squares, averaging). A I (m, x) : A I (x, z n ) A(m, n), d I m(x) : d I m(z n ) d mn, d mn = z m z n Would in general lead to measure valued transport coefficients. (Bouchitte, Fragala, 2001.) One approach: compute basis functions for a finite element method from microscopic models in each cell. ( DellaRossa, D Angelo, Quarteroni 2010). Requires some uniformity or quasi-periodicity. More applicable to problems where distances in the network have a physical meaning (porous media...)

24 ECONOMIC AND SOCIAL NETWORKS 22 IDEA: Re- arrange the nodes (compute a metric) such that d I (x), A I (x) become as uniform as possible. original arcs

25 Given A(m, n), T(a, m, n), u(τ, m, n), choose the z n such that the transport coefficients in the convection - diffusion equation become uniform. (Nonlinear) least squares or optimization problem for the node coordinates z n.

26 Cost functionals 23 z n z m should be proportional to the average time it takes to get from node n to node m, weighted by the probability of the path. 1 step velocity: J 1 = n { m A mn ( z m z n v 0 E[a + τ] mn )} 2 v 0 = O(1) gives scale of the graph. 2 step velocity: J 2 = n { mkn A mk A kn [ z m z n v 0 (E[a + τ] mk + E[a + τ] kn )]} 2 V should be small to give diffusion regime J V = V 2 minimize J 1 + J 2 + J 3 + J 4 + J V

27 Small World with 10 nodes 24 original arcs

28 Small World with 10 nodes reordered. Topologically equivalent!! reordered arcs

29 Flow field for Small World with 10 nodes originalall velocities

30 Flow field for Small World with 10 nodes reordered reorderedall velocities

31 Large World with 6000 nodes original arcs original arcs

32 Large World with 6000 nodes reordered 55 reordered arcs reordered arcs

33 Large World flow field with 6000 nodes reordered 55 reordered mean velocities

34 Map onto a carthesian grid by averaging over grid cells 27 o: holes in the grid with zero flux 55 reorderedgrid velocities

35 Relative Diffusivity per node ελ max(d) V 3 RELATIVE DIFFUSIVITY

36 Conclusion Two ideas: 1 Accurate microscopic model for transport on networks Quantitatively correct transport coefficients for macroscopic models. 2 Reorganizing the geometry of abstract complex networks to replace measure valued transport coefficients by relatively smooth functions. Extensions: Nonlinear problem via mean fields, V(x, ρ) = E[d(x)] E[a+τ](x,ρ). Transport coefficients dependent on density via clearing functions (Graves, Missbauer, Degond +CR).