Random Averaging. Eli Ben-Naim Los Alamos National Laboratory. Paul Krapivsky (Boston University) John Machta (University of Massachusetts)

Transcription

1 Random Averaging Eli Ben-Naim Los Alamos National Laboratory Paul Krapivsky (Boston University) John Machta (University of Massachusetts) Talk, papers available from:

2 Plan I. Averaging II. Restricted averaging III.Diffusive averaging IV.Orientational averaging

3 Themes 1. Scaling and multiscaling 2. Cascades 3. Pattern formation and bifurcations 4. Phase transitions and synchronization

4 1. Averaging

5 The basic averaging process N identical particles (grains, billiard balls) Each particle carries a number (velocity) v i Particles interact in pairs (collision) Both particles acquire the average (inelastic) (v 1, v 2 ) ( v1 + v 2 2, v 1 + v 2 2 ) Melzak 76

6 Conservation laws & dissipation Total number of particles is conserved Total momentum is conserved N i=1 v i = constant Energy is dissipated in each encounter E i = 1 2 v2 i E = 1 4 (v 1 v 2 ) 2 We expect the velocities to shrink

7 Some details Dynamic treatment Each particle collides once per unit time Random interactions The two colliding particles are chosen randomly Infinite particle limit is implicitly assumed N Process is galilean invariant Set average velocity to zero x x + x 0 x = 0

8 The temperature Definition T = v 2 Time evolution = exponential decay dt dt = λ T λt T = T 0 e λ = 1 2 All energy is eventually dissipated Trivial steady-state P (v) δ(v)

9 The moments Kinetic theory P (v, t) t = dv 1 dv 2 P (v 1, t)p (v 2, t) [ δ ( v v 1 + v 2 2 ) ] δ(v v 1 ) Moments of the distribution M n = dv v n P (v, t) M 0 = 1 M 2n+1 = 0 Closed nonlinear recursion equations dm n dt Asymptotic decay n 2 + λ n M n = 2 n m=2 ( n m ) M m M n m λ n < λ m + λ n m M n e λ nt with λ n = 1 2 (n 1)

10 Multiscaling Nonlinear spectrum of decay constants λ n = 1 2 (n 1) Spectrum is concave, saturates λ n < λ m + λ n m Each moment has a distinct behavior M n M m M n m as t Multiscaling Asymptotic Behavior

11 The Fourier transform The Fourier transform F (k) = Obeys closed, nonlinear, nonlocal equation F (k) t Scaling behavior, scale set by second moment Nonlinear differential equation + F (k) = F 2 (k/2) dv e ikv P (v, t) F (k, t) f ( ke λt) λ = λ 2 2 = 1 4 λ z f (z) + f(z) = f 2 (z/2) f(0) = 1 f (0) = 0 Solution f(z) = (1 + z )e z

12 The velocity distribution Self-similar form P (v, t) e λt p ( ve λt) Obtained by inverse Fourier transform p(w) = 2 π 1 (1 + w 2 ) 2 Power-law tail p(w) w 4 1. Temperature is the characteristic velocity scale 2. Multiscaling is consequence of diverging moments of the power-law similarity function

13 Stationary Solutions Stationary solutions do exist! F (k) = F 2 (k/2) Family of exponential solutions F (k) = exp( kv 0 ) Lorentz/Cauchy distribution P (v) = 1 πv (v/v 0 ) 2 How is a stationary solution consistent with energy dissipation?

14 Extreme Statistics Large velocities, cascade process v ( v 2, v 2 ) (v 1, v 2 ) ( v1 + v 2 2, v 1 + v 2 2 ) Linear evolution equation P (v) t = 4P ( v 2 ) P (v) Steady-state: power-law distribution P (v) v 2 4P ( v 2 ) = P (v) Divergent energy, divergent dissipation rate

15 Injection, Cascade, Dissipation Experiment: rare, powerful energy injections Lottery MC: award one particle all dissipated energy ln P ( v ) v 0 ln v V Injection selects the typical scale!

16 I. Conclusions Moments exhibit multiscaling Distribution function is self-similar Power-law tail Stationary solution with infinite energy exists Driven steady-state Energy cascade

17 1I. Restricted Averaging

18 The compromise process Opinion measured by a continuum variable Compromise: reached by pairwise interactions (x 1, x 2 ) < x < ( x1 + x 2 Conviction: restricted interaction range 2 x 1 x 2 < 1, x 1 + x 2 2 ) Minimal, one parameter model Mimics competition between compromise and conviction Weisbuch 2001

19 Problem set-up Given uniform initial (un-normalized) { distribution P 0 (x) = Find final distribution Multitude of final steady-states P 0 (x) = P (x) =? N i=1 1 x < 0 x > m i δ(x x i ) x i x j > 1 Dynamics selects one (deterministically) Multiple localized clusters

20 Numerical methods, kinetic theory Same master equation, restricted integration P (x, t) t = x 1 x 2 < 1 dx 1 dx 2 P (x 1, t)p (x 2, t) Direct Monte Carlo simulation of stochastic process Numerical integration of rate equations [ δ ( x x 1 + x 2 2 ) δ(x x 1 ]

21 Rise and fall of central party 0 < < < < Central party may or may not exist!

22 Resurrection of central party < < < < Parties may or may not be equal in size

23 Bifurcations and Patterns

24 Self-similar structure, universality Periodic sequence of bifurcations 1. Nucleation of minor cluster branch 2. Nucleation of major cluster brunch 3. Nucleation of central cluster Alternating major-minor pattern Clusters are equally spaced Period L gives major cluster mass, separation x( ) = x( ) + L L = 2.155

25 How many political parties? frequency number of parties Data: CIA world factbook countries with multi-party parliaments Average=5.8; Standard deviation=2.9

26 Cluster mass Masses are periodic m( ) = m( + L) Major mass M L = Minor mass m Why are the minor clusters so small? gaps?

27 Scaling near bifurcation points Minor mass vanishes m ( c ) α Universal exponent α = { 3 type1 4 type3 L-2 is the small parameter explains small saturation mass

28 Heuristic derivation of exponent Perturbation theory Major cluster Minor cluster Rate of transfer from minor cluster to major cluster dm = 1 + ɛ x( ) = 0 x( ) = ±(1 + ɛ/2) dt = m M Process stops when x e t f /2 ɛ Final mass of minor cluster m ɛ e t x 2 e t m( ) m(t f ) ɛ 3 α = 3

29 Linear stability analysis Fastest growing mode Pattern selection P 1 e i(kx+wt) = w(k) = 8 k sin k 2 2 k sin k 2 dw dk = L = 2π k = Traveling wave (FKPP saddle point analysis) dw dk = Im(w) Im(k) = L = 2π k = Patterns induced by wave propagation from boundary However, emerging period is different < L < Pattern selection is intrinsically nonlinear

30 II. Conclusions Clusters form via bifurcations Periodic structure Alternating major-minor pattern Central party does not always exist Power-law behavior near transitions Nonlinear pattern selection

31 III. Diffusive Averaging

32 Diffusive Forcing Two independent competing processes 1. Averaging (nonlinear) (v 1, v 2 ) ( v1 + v 2 2, v 1 + v 2 2 ) 2. Random uncorrelated white noise (linear) dv j dt = η j(t) η j (t)η j (t ) = 2Dδ(t t ) Add diffusion term to equation (Fourier space) (1 + Dk 2 )F (k) = F 2 (k/2) System reaches a nontrivial steady-state Energy injection balances dissipation

33 Infinite product solution Solution by iteration F (k) = Dk 2 F 2 (k/2) = Dk 2 (1 + D(k/2) 2 ) 2 F 4 (k/4) = Infinite product solution F (k) = [ 1 + D(k/2 i ) 2] 2 i i=0 Exponential tail Also follows from v ( ) P (v) exp v / D D 2 P (v) v 2 = P (v) P (k) Dk 2 1 k i/ D Non-Maxwellian distribution/overpopulated tails

34 Cumulant solution Steady-state equation F (k)(1 + Dk 2 ) = F 2 (k/2) Take the logarithm ψ(k) = ln F (k) ψ(k) + ln(1 + Dk 2 ) = 2ψ(k/2) Cumulant solution [ ] F (k) = exp n=1 ψ n ( Dk 2 ) n /n Generalized fluctuation-dissipation relations ψ n = λ 1 n = [ n] 1

35 Experiment A shaken box of marbles Menon 01 Aronson 05

36 III. Conclusions Nonequilibrium steady-states Energy pumped and dissipated by different mechanisms Overpopulation of high-energy tail with respect to equilibrium distribution

37 IV. Orientational Averaging

38 Orientational Averaging Each rod has an orientation 0 θ π Alignment by pairwise interactions (θ 1, θ 2 ) {( θ1 +θ 2, θ 1+θ ) ( θ1 +θ 2 +2π, θ 1+θ 2 +2π 2 2 ) θ 1 θ 2 < π θ 1 θ 2 > π Diffusive wiggling Kinetic theory P t = D 2 P θ 2 + π dθ j dt = η j(t) π dφ P ( θ φ 2 η j (t)η j (t ) = 2Dδ(t t ) ) ( P θ + φ ) P. 2

39 Fourier analysis Fourier transform P k = e ikθ = π π dθe ikθ P (θ) P (θ) = 1 2π k= P k e ikθ Order parameter Probes state of system R = R = e iθ = P 1 { 0 disordered state 1 perfectly ordered state Closed equation for Fourier modes P k = i+j=k G i,j P i P j G i,j = 0 when i j = 2n

40 Nonequilibrium phase transition Critical diffusion constant Subcritical: ordered phase D c = 4 π 1 R > 0 Supercritical: disordered phase R = 0 Critical behavior R (D c D) 1/2

41 Distribution of orientation Fourier modes decay exponentially with R P k R k Small number of modes sufficient

42 Partition of Integers Iterate the Fourier equation P k = G i,j P i P j = i+j=k i+j=k l+m=j G i,j G l,m P i P l P m = Series solution R = r 3 R 3 + r 5 R 5 + Partition rules k = i + j i 0 j 0 G i,j 0 r 3 = G 1,2 G 1,

43 Experiments A shaken dish of toothpicks

44 IV. Conclusions Nonequilibrium phase transition Weak noise: ordered phase (nematic) Strong noise: disordered phase Solution relates to iterated partition of integers Only when Fourier spectrum is discrete: exact solution possible for arbitrary averaging rates