Random Averaging Eli Ben-Naim Los Alamos National Laboratory Paul Krapivsky (Boston University) John Machta (University of Massachusetts) Talk, papers available from: http://cnls.lanl.gov/~ebn
Plan I. Averaging II. Restricted averaging III.Diffusive averaging IV.Orientational averaging
Themes 1. Scaling and multiscaling 2. Cascades 3. Pattern formation and bifurcations 4. Phase transitions and synchronization
1. Averaging
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
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
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
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)
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)
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
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
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
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 0 1 1 + (v/v 0 ) 2 How is a stationary solution consistent with energy dissipation?
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
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!
I. Conclusions Moments exhibit multiscaling Distribution function is self-similar Power-law tail Stationary solution with infinite energy exists Driven steady-state Energy cascade
1I. Restricted Averaging
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
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
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 ]
Rise and fall of central party 0 < < 1.871 1.871 < < 2.724 Central party may or may not exist!
Resurrection of central party 2.724 < < 4.079 4.079 < < 4.956 Parties may or may not be equal in size
Bifurcations and Patterns
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
How many political parties? frequency number of parties Data: CIA world factbook 2002 120 countries with multi-party parliaments Average=5.8; Standard deviation=2.9
Cluster mass Masses are periodic m( ) = m( + L) Major mass M L = 2.155 Minor mass m 3 10 4 Why are the minor clusters so small? gaps?
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
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
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 = 2.2515 Traveling wave (FKPP saddle point analysis) dw dk = Im(w) Im(k) = L = 2π k = 2.0375 Patterns induced by wave propagation from boundary However, emerging period is different 2.0375 < L < 2.2515 Pattern selection is intrinsically nonlinear
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
III. Diffusive Averaging
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
Infinite product solution Solution by iteration F (k) = 1 1 + Dk 2 F 2 (k/2) = 1 1 1 + 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) 1 1 + Dk 2 1 k i/ D Non-Maxwellian distribution/overpopulated tails
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 = [ 1 2 1 n] 1
Experiment A shaken box of marbles Menon 01 Aronson 05
III. Conclusions Nonequilibrium steady-states Energy pumped and dissipated by different mechanisms Overpopulation of high-energy tail with respect to equilibrium distribution
IV. Orientational Averaging
Orientational Averaging Each rod has an orientation 0 θ π Alignment by pairwise interactions (θ 1, θ 2 ) {( θ1 +θ 2, θ 1+θ 2 2 2 ) ( 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
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
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
Distribution of orientation Fourier modes decay exponentially with R P k R k Small number of modes sufficient
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 1,1 3 1 2 1 1
Experiments A shaken dish of toothpicks
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