Extensions of the Kac N-particle model to multi-linear interactions

1. Extensions of the Kac N-particle model to multi-linear interactions Irene M. Gamba Department of Mathematics and ICES The University of Texas at Austin Classical and Random Dynamics in Mathematical Physics CoLab UT Austin-Portugal, April 2010

2. Drawing from classical statistical transport of interactive/collisional kinetic models • Rarefied ideal gases-elastic:classical conservativeBoltzmann Transport eq. • Energy dissipative phenomena: Gas of elastic or inelastic interacting systems in the presence of a thermostat with a fixed background temperature өb or Rapid granular flow dynamics: (inelastic hard sphere interactions): homogeneous cooling states, randomly heated states, shear flows, shockwaves past wedges, etc. • (Soft) condensed matter at nano scale: Bose-Einstein condensates models and charge transport in solids: current/voltage transport modeling semiconductor. • Emerging applications from stochastic dynamics for multi-linear Maxwell type interactions : Multiplicatively Interactive Stochastic Processes: • information percolation models, particle swarms in population dynamics, • Goals: • Understanding of analytical properties: large energy tails • Long time asymptotics and characterization of asymptotics states • A unified approach for Maxwell type interactions and generalizations. • Spectral-Lagrangian solvers for dissipative interactions

3. Motivation:Connection between the kinetic Boltzmann equations and Kac probabilistic interpretation of statistical mechanics (Bobylev, Cercignani and IMG, arXiv.org’06, 09, CMP’09) Consider a spatially homogeneous d-dimensional ( d ≥ 2) “rarefied gas of particles” having a unit mass. Let f(v, t), where v ∈ Rdand t ∈ R+, be a one-point pdf with the usual normalization Assumptions: I – interaction (collision) frequency is independent of the phase-space variable (Maxwell-type) II - the total “scattering cross section” (interaction frequency w.r.t. directions) is finite. Choose such units of time such that the corresponding classical Boltzmann eqs. reads as a birth-death rate equation for pdfs with Q+(f) is the gain term of the collision integral which Q+ transforms finto another probability density

4. The same stochastic model admits other possible generalizations. For example we can also include multiple interactions and interactions with a background (thermostat). This type of model will formally correspond to a version of the kinetic equation for some Q+(f). where Q(j)+ , j = 1, . . . ,M, are j-linear positive operators describing interactions of j ≥ 1 particles, and αj ≥ 0 are relative probabilities of such interactions, where Assumption: Temporal evolution of the system is invariant under scaling transformations in phase space: if St is the evolution operator for the given N-particle system such that St{v1(0), . . . , vM(0)} = {v1(t), . . . , vM(t)} , t ≥ 0 , then St{λv1(0), . . . , λ vM(0)} = {λv1(t), . . . , λvM(t)} for any constant λ> 0 which leads to the property Q+(j) (Aλ f) = Aλ Q+(j) (f), Aλ f(v) = λd f(λ v) , λ > 0, (j = 1, 2, .,M) Note that the transformation Aλ is consistent with the normalization of f with respect to v. Note: this property on Q(j)+ is needed to make the consistent with the classical BTE for Maxwell-type interactions

5. Property:Temporal evolution of the system is invariant under scaling transformations of the phase space:Makes the use of the Fourier Transform a natural tool so the evolution eq. is transformedinto an evolution eq. for characteristic functions which is also invariant under scaling transformations k → λ k, k ∈ Rd If solutions are isotropic then -∞ -∞ where Qj(a1, . . . , aj) can be an generalized functions of j-non-negative variables. • All these considerations remain valid for d = 1, the only two differences are: • The evolving Boltzmann Eq should be considered as the one-dimensional Kac master equation, and one uses the Laplace transform • 2. We discussed a one dimensional multi-particle stochastic model with non-negative phase • variables v in R+,

6. The structure of this equation follows from the well-known probabilistic interpretation by M. Kac:Consider stochastic dynamics of N particles with phase coordinates (velocities) VN=vi(t) ∈ Ωd, i = 1..N , with Ω= R or R+ A simplified Kac rules of binary dynamics is:on each time-step t = 2/N , choose randomly a pair of integers 1 ≤ i < l ≤ N and perform a transformation (vi, vl) →(v′i , v′l) which corresponds to an interaction of two particles with ‘pre-collisional’ velocities vi and vl. Then introduce N-particle distribution function F(VN, t) and consider a weak form of the Kac Master equation (we have assumed thatV’ Nj= V’N j ( VNj , UN j · σ) for pairs j=i,l with σ in a compact set) dσ 2 ΩdN ΩdN x Sd-1 Introducing a one-particle distribution function (by setting v1 = v) and the hierarchy reduction for B= -∞ or B=0 B B B The assumed rules lead (formally, under additional assumptions) to molecular chaos, that is The corresponding “weak formulation” for f(v,t) for any test function φ(v) where the RHS has a bilinear structure from evaluating f(vi’,t) f(vl’, t) M. Kac showed yields the the Boltzmann equation of Maxwell type in weak form (or Kac’s walk on the sphere)

7. Recall A general formstatistical transport : The space-homogenous BTE with external heating sources Important examples from mathematical physics and social sciences: The term models external heating sources: • background thermostat (linear collisions), • thermal bath (diffusion) • shear flow (friction), • dynamically scaled long time limits (self-similar solutions). ‘v v η v* ‘v* u’= (1-β) u + β |u| σ , with σthe direction of elastic post-collisional relativevelocity Inelastic Collision

8. The collision frequency is given by Qualitative issues on elastic: Bobylev,78-84, and inelastic: Bobylev, Carrillo I.G, JSP2000, Bobylev, Cercignani 03-04,with Toscani 03, with I.M.G. JSP’06, arXiv.org’06, CMP’09 Classical work of Boltzmann, Carleman, Arkeryd, Shinbrot,Kaniel, Illner,Cercignani, Desvilletes, Wennberg, Di-Perna, Lions, Bobylev, Villani, (for inelastic as well),Panferov, I.M.G,Alonso (spanning from 1888 to 2009) Qualitative issues on variable hard spheres, elastic and inelastic: I.G., V.Panferov and C.Villani, CMP'04, Bobylev, I.G., V.Panferov JSP'04, S.Mishler and C. Mohout, JSP'06, I.G.Panferov, Villani 06 -ARMA’09, R. Alonso and I.M. G., 07. (JMPA ‘08, and preprints 09)

9. Recall self-similarity:

10. Energy dissipation implies the appearance of Non-Equilibrium Stationary Statistical States

11. Back to molecular models of Maxwell type (as originally studied) so is also a probability distribution function in v. Then: work in the space of “characteristic functions” associated to Probabilities: “positive probability measures in v-space are continuous bounded functions in Fourier transformedk-space” The Fourier transformed problem: Γ Fourier transformed operator characterized by One may think of this model as the generalization original Kac (’59) probabilistic interpretation of rules of dynamics on each time step Δt=2/N of N particles associated to system of vectors randomly interchanging velocities pairwise while preserving momentum and local energy, independently of their relative velocities. Bobylev, ’75-80, for the elastic, energy conservative case. Drawing from Kac’s models and Mc Kean work in the 60’s Carlen, Carvalho, Gabetta, Toscani, 80-90’s For inelastic interactions: Bobylev,Carrillo, I.M.G. JSP’00 Bobylev, Cercignani,Toscani, 03, Bobylev, Cercignani, I.M.G’06 and 09, for general non-conservative problem

12. Recall fromFourier transform: nthmoments of f(., v) are nth derivatives of φ(.,k)|k=0 Θ For isotropic (x = |k|2/2 ) or self similar solutions (x = |k|2/2 eμt), μ is the energy dissipation rate, that is: Θt = - μΘ ,and with , the Fourier transformed collisional gain operator is written Kd accounts for the integrability of the function b(1-2s)(s-s2)(N-3)/2 For isotropic solutions the equation becomes (after rescaling in time the dimensional constant) φt + φ = Γ(φ , φ ) ; φ(t,0)=1, φ(0,k)=F (f0)(k), Θ(t)= - φ’(0) In this case, using the linearization of Γ(φ , φ ) about the stationary state φ=1, we can inferred the energy rate of change by looking at λ1defined by kinetic energy is dissipated < 1 λ1:=∫10 (aβ(s) + bβ(s)) G(s) ds= 1kinetic energy is conserved > 1 kinetic energy is generated

13. Examples Existence, asymptotic behavior - self-similar solutions and power like tails: From a unified point of energy dissipative Maxwell type models: λ1energy dissipation rate(Bobylev, I.M.G.JSP’06, Bobylev,Cercignani,I.G. arXiv.org’06- CMP’09)

14. Classical Existence approach : Wild's sum in the Fourier representation. The existence theorems for the classical elastic case ( β=e = 1) of Maxwell type of interactions were proved by Morgenstern, ,Wild 1950s, Bobylev 70s and for inelastic ( β<1) by Bobylev,Carrillo, I.M.G. JSP’00 using the Fourier transform • rescale time t →τ and solve the initial value problem Γ Γ 1-β/2 β/2 β/2 by a power series expansion in time where the phase-space dependence is in the coefficients Wild's sum in the Fourier representation. Γ Note that if the initial coefficient |φ0|≤1, then |Фn|≤1 for any n≥ 0. the series converges uniformly for τϵ[0; 1).

15. Applications to agent interactions • Two examples: • M-game multi agent model(Bobylev Cercignani, Gamba, CMP’09) • A couple ofinformation percolation models • (Dauffie, Malamud and Manso, 08-09)

16. An example for multiplicatively interacting stochastic process: • M-game multi linear models (Bobylev, Cercignani, I.M.G.; CMP’09): • particles: j ≥ 1indistinguishable players • phase state: individual capitals (goods) ischaracterized by a vector Vj= (v1;…; vj) ϵRj+ • A realistic assumption is that a scaling transformation of the phase variable (such as a change of • goods interchange) does not influence a behavior of player. • The game of these n partners is understood as a random linear transformation (j-particle collision) Set V’j= G Vj ; Vj= (v1; …; vj) ; V’j= (v’1, … ; v’j ) ; where G is a square j x j matrix with non-negative random elements such that the model does not depend on numeration of identical particles. Simplest class: a 2-parameter family G = {gik , i, k = 1, . . . , j} , such that gik = a, if i = k and gik= b, otherwise, The parameters (a,b) can be fixed or randomly distributed in R+2 with some probability density Bn(a,b). The corresponding transformation is

17. Model of M players participating in a M-linear ‘game’ according to the Kac rules (Bobylev, Cercignani,I.M.G.) CMP’09 • Assume • VN(t), N >> M undergoes random jumps caused by interactions. • Intervals between two successive jumps have the Poisson distribution with the average • ΔtM = l = Θ/N, interaction frequency with • Then we introduce M-particle distribution function F(VN; t) and consider a weak form as in the Kac Master eq: • Jumps are caused by interactions of 1 ≤ j ≤ M ≤ N particles (the case M =1 is understood as a interaction with background) • Relative probabilities of interactions which involve 1; 2; …;M particles are given by non-negative real numbers β1; β2 ; …. βM such that β1 + β2 + …+ βM = 1 , • So, it is possible to reduce the hierarchy of the system to

18. Taking the Laplace transform of the probability f: • Taking the test function on the RHS of the equation for f: • And assuming the “molecular chaos” assumption (factorization) ∞ In the limit N The evolution of the corresponding characteristic function is given by with initial condition u|t=0 = e−x (the Laplace transformed condition from f|t=0 =δ(v − 1) )

19. Another Example: Information aggregation model with equilibrium search dynamics (Duffie, Malamud & Manso 08) For any search-effort policy function C(n), the cross-sectional distribution ft of precisions and posterior means of the i-agents is almost surely given by ft(n; x; w) = μC(n,t) pn(x |Y (w)) where μt(n)is the fraction of agents with information precision n at time t, which is the unique solution of the differential equation below (of generalized Maxwell type) and pn( x| Y(w) ) is the Y-conditional Gaussian density of E(Y |X1; …. ;Xn), for any n signals X1; … ;Xn. This density has conditional mean and conditional variance mt(n)satisfies the dynamic equation withπ(n) a given distribution independent of any pair of agents Where μtC(n) = C(n) μ(n,t) is the effort-weighted measure such that: C(n) is the search-effort policy function

20. Example from information search (percolation) model not of Maxwell type!! For μt(n) for the fraction of agents with precision n (related to the cross-sectional distribution μtof information precision at time t in a given set) its the evolution equation is given by Where μtC(n) = C(n) μ(n,t) is the effort-weighted measure such that: C(n) is the search-effort policy function Linear term: represents the replacement of agents with newly entering agents. Gain Operator:The convolution of the two measuresμtC*μtCrepresents the gross rate at which new agents of a given precision are created through matching and information sharing. Loss operator: The last term of captures the rate μtCμtC(N) of replacement of agents with prior precision n with those of some new posterior precision that is obtained through matching and information sharing, where is the cross-sectional average search effort Remark:The stationary model can be viewed as a form of algebraic Riccati equation.

21. Another Example: Information aggregation model II (Duffie, Malamud & Giroux 09) • A random variable X of potential concern to all agents has 2 possible outcomes, • H (“high”) with probability n , and L (“low”) 1 − n. • Each agent is initially endowed with a sequence of signals {s1, . . . , sn}that may be informative about X. • The signals {s1, . . . , sn} (primitively observed by a particular agent are, conditional on X, independent with outcomes 0 and 1 (Bernoulli trials). • W.l.g assume P(si = 1|H) r P(si = 1|L). A signal i is informative if P(si = 1|H) > P(si = 1|L). Definition of “phase space” Basic probability by Bayes’ rule: the logarithm of the likelihood ratio between states H and L conditional on signals {s1, . . . , sn} “type” q of the set of signals • The higher the type f of the set of signals, the higher is the likelihood ratio between states H and L and the higher the posterior probability that X is high. • Any particular agent is matched to other agents at each of a sequence of Poisson arrival times with a mean arrival rate (intensity) l , which is the same across agents. • At each meeting time, m−1 other agents are randomly selected from the population of agents

22. Interaction law : The meeting group size m is a parameter of the information model that varies • Binary: for almost every pair of agents, the matching times and counterparties of one agent are • independent of those of the other: whenever an agent of type q meets an agent with type f and they communicate to each other their posterior distributions of X, • they both attain the posterior type q+f. • m-ary : whenever m agents of respective types 1, . . . , m share • their beliefs, they attain the common posterior type 1 + · · · + m. Aggregation model Statistical equation: (Smolukowski type) We let μt denote the cross-sectional distribution of posterior types in the population at time t. • The initial distribution μ0 of types induced by an initial allocation of signals to agents. • Assume that there is a positive mass of agents that has at least one informative signal. • That is, the first moment m1(μ0(q) ) > 0 if X = H, and m1(μ0(q) ) < 0 if X = L. • Assume that the initial law μ0 has a moment generating function, finite on a neighborhood of • z = 0 , where z = ⌠ ezqd(μ0(q))(Laplace transform) or Binary “m-ary” Multi-agent aggregation equation in integral form Existence by ‘Wild sums’ methods Self-similarity, Pareto tails formation and dynamically scaled solutions (with Ravi Srinivasan)

23. We notice the similarity with the the Kac model: let the type signals Vm and its posterior V’m with V’m= G Vm ; Vm= (v1; …; vm) ; V’m= (v’1, … ; v’m) ; where G is a square m x m matrix with entries G = {gik = 1 , for all i, k = 1, . . . , m} , Then the m-particle distribution function F(VN, t) and the weak form of the Kac Master equation for N=m 2 Introducing a one-particle distribution function (by setting v1 = v) and the hierarchy reduction The assumed rules lead (formally, under additional assumptions) to molecular chaos, that is Then the aggregation models hold for f(vm , t ) for either binary or multi-agent forms The approach extends to more general information percolation models where thesignal type do not necessarily aggregatebut “distributes ” itself between the posterior types (in collaboration with Ravi Srinivasan)

24. Rigorous results for Maxwell type interacting models (Bobylev, Cercignani, I.M.G.;.arXig.org ‘06 - CPAM 09) Existence, stability,uniqueness, Θ with 0 < p < 1 infinity energy, or p ≥ 1 finite energy

25. (for initial data with finite energy) Relates to the work of Toscani, Gabetta,Wennberg, Villani,Carlen, Carvallo,…..

26. - I Wealth distribution Spectrum Boltzmann Spectrum Aggregation Spectrum

27. Stability estimate for a weighted pointwise distance for finite or infinite initial energy These estimates are a consequence of the L-Lipschitz condition associated to Γ: they generalized Bobylev, Cercignani and Toscani,JSP’03 and later interpeted as “contractive distances” (as originally by Toscani, Gabetta, Wennberg, ’96) These estimates imply, jointly with the property of the invariance under dilations for Γ, selfsimilar asymptotics and the existence of non-trivial dynamically stable laws.

28. Existence of Self-Similar Solutions with initial conditions REMARK: The transformation , forp > 0 transforms the study of the initial value problem touo(x) = 1+x and ||uo|| ≤1 so it is enough to study the casep=1

29. In addition, the corresponding Fourier Transform of the self-similar pdf admits an integral representation by distributionsMp(|v|)with kernels Rp(τ) , for p = μ−1(μ∗). They are given by: Similarly, by means of Laplace transform inversion, for v ≥0 and 0 < p ≤ 1 with These representations explain the connection of self-similar solutions with stable distributions

30. Theorem: appearance of stable law(Kintchine type of CLT)

32. Recall the self similar problem Then,

33. ms> 0 for all s>1.

35. Study of the spectral function μ(p) associated to the linearized collision operator For any initial state φ(x) = 1 – xp + x(p+Є) , p ≤ 1. Decay rates in Fourier space: (p+Є)[ μ(p) - μ(p +Є) ] For finite (p=1) or infinite (p<1) initial energy. μ(p) For p0 >1 and 0<p< (p +Є) < p0 For μ(1) = μ(s*), s* >p0 >1 Power tails CLT to a stable law Self similar asymptotics for: p0 1 s* p μ(s*)= μ(1) μ(po) Finite (p=1) or infinite (p<1) initial energy No self-similar asymptotics with finite energy For p0< 1 and p=1

36. )

37. In general we can see that 1. For more general systems multiplicatively interactive stochastic processes the lack of entropy functional does not impairs the understanding and realization of global existence (in the sense of positive Borel measures), long time behavior from spectral analysis and self-similar asymptotics. 2. “power tail formation for high energy tails” of self similar states is due to lack of total energy conservation, independent of the process being micro-reversible (elastic) or micro-irreversible (inelastic). It is also possible to see Self-similar solutions may be singular at zero. 3. The long time asymptotic dynamics and decay rates are fully described by the continuum spectrum associated to the linearization about singular measures. 4. Recent probabilistic interpretation was given by F. Bassetti and L. Ladelli (preprint 2010)

38. Back to the M-game model with initial condition u|t=0 = e−x • Example of (a,b) pair choice: The game of j ≥ 1 players is played in three steps: • 1- the participants collect all their goods and form a sum S = v1 + v 2 + · · · + vj ; • 2- the sum is multiplied by a random number θ≥ 0 distributed with given probability • density q(θ) in R+; • 3- the resulting sum S’ = θS = v′1 +· · ·+v′j is given back to the players in accordance • with the following rule: for some fixed or random parameter 0 ≤ g ≤ 1. • a part of it S’1 = (1-g) S’ is divided proportionally to initial contributions, • whereas the rest S′2 = gS′ is divided among all players equally, Simple algebra shows that this “game” is equivalent to chose (a,b)

39. Interpretation of the involved parameters in the (a,b) pair The meaning of the parameter qmay be given by: something was bought (or produced) for the value S and then sold for S ′ =qS (with gain if q > 1 or loss if q < 1). An interesting example arises from assuming the following probability density for : q(q) = q(q) + (1 − q) d(q − s) , s > 1, 0 ≤ q ≤ 1 where q characterizes a risk of complete loss. The parameter g can be interpreted as a fixed control parameter to give more chances to losers, (may be introduced in the game in order to prevent large differences between affluent and destitute in the future). In particular, our model explains how exactly these differences depend on the parameter .

40. In particular the M-game model reduces to for j ≥ 2, whose spectral function is , It is possible to prove that : μ(p) is a curve with a unique minima at p0>1 and approaches + ∞ as p 0 andμ’(1) < 0 for In addition it is possible to find . g* < g < 1 for which there are selfsimilar asymptotics and . another g** < 1 , such that g*<g<g**corresponding to second root conjugate to μ(1) So a self-similar attracting state with a power law exists and it is an attractor

41. A.V. Bobylev, C. Cercignani and I. M. Gamba, On the self-similar asymptotics for generalized  non-linear kinetic Maxwell models, Comm. Math. Phys. 291 (2009), no. 3, 599--644. A.V. Bobylev, C. Cercignani and I. M. Gamba, Generalized kinetic Maxwell models of granular gases; Mathematical models of granular matter Series: Lecture Notes in Mathematics Vol.1937, Springer, (2008). A.V. Bobylev, C. Cercignani and I. M. Gamba, On the self-similar asymptotics for generalized non-linear kinetic Maxwell models, arXiv:math-ph/0608035 A.V. Bobylev and I. M. Gamba, Boltzmann equations for mixtures of Maxwell gases: exact solutions and power like tails.J. Stat. Phys. 124, no. 2-4, 497--516. (2006). A.V. Bobylev, I.M. Gamba and V. Panferov, Moment inequalities and high-energy tails for Boltzmann equations wiht inelastic interactions, J. Statist. Phys. 116, no. 5-6, 1651-1682.(2004). A.V. Bobylev, J.A. Carrillo and I.M. Gamba, On some properties of kinetic and hydrodynamic equations for inelastic interactions, Journal Stat. Phys., vol. 98, no. 3?4, 743?773, (2000). D. Duffie, S. Malamud, and G. Manso,Information Percolation with Equilibrium Search Dynamics, Econometrica, 2009. D. Duffie, G. Giroux, and G. Manso, Information Percolation, preprint 2009 I.M. Gamba and Sri Harsha Tharkabhushaman, Spectral - Lagrangian based methods applied to computation of Non - Equilibrium Statistical States.Journal of Computational Physics 228 (2009) 2012–2036 I.M. Gamba and Sri Harsha Tharkabhushaman, Shock Structure Analysis Using Space Inhomogeneous Boltzmann Transport Equation, To appear in Jour. Comp Math. 09 And references therein Thank you very much for your attention