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
Drawing from classical statistical transport of interactive/collisional kinetic models
• Rarefied ideal gases-elastic: classical conservative Boltzmann 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
•Spectral-Lagrangian solvers for dissipative interactions
generalizations.
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 ∈ Rd and 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 f into another
probability density
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)} ,
then
St{λv1(0), . . . , λ vM(0)} = {λv1(t), . . . , λvM(t)}
t≥0,
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
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 transformed into 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:
1. 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+,
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 that V’ N j= V’N j ( VN j , UN j · σ) for pairs j=i,l with σ
in a compact set)
dσ
Ωd
2
N
Introducing a one-particle distribution function (by setting v1 = v) and the hierarchy reduction
B
B
B
for B= -∞ or B=0
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)
Recall A general form statistical 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 (selfsimilar solutions).
‘v
v
η
‘v*
Inelastic Collision
v*
u’= (1-β) u + β |u| σ , with σ the direction of elastic post-collisional relative velocity
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)
Recall self-similarity:
Energy dissipation implies the appearance of Non-Equilibrium Stationary Statistical States
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 transformed k-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
Recall from Fourier 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 = - μ Θ , 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(0,k)=F (f0)(k),
φ(t,0)=1,
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 λ1 defined by
λ1 := ∫ 0 (aβ(s) + bβ(s)) G(s) ds
< 1
kinetic energy is dissipated
= 1
kinetic energy is conserved
kinetic energy is generated
1
> 1
Examples
Existence, asymptotic behavior - self-similar solutions and power like tails:
From a unified point of energy dissipative Maxwell type models: λ1 energy
dissipation rate (Bobylev, I.M.G.JSP’06, Bobylev,Cercignani,I.G. arXiv.org’06- CMP’09)
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 τ ϵ [0; 1).
Applications to agent interactions
Two examples:
•M-game multi agent model
(Bobylev Cercignani, Gamba, CMP’09)
• A couple of information percolation models
(Dauffie, Malamud and Manso, 08-09)
An example for multiplicatively interacting stochastic process:
M-game multi linear models (Bobylev, Cercignani, I.M.G.; CMP’09):
particles: j ≥ 1 indistinguishable players
phase state: individual capitals (goods) is characterized by a vector Vj = (v1;…; vj) ϵ Rj+
•
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
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
• 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) )
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
with π(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
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 μt of
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 measures μtC * μtC represents 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 μtC μtC(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.
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
1. 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.
2. 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.
3. At each meeting time, m−1 other agents are randomly selected from the population of agents
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 .
Aggregation model
• m-ary : whenever m agents of respective types 1, . . . , m share
their beliefs, they attain the common posterior type 1 + · · · + m.
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 = ⌠ ezq
d(μ0(q)) (Laplace transform)
or
Binary
aggregation equation in integral form
“m-ary”
Multi-agent
Existence by ‘Wild sums’ methods
Self-similarity, Pareto tails formation and dynamically scaled solutions (with Ravi Srinivasan)
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) ;
G is a square m x m matrix with entries
V’m = (v’1, … ; v’m) ;
where
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 the signal type
do not necessarily aggregate but “distributes ” itself between the posterior types
(in collaboration with Ravi Srinivasan)
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
(for initial data with finite energy)
Relates to the work of Toscani, Gabetta,Wennberg, Villani,Carlen, Carvallo,…..
-I
Aggregation Spectrum
Boltzmann Spectrum
Wealth distribution
Spectrum
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.
Existence of Self-Similar Solutions
with initial conditions
REMARK: The transformation
problem to
uo(x) = 1+x and
, for p > 0 transforms the study of the initial value
||uo|| ≤ 1
so it is enough to study the case p=1
In addition, the corresponding Fourier Transform of the self-similar pdf admits an integral
representation by distributions Mp(|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
Theorem: appearance of stable law
(Kintchine type of CLT)
Recall the self similar problem
Then,
ms> 0 for all s>1.
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.
μ(p)
Decay rates in Fourier space: (p+Є)[ μ(p) - μ(p +Є) ]
For finite (p=1) or infinite (p<1) initial energy.
For p0 >1 and 0<p< (p +Є) < p0
Self similar asymptotics for:
For μ(1) = μ(s*) , s* >p0 >1
Power tails
CLT to a stable law
1
p0
s*
p
μ(s*) = μ(1)
μ(po)
Finite (p=1) or infinite (p<1) initial energy
For p0< 1 and p=1
No self-similar asymptotics with finite energy
)
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)
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’ = θ S
=
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)
Interpretation of the involved parameters in the (a,b) pair
The meaning of the parameter q
may 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 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
and μ’(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
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Maxwell models, arXiv:math-ph/0608035
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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