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Classical solutions for the Boltzmann transport equations
for soft potentials with initial data near local Maxwellians
Irene M. Gamba
The University of Texas at Austin
Department of Mathematics and ICES
FRG - kinetic models
Brown University -- May 2010
Collaborators
Ricardo Alonso, Rice University, (Applied Math)
Emanuel Carneiro, Institute for Advanced Studies (Math)
Alexander Bobylev, Karlstad University,
Carlo Cercignani, Politecnico di Milano,
Ravi Srinivasan, UT Austin
Part II: new applications in kinetic collisional theory to multi agent dynamics
‘v
‘v*
v
v*
C = number of particle in the box
a = diameter of the spheres
d = space dimension
η
elastic
inelastic
u · η = uη := impact velocity
η:= impact direction
(random in S+d-1)
u · η = (v-v*) . η = - e ('v-'v*) · η = -e 'u . η
u · η ┴ = (v-v*) · η┴
= ('v-'v*) · η ┴ = 'u · η ┴
for hard spheres in 3-d: ( L. Boltzmann 1880's), in strong form:
For f (t; x; v) = f and f (t; x; v*) = f* describes the evolution of a
probability distribution function (pdf) of finding a particle centered at x ϵd,
with velocity v ϵd, at time t ϵ+ , satisfying
e := restitution coefficient : 0 < e ≤ 1
e = 1 elastic interaction , 0 < e < 1 inelatic interaction, ( e=0 „sticky‟ particles)
u = v-v* := relative velocity
|u · η| dη := collision rate
The classical Elastic/Inelastic Boltzmann Transport Equation
γ
θ
'v* and 'v are called pre-collisional velocities, and
v* and v are the corresponding post-collisional velocities
Grad’s assumption allows to split the collision operator in a gain and a loss part, Q( f, g) = Q+( f, g) − Q−( f, g) = Gain - Loss
The loss operator Q−( f, g) = f R(g), with R(g), called the collision frequency, given by
NOTE: The loss bilinear form is local in f and a weighted convolution in g.
while the gain is a bilinear form with a weighted symmetric convolution structure
Consider the Cauchy Boltzmann problem (Maxwell, Boltzmann 1860s-80s); Grad 1950s;
Cercignani 60s; Kaniel Shimbrot 80’s, Di Perna-Lions late 80’s):
Find a function f (t, x, v) ≥ 0 that solves the equation
Assumption on the model: A more general collision kernel reflecting intra molecular potentials and
higher dimensional spaces takes the form B(u, û · σ) with
(i) B(u, û · σ) = |u|λ b(û · σ) with -n < λ ≤ 1 ; we call soft potentials: -n < λ < 0
(ii) Grad‟s assumption: b(û · σ) є L1(S n−1), that is
Exchange of velocities in center of mass-relative velocity frame
Energy dissipation parameter β (or restitution parameter e)
Recall: Q+(v) operator in weak (Maxwell) form, and then it can easily be extended to
dissipative (inelastic) collisions
with
λβ
1-β
γ
Grad Cut-off
condition
λ = 0 for Maxwell Type (or Maxwell Molecule) models γ
= 1 for hard spheres models;
0< λ <1 for variable hard potential models,
-d < λ < 0 for variable soft potential models.
Average angular estimates & weighted Young‟s inequalities &
Hardy Littlewood Sobolev inequalities & sharp constantsR. Alonso and E. Carneiro’08 (to appear in Adv. Math.), and R. Alonso and E. Carneiro, IG, 09 (Comm.Math.Phys,10)
by means of radial symmertrization (rearrangement) techniques
and the angular mixing operator
Bobylev‟s (‟75) The angular operator on Maxwell type interactions λ=0
is the well know identity for the Fourier transform of the Q+
translations and reflections
is invariant under rotations
The weak formulation of the gain operator is a symmetric weighted convolution
Where the weight is an invariant under rotation operator involving
Young‟s inequality for variable hard potentials : 0 ≤ λ ≤ 1
Hardy-Littlewood-Sobolev type inequality for soft potentials : -n < λ < 0
for
0 ≤ λ ≤ 1
-n < λ < 0
Sketch of proof: important facts
1- Radial symmetrization
2- Preservation of the Lp - norm
and for set
α corresponds to moments weights
3- angular mixing operator on radial functions
Angular averaging lemma4-
5- Young‟s inequality for hard potentials for general 1 ≤ p , q, r ≤ ∞
The main idea is to establish a connection between the Q+ and P operators, and then use the knowledge
from the previous estimates. For α = 0 = λ (Maxwell type interactions) no weighted norms
The exponents p, q, r in Theorem 1 satisfy
Regroup and use Holder’s inequality and the angular averaging estimates on Lr’/q’ to obtain
These estimates resemble a Beckner and Brascamp-Lieb type inequality argument (for a nonlinear
weight) with best/exact constants approach to obtain Young‟s inequality
Remark: 1- Previous LP estimates by Gustafsson 88, Villani-Mouhot ‘04 for pointwise bounded b(u . σ),
I.M.G-Panferov-Villani ’03 for (p,1,p) with σ -integrable b(u . σ) in Sn-1.
2-The dependence on the weight α may have room to improvement. One may expect estimates with
polynomial (?) decay in α , like in L1α as shown Bobylev,I.M.G, Panferov and recently with Villani (97, 04,08)
(also previous work of Wennberg ’94, Desvilletes, 96, without decay rates.)
Maxwell type interactions λ=0 with β constant: the constants are sharp in (1,2, 2) and in (2,1,2)
Corollary:
The constant is achieved by the sequences:
and
So approximate
a Dirac
in x
(see Alonso and Carneiro, to appear in Adv Math 2009)
Hardy-Littlewood-Sobolev inequality for soft potentials -n < λ < 0 :
Applying Holder’s inequality and then the angular averaging lemma to the inner integral with (p, q, r) =
(a,1, a), a to be determined, one obtains
The choice of integrability exponents allowed to get
rid of the integrand singularity at s = −1, producing a
uniform control with respect to the inelasticity β.
Indeed, combining with the complete integral above, using triple Holder’s inq. yields
Is it possible to make such choice of a ?
This is a type of Brascamp-Lieb inequality argument (for a non-linear coordinate transformation)
Then: for
Using the classical Hardy-Littlewood-Sobolev inequality to obtain (Lieb ’83) with explicit constants
where the exponents satisfy
In fact, it is possible to find
1/a in the non-empty interval
so with C= D1 = C1 C21/a C3
1/a’
and
provided r < q !!
and r < q
The case r < p follows using the symmetric convolution structure Q+(f,g) (crucial for relating
to the classical HLS inequality for convolutions with singular kernels
Finally, the case r r max{ p,q } follows using the Riez-Thorin interpolation theorem
so with C=D2= C4 C51/a C6
1/a’ and r < p
so with C=D1t D2
1-t
However the loss operator Q- (f,g) lacks the symmetric convolution structure !!
• The locality in f does not support a commutative convolution structure in its weak form.
--
for r < q
Counterexample for the HLS loss operator Q-(f,g) estimates:
One should not expect the HLS type inequality for the loss term Q-(f,g) to hold outside the
range 1 b p b n/(n-l) .
For instance, set p = q = r = 2 and l = n/2, ( so n/(n- l) = 2)
Then taking the potential F(x) = |x|-n/2 and the function g(x)
Then Q-(f,g) (v) = f(v) has a pointwise blow up.
Then, compute
Consider the Cauchy Boltzmann problem:
B(u, û · σ) = |u|−λ b(û · σ) with 0 ≤ λ < n-1 with the Grad‟s assumption:
Q−( f, g) = f
Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials
with integrable angular cross section (Ricardo Alonso & I.M.G., Jour. Stat Phys. 09 )
with
Notation and spaces: For
Set with the norm
(1)
and
Set , so problem one reduces to
Theorem: Let B(u, û · σ) = |u|−λ b(û · σ) with -n < λ ≤ 0 with the Grad‟s assumption
In addition, assume that f0 is ε–close to the local Maxwellian distribution
M(x, v) = C Mα,β(x − v, v ) (0 < α, 0 < β).
Then, for sufficiently small ε the Boltzmann equation has a unique solution satisfying
C1(t) Mα1,β1(x − (t + 1)v, v ) ≤ f ( t, x-v t , v ) ≤ C2(t) Mα2,β2
(x − (t + 1)v, v )
for some positive functions 0 < C1(t) ≤ C ≤ C2(t) < ∞, and parameters 0 < α2 ≤ α ≤ α1 and
0 < β2 ≤ β ≤ β1.
Moreover, the case β = 0 (infinite mass) is permitted as long as β1 = β2 = 0.
(this last part extends the result of Mishler & Perthame ’97 to soft potentials)
Distributional solutions near local Maxwellians : Ricardo Alonso, IMG’08
Previous work by Toscani ’88, Goudon’97, Mischler –Perthame ‘97
As a consequence, one concludes that the distributional solution f is controlled by a traveling Maxwellian,
and that
It behaves like the heat equation, as
mass spreads as t grows
Classical solutions (Different approach from Guo’03, our methods follow some of the those by
Boudin & Desvilletes ‘00, plus new ones )
Definition. A classical solution in [0, T] of problem our is a function such that
,
Theorem (Application of HLS inequality to Q+ for soft potentials) : Let the collision kernel satisfying
assumptions λ < n and the Grad cut-off,
Loss
Gain
Then, for 1 < p, q, r < ∞, with 1/p + 1/q + /n = 1 + 1/r,
However, for 1 < p, q, r < ∞, with 1/p + 1/q + /n = 1 + 1/r and r < p
Remark: the singularity at s = 1 that appears in the constant formula is treated by
symmetrazing b(s) when f = g, so that
• The corresponding HLS estimate for general kernels B(u, û · σ) = F(u) b(û · σ),
with F e Ls radially symmetric and non-increasing.
Then, for 1 ≤ p, q, r ≤ ∞ , with 1/p + 1/q + 1/s = 1 + 1/r and r ≤ p
Therefore, in order to estimate the gradients of the constructed solutions in L∞-Maxwellian
weighted space we need to impose more integrability on the angular part of cross section
when we use to study the regularity of the constructed solution
Remark: the estimate for Q + is the analog one with just 1/p + 1/q + 1/s = 1 + 1/r
Gradient estimates
Theorem (R. Alonso, I.M.G): Assume b e La (Sn-1) for some a >1 , and that the initial
state f0 is small or is near a local Maxwellian. Assume also that f0e (L1 ∩ Lp0) for p0
>1.
Then, there is a unique classical solution f to the Cauchy problem in the interval [0, T]
satisfying the L∞-Maxwellian weighted estimates. Furthermore, there exists
1< b(a)<min(p0, n/(n-l) ) such that for any p e [1, b(a)] the following
estimates hold
Remarks: b e La secures C+ is finite. There is no constrain on the size of f0
Proof: set
: ∫
Case p>1:
1. Estimate the Q+ part of the operator by HLS
for
now taking 1< b(a) < min(p0, n/(n-l) ) such that for any p e (1, b(a)) implies
2. The corresponding estimate for Q+( tf, Df) is direct since b(·) was chosen to have
support in [0, 1].
3. The estimate for Q−(tf,Df) follows using the HLS theorem for the loss operator by
choosing b(a) < min{p0, n/ (n-l)}.
with
By Gronwall inequality with a = n/(n−λ)
4.
p e (1, b(a))
Case p = 1:
1. Separate the potential in two radially symmetric potentials
2. Then
3. Where, using the upper Mawxellian control on f
with n/s’ > 1 for s e ( n/(n−1) , n/(n – l) ).
4. Thenby Gronwall
Velocity regularity
Proof : Take for a fix h > 0 and ˆv ∈ S n−1 and the
corresp. translation operator and transforming
v∗ → v∗ + hˆv in the collision operator.
: ∫
Theorem Let f be a classical solution in [0, T] with f0 satisfying the condition of smallness
assumption or is near to a local Maxwellian and ∇x f0 ∈Lp(R2n) for some
1 < p < b(a) < min{p0 , n/(n-l) ).
In addition assume that ∇v f0 ∈ Lp(R2n). Then, f satisfies
the estimate
with
then
Which is solved by
Just set
Then, by the regularity estimate
with 0 < λ < n-1
Bernoulli ODE
for 1 ≤ p < b(a) < min{ p0 , n/(n-l) }
Lp and Mα,β stability
Set
Now, since f and g are controlled by traveling Maxwellians one has
Theorem Let f and g distributional solutions of problem associated to the initial datum
f0 and g0 respectively. Assume that these datum satisfies the condition of theorems for small data or
near Maxwellians solutions (0 < λ < n-1) . Then, there exist C > 0 independent of time such that
Moreover, for f0 and g0 sufficiently small in Mα,β
Our result is for integrable b(û · σ)
with 0 < λ < n-1
for 1 b p < b(a) < min{ p0 , n/(n-
l) }
Consider a spatially homogeneous d-dimensional ( d ≥ 2) “rarefied gas of particles” having a
unit mass.
Let f(v, t), where v ∈d and t ∈+, 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
Motivation: Connection between the kinetic Boltzmann equations and Kac probabilistic
interpretation of statistical mechanics (Bobylev, Cercignani and IMG, arXiv.org’06, 09, CMP’09)
Part II : Extensions of the Kac N-particle model to multi-linear interactions connection to
dynamics of information
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
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 ∈d
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+,
If solutions are isotropic
then
where Qj(a1, . . . , aj) can be an generalized functions of j-non-negative variables.
-∞-∞
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= or +
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)
The assumed rules lead (formally, under additional assumptions)
to molecular chaos, that is
Introducing a one-particle distribution function (by setting v1 = v) and the hierarchy reduction
The corresponding “weak formulation” for f(v,t) for any test function υ(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)
2Ωd
N
B BBfor B= -∞ or B=0
dσ
Molecular models of Maxwell type (as originally studied)
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
characterized by
so is also a probability distribution function in v.
The Fourier transformed problem:
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.
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”
Fourier transformed operatorΓ
accounts for the integrability of the function b(1-2s)(s-s2)(N-3)/2
λ1 := ∫1
0 (aβ(s) + bβ(s)) G(s) ds = 1 kinetic energy is conserved
< 1 kinetic energy is dissipated
> 1 kinetic energy is generated
For isotropic solutions the equation becomes (after rescaling in time the dimensional constant)
υt + υ = Γ(υ , υ ) ; υ(t,0)=1, υ(0,k)=F (f0)(k), Θ(t)= - υ‟(0)
In this case, using the linearization of Γ(υ , υ ) about the stationary state υ=1, we can inferred the
energy rate of change by looking at λ1 defined by
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
Recall from Fourier transform: nthmoments of f(., v) are nth derivatives of υ(.,k)|k=0
Θ
the Fourier transformed collisional gain operator is written
, with
Kd
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
Note that if the initial coefficient |υ0|≤1, then |Фn|≤1 for any n≥ 0.
the series converges uniformly for τ ϵ [0; 1).
Classical Existence approach : Wild's sum in the Fourier representation.
Γ
Γ
Γ
• rescale time t → τ
and solve the initial value problem
by a power series expansion in time where the phase-space dependence is in the coefficients
Wild's sum in the Fourier representation.(It can be also obtained by a Picard interation
of the integral ODE above)
β/2β/21-β/2
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)
Examples
Applications to agent interactions
Two examples:
• A couple of information percolation models (Dauffie, Malamud and Manso, 08-09)
•M-game multi agent model (Bobylev Cercignani, Gamba, CMP‟09)
Example 1 : Information aggregation model I (Duffie, Malamud & Giroux 09)
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
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
• 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).
• 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 qmeets 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.
Interaction law : The meeting group size m is a parameter of the information model that varies
Aggregation model
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)
aggregation equation in integral formBinary
or
“m-ary”
Multi-agent
Existence by „Wild sums‟ methods
Self-similarity, Pareto tails formation and dynamically scaled solutions (with Ravi Srinivasan)
Statistical equation: (Smolukowski type)
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} ,
We notice the similarity with the the Kac model: let the type signals Vm and its posterior V‟m
Then the m-particle distribution function F(VN, t) and the weak form of the
Kac Master equation
The assumed rules lead (formally, under additional assumptions)
to molecular chaos, that is
Introducing a one-particle distribution function (by setting v1 = v) and the hierarchy reduction
2for N=m
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)
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))
Another Example: Information aggregation model with equilibrium search dynamics
(Duffie, Malamud & Manso 08)
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
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.
Example from information search (percolation) model not of Maxwell type!!
Loss operator: The last term of captures the rate μtC μt
C(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.
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 j+
• 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)
The parameters (a,b) can be fixed or randomly distributed in +2 with some probability density
Bn(a,b).
The corresponding transformation is
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,
• 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 ,
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:
Model of M players participating in a M-linear „game‟ according to the Kac rules(Bobylev, Cercignani,I.M.G.) CMP’09
So, it is possible to reduce the hierarchy of the system to
In the limit N ∞
• Taking the test function on the RHS of the equation for f:
• Taking the Laplace transform of the probability f:
• And assuming the “molecular chaos” assumption (factorization)
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) )
Existence, stability,uniqueness,
(Bobylev, Cercignani, I.M.G.;.arXig.org „06 - CPAM 09)
with 0 < p < 1 infinity energy,
or p ≥ 1 finite energy
Θ
Rigorous results for Maxwell type interacting models
Relates to the work of Toscani, Gabetta,Wennberg, Villani,Carlen, Carvallo,…..
(for initial data with finite energy)
Boltzmann Spectrum
- I
Aggregation Spectrum Wealth distribution
Spectrum
References and preprints http://rene.ma.utexas.edu/users/gamba/publications-web.htm
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