Transcript Maxwellian models

Evolution of statistical models of non-conservative
particle interactions
Irene M. Gamba
Department of Mathematics and ICES
The University of Texas at Austin
Kinetics and Statistical Methods for Complex Particle Systems
Lisbon, July 2009
Collaborators: A. Bobylev, Karlstad University.
Ricardo Alonso, UT Austin-Rice University,
Carlo Cercignani, Politecnico di Milano
Vladislav Panferov, CSU, Northridge, CA,
Cedric Villani, ENS Lyon, France.
S. Harsha Tharkbushanam, ICES and PROS
Overview
•
Introduction to classical kinetic equations for elastic and inelastic
interactions:
The Boltzmann equation for binary elastic and inelastic collisions
* Description of interactions, collisional frequency and potentials
* Energy dissipation & heat source mechanisms
* Self-similar models
• Interactions of Maxwell type – The Fourier transform Boltzmann
problem
* Initial value problem in the space of characterictic functions (Fourier
transformed probabilities)
* Connection to the Kac – N particle model
* Extensions of the Kac N-particle model to multi-particle interactions
* construction of self-similar solutions and their asymptotic properties.
* characterizations of their probability density functions: Power tails
* Applications to agent interactions: information percolation and M-game
multilinear model
* Explicit self similar solutions to a non-linear equation with a cooling
background thermostat
Some issues of variable hard and soft potential interactions
•
Dissipative models for Variable hard potentials with
heating sources:
All moments bounded
Stretched exponential high energy tails
Spectral - Lagrange solvers for collisional problems
• Deterministic solvers for Dissipative models - The space
homogeneous problem
• FFT application - Computations of Self-similar solutions
•
Space inhomogeneous problems
Time splitting algorithms
Simulations of boundary value – layers problems
Benchmark simulations
Statistical transport from 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,
Simulations of granular flows from
UT Austin and CalTech groups
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
Part I
• Introduction to classical kinetic equations for
elastic and inelastic interactions:
The Boltzmann equation for binary elastic and
inelastic collisions
* Description of interactions, collisional frequency
and potentials
* Energy dissipation & heat source mechanisms
* Self-similar models
Part I
The classical Elastic/Inelastic Boltzmann Transport Equation
for hard spheres: ( 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 ϵ Rd, with velocity v ϵ Rd, at time t ϵ R+ , satisfying
u = v-v* := relative velocity
|u · η| dη := collision rate
‘v
u · η = uη := impact velocity
η := impact direction (random in S+d-1)
v
inelastic
η
‘v*
v*
u · η = (v-v*) . η = - e ('v-'v*) · η = -e 'u . η
u · η ┴ = (v-v*) · η ┴ = ('v-'v*) · η ┴ = 'u · η ┴
C = number of particle in the box
a = diameter of the spheres
d = space dimension
e := restitution coefficient : 0 ≤ e ≤ 1
e = 1 elastic interaction ,
0 < e < 1 inelatic interaction, ( e=0
elastic
γ
θ
‘sticky’ particles)
:= mass density
:= statistical correlation function (sort of mean field ansatz,i.e. independent of v)
= for elastic interactions (e=1)
i.e. enough intersitial space
May be extended to multi-linear interactions (in some special cases to see later)
it is assumed that the restitution coefficient is only a function of the impact velocity e = e(|u·n|).
The properties of the map z  e(z) are
v' = v+ (1+e) (u . η) η
2
and
v'* = v* + (1+e) (u . η) η
2
The notation for pre-collision perspective uses symbols 'v, 'v* : Then,
for 'e = e(| 'u · n|) = 1/e, the pre-collisional velocities are clearly given by
'v = v+ (1+'e) ('u . η) η
2
and 'v* = v* + (1+'e) ('u . η) η
In addition, the Jacobian of the
transformation is then given by
2
J(e(z)) =
However, for a ‘handy’ weak formulation we need to write
the equation in a different set of coordinates involving
σ := u'/|u| the unit direction of the specular (elastic) reflection
of the postcollisional relative velocity, for d=3
e(z) + zez(z) = θz(z) =( z e(z) )z
γ
σ
θ
Goal: Write the BTE in ( (v +v*)/2 ; u) =
(center of mass, relative velocity) coordinates.
Let u = v – v* the relative velocity associated to an elastic
interaction. Let P be the orthogonal plane to u.
Spherical coordinates to represent the d-space spanned by
{u; P} are {r; φ; ε1; ε2;…; εd-2}, where r = radial
coordinates, φ = polar angle, and
{ε1; ε2;…; εd-2}, the n-2 azimuthal angular variables.
σ
then
•
, θ = scattering angle
with
0 ≤ sin γ = b/a ≤ 1, with b = impact parameter, a = diameter of particle
• Assume scattering effects are symmetric with respect to θ = 0 → 0 ≤ θ ≤ π ↔ 0 ≤ γ ≤ π/2
• The unit direction σ is the specular reflection of u w.r.t. γ, that is
|u|σ = u-2(u · η) η
• Then write the BTE collisional integral with the σ-direction dη dv* → dσ dv* , η, σ in Sd-1
using the identity
So the exchange of coordinates can be performed.
In addition, since dσ = |Sd-2| sind-2 θ dθ , then any function b(u · σ) defined on Sd-1 satisfies
|u|
∫ Sd-1
b(|u · σ| )dσ =
|u|
|Sd-2|
∫0
1
b(z) (1-z2) (d-3)/2 dz
, z=cosθ
Interchange of velocities during a binary collision or interaction
σ = uref/|u| is the unit vector in the direction
of the relative velocity with respect to an elastic collision
v'
u'
σ
η
θ
.
v'
u'
σ
v
u
η
γ
γ
.
.
.
e
1-β
β
v*
v*
v'*
v'*
Elastic collision
Inelastic collision
1- β +e = β
Remark:
θ ≈ 0 grazing
and θ ≈ π head on collisions or interactions
v
u
Weak (Maxwell) Formulation: center of mass/ (specular reflected) relative velocity
Due to symmetries of the collisional integral one can obtain (after interchanging the variables of integration)
Both Elastic/inelastic formulations: The inelasticity shows only in the exchange of velocities.
Center of mass-relative velocity coordinates for Q(f; f):
1-β
σ = uref/|u| is the unit vector in the direction
of the relative velocity with respect to an
elastic collision
γ
γ = 0 for Maxwell Type (or Maxwell Molecule) models
γ = 1 for hard spheres models;
0< γ <1 for variable hard potential models,
-d < γ < 0 for variable soft potential models.
β
Collisional kernel or transition probability of interactions is calculated using intramolecular
potential laws:
is the angular cross section
satisfies
In addition, we shall use the α-growth condition
which is satisfied for angular cross section function
for α > d-1 (in 3-d is for α>2)
Weak Formulation & fundamental properties of the collisional integral
and the equation:
Conservation of moments & entropy inequality
x-space homogeneous (or periodic boundary condition) problem: Due to symmetries of the
collisional integral one can obtain (after interchanging the variables of integration)
Invariant quantities (or observables) - These are statistical moments of the ‘pdf’
Time irreversibility is expressed in this inequality
stability
In addition:
The Boltzmann Theorem:
there are only N+2 collision invariants
→yields the compressible Euler eqs → Small perturbations of Mawellians yield CNS eqs.
Exact energy identity for a Maxwell type interaction models
Then f(v,t) → δ0 as t → ∞ to a singular concentrated measure (unless there is ‘source’)
Current issues of interest regarding energy dissipation: Can one tell the shape or classify
possible stationary states and their asymptotics, such as self-similarity?
Non-Gaussian (or Maxwellian) statistics!
Reviewing inelastic properties
INELASTIC Boltzmann collision term: No classical H-Theorem if e = constant < 1
However, it dissipates total energy for e=e(z) < 1 (by Jensen's inequality):
 Inelasticity brings loss of micro reversibility
but keeps time irreversibility !!: That is, there are stationary states and, in some particular
cases we can show stability to stationary and self-similar states (Multi-linear Maxwell molecule
equations of collisional type and variable hard potentials for collisions with a background thermostat)
 However: Existence of NESS: Non Equilibrium Statistical States (stable stationary states
are non-Gaussian pdf’s)
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 (self-similar
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)
Next we need to recall self-similarity:
Energy dissipation implies the appearance of Non-Equilibrium Stationary Statistical States
Part II
• Interactions of Maxwell type – The Fourier transform
Boltzmann problem
* initial value problem in the space of characterictic
functions (Fourier transformed probabilities)
* Connection to the Kac – N particle model
* Extensions of the Kac N-particle model to multi-particle
interactions
* construction of self-similar solutions and their asymptotic
properties.
* characterizations of their probability density functions:
Power tails
* Applications to agent interactions: information percolation
and M-game multilinear model
Motivation of maxwell type models for inelastic interactions (or Pseudo Maxwell molecule models)
They can always be obtained by assuming that the relative speed |u| scales by a mean field quantit
Example:
Then, one obtains the Energy Identity
So it is possible to obtain the (expected)
polynomial time decay rate for the
kinetic energy
•In addition, we (Bobylev, Carrillo and I.M.G., JSP’00) were able to solve the initial value problem
by the method of Wild sums →
Question: Is the kinetic decay rate what it matters for hydrodynamics?
Not quite, also the behavior
of the kinetic solution is
relevant as well
Maxwell type of elastic or inelastic interactions (or Pseudo Maxwell molecule models)
They can always be obtained by assuming that the relative speed |u| scales by a mean field quantit
Example:
Energy Identity
And for e constant we showed that:
• large even moments of self-similar solutions become negative. (also in BCG JSP'00)
• Existence of solution with power like velocity tails for a set of 0 < e < 1 and corresponding selfsimilar asymptotics and decay estimates.(Ernst-Brito JSP'02; Bobylev-Cercignani JSP'02; with Toscani; JSP'03)
•for any 0 < e < 1: NOT all even moments can be bounded for initial data in L1k(Rd), for all e,
(Bobylev,I.M.G.JSP'06 )
•Generalization to multi-linear energy conservative or dissipative collisional forms in Maxwell
type model formulation with applications to kinetic mixtures with sources, social dynamical
interactions, and more (Bobylev,Cercignani, I.M.G. '06)
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
Bobylev 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/M of M 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. 00
Bobylev, Cercignani,Toscani, 03, Bobylev, Cercignani, I.M.G’06 and 08, for general non-conservative problem
Recall from Fourier transform: nthmoments of f(., v) are nth derivatives of φ(.,k)|k=0
Θ
And, 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 ), by performing the operations
with
,
then, the Fourier transformed collisional 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
< 1
λ1 := ∫ 0 (aβ(s) + bβ(s)) G(s) ds
1
=1
> 1
kinetic energy is dissipated
kinetic energy is conserved
kinetic energy is generated
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’08)
Existence: 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 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
the series converges uniformly for τ ϵ [0; 1).
for any n≥ 0.
Problem for (elastic) inelastic interaction
(B-C-G, JSP’00)
near a Dirac delta
Spherical harmonic expansions
For compact operators invariant
under rotations
Problem for (elastic) inelastic interaction
(B-C-G, JSP’00)
Problem for (elastic) inelastic interaction
(B-C-G, JSP’00)
Problem for (elastic) inelastic interaction
(B-C-G, JSP’00)
such that
Thus, as t →∞, it recovers
conservation of energy
Remark: Variable restitution coefficient: there are no self-similar solutions, but for small temperature or
restitution coefficient uniformly close to 1, the homogeneous solution is close to the Maxwellian distribution as
described before.
Problem for (elastic) inelastic interaction
(B-C-G, JSP’00)
Remarks:
-- Power like tails for e constant and self-similar asymptotics. (Ernst-Brito,
Bobylev-Cercignani- JSP'02, with Toscani-JSP'03, Bobylev I.M.G, JSP’06)
-- Generalization
to global dissipative Kac-type models with multi linear
interactions by Spectral Characteristic methods (Bobylev-Cercignani-I.M.G
arXiv.org’06, 08,CMP’09)
Generalization of Maxwell to multi-linear interacting models
Motivation: Lays on the observation that quite different equations for probability dynamics
leads to the same class of equations in the evolution equation for the Fourier (Laplace)
representation for their characteristic (generating) functions.
Examples:
• Kac caricature models for elastic particles
• elastic or inelastic homogenous Boltzmann equation of Maxwell type interactions in
higher dimensions
• models for slow down processes: background cooling (soft condensed matter phenomena)
• statistical evolution in social dynamics by binary interactions
We present a canonical probabilistic model equivalent to generalized Maxwell molecule
models:
Ideas follow from the ‘same line of thought’ where only games with two players
were considered in MISP (or random interactive processes) ben-Avraham, Ben-Naim,
Lindenberg & Rosas '03; Pareschi & Toscani '05-06, and Fujihara, Ohtsuki, & Yamamoto '06:
More generally (Bobylev, Cercignani and IMG, arXiv.org’06, 09, CMP’09)
Connection between the kinetic Boltzmann equations and Kac probabilistic
interpretation of statistical mechanics
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
Assumption: I - collision frequency is independent of velocities of interacting particles (Maxwell-type)
II - the total scattering cross section is finite.
Hence, one can choose such units of time such that the corresponding classical Boltzmann eqs. reads
with
Q+(f) is the gain term of the collision integral and Q+ transforms f to 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
What properties of Q(j)+ are needed to make them consistent with the Maxwell-type interactions?
1.
Temporal evolution of the system is invariant under scaling transformations of the 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.
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
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
( and connects to the lecture of R. Pego)
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
N
2
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 (as in E. Carlen lecture) (or Kac’s walk on the sphere)
(Bobylev, Cercignani, I.M.G.;.arXig.org ’06, ’09,- CMP’09)
Rigorous results
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
Boltzmann Spectrum
(Bobylev, Cercignani, I.M.G.;.arXig.org ’06, ’09,- CMP’09)
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 φ0(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
)
Explicit solutions an elastic model in the presence of a thermostat for d ≥ 2
Mixtures of colored particles (same mass β=1 ): (Bobylev & I.M.G., JSP’06)
=
Set β=1
=
,
with
and set
Transforms
1.
Laplace transform of ψ:
The eq. into
and y(z) =z-2 u(zq) + B
2- set
, B constant
Transforms
The eq. into
and
3-
Hence, choosing α=β=0 = B(B-1)
=0
Painleve eq.
with θ=μ
-1 -5μq and 6μq2 = ± 1
Theorem: the equation for the slowdown process in Fourier space, has exact self-similar
solutions satisfying the condition
for the following values of the parameters θ(p) and μ(p):
Case 1:
Case 2:
where the solutions are given by equalities
with
and
Case 1:
Infinity energy
SS solutions
Case 2:
Finite energy
SS solutions
For p = 1/3 and p=1/2 then θ=0  the Fourier
transf. Boltzmann eq. for one-component gas 
These exact solutions were already obtained by
Bobylev and Cercignani, JSP’03
after transforming Fourier back in phase space
, both, for infinite and finite energy cases
Qualitative results for Case 2 with finite energy:
Also, rescaling back w.r.t. to M^(k) and Fourier transform back f0ss(|v|) = MT(v) and
the similarity asymptotics holds as well.
Computations: spectral Lagrangian methods in collaboration with Harsha Tharkabhushaman
JCP’09 and JCM’09
Applications to agent interactions: information percolation and M-game multi linear models
Model of N players participating in a M-linear ‘game’ according to the Kac rules (Bobylev, Cercignani,I.M.G.):
Assume VN(t), N≥ M undergoes random jumps caused by interactions.
Intervals between two successive jumps have the Poisson distribution with the average ΔtM = Θ /N, Θ const.
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 ≤ n ≤ 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 respectively by non-negative real
numbers β1; β2 ; …. βM such that β1 + β2 + …+ βNM = 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
∞
M
Example: For the choice of
rules of random interaction
With a jump process for θ a random
variable with a pdf
So we obtain a model of the class being under discussion where self-similar asymptotics is possible
,
M
M
So
whose spectral function is
Where
is a curve
with a unique
minima
p0>1 and
approaches
∞ asstudied
p
0
is a μ(p)
multi-linear
algebraic
equation
whoseatspectral
properties
can be+well
and it is possible to find a second root conjugate to μ(1) for γ<γ*<1
Also μ’(1) < 0 for
So a self-similar attracting state with a power law exists
M-game model
M
In the limit M
∞
So we obtain a model of the class being under discussion where self-similar asymptotics is possible:
M
,
M
whose spectral function is
Where μ(p) is a curve with a unique minima at p0>1 and approaches + ∞ as p
And it is possible to find a second root conjugate to μ(1) for γ<γ*<1
So a self-similar attracting state with a power law exists and it is an attractor
0 and μ’(1) < 0 for
Part III
Some issues of variable hard and soft potential interactions
•
Dissipative models for Variable hard potentials with
heating sources:
All moments bounded
Stretched exponential high energy tails
Spectral - Lagrange solvers for collisional problems
•
Deterministic solvers for Dissipative models - The space
homogeneous problem
• Computations of Self-similar solutions
•
Space inhomogeneous problems – Simulations of boundary value
problems – boundary layers
Non-Equilibrium Stationary Statistical States
Key property:
Summability of series of moments of BTE solutions
Estimates for Existence theory:
Average angular estimates & weighted Young’s inequalities
R. Alonso and E. Carneiro’08, and R. Alonso and E. Carneiro, IG, 08
with
Angular average inequality
Young’s inequality for variable hard potentials : 1 ≥ λ ≥ 0
Hardy-Littlewood-Sobolev type inequality for soft potentials : 0 > λ ≥ -n
These two constants C depends linearly of the expression given above for the constant of the
angular averaging lemma
Spectral - Lagrange solvers for collisional problems
Collision Integral Algorithm
~
~
~
Discrete Conservation operator
‘conserve’ algorithm
Stabilization property
tr = reference time = mft
Δt= 0.25 mft.
Soft condensed matter
phenomena
Remark: The numerical algorithm is based on the evolution of the continuous spectrum of the solution
as in Greengard-Lin’00 spectral calculation of the free space heat kernel, i.e. self-similar of the heat
equation in all space.
Isomoment estimates
Shannon Sampling theorem
Space inhomogeneous simulations
mean free time := the average time between collisions
mean free path := average speed x mft (average distance traveled between collisions)
 Set the scaled equation for 1= Kn := mfp/geometry of length scale
Spectral-Lagrangian methods in 3D-velocity space and 1D physical space discretization in the
simplest setting:
Spatial mesh size Δx = O.O1 mfp
Time step Δt = r mft ,
N= Number of Fourier modes in each j-direction in 3D
mft= reference time
Resolution of discontinuity ’near the wall’ for diffusive boundary conditions:
(K.Aoki, Y. Sone, K. Nijino, H. Sugimoto, 1991)
Sudden heating: Constant moments initial state with a discontinuous pdf at the boundary wall,
with wall kinetic temperature increased by twice its magnitude:
Boundary Conditions for sudden heating:
Calculations in the next two pages:
Mean free path l0 = 1.
Number of Fourier modes N = 243,
Spatial mesh size Δx = 0.01 l0 .
Time step Δt = r mft
Formation of a shock wave by an initial sudden change of wall temperature from T0 to 2T0.
Sudden heating problem (BGK eq. with lattice Boltzmann solvers)
K.Aoki, Y. Sone, K. Nijino, H. Sugimoto, 1991
Comparisons with K.Aoki, Y. Sone, K. Nijino, H. Sugimoto, 1991 (Lattice Boltzmann on BGK)
Jump in pdf
Sudden heating problem
Heat transfer problem:
Diffusive boundary conditions
Temperature: T0 given at xo=0
and T1 = 2T0 at x1 = 1.
Knudsen Kn = 0.1, 0.5, 1, 2, 4
Comments:
. Tails are important to understand evolution of moments (well known….!!!) They
depend on the rate of collision as a function of velocity.
(Decay rates to equilibrium states depend on the angular cross section as one can get
exact and best constant depending on b(θ) )
• Tails control methods to space inhomogeneous problems: may lead to local in
x-space, global in v-space control of the solution BTE,
…. but we do not how to do it yet…
•The use of Young and Hardy Littlewood Sobolev type of inequalities allows to revisit
and/or extend the existence and regularity results of the space inhomogeneous BTE
with soft potentials and angular cross sections that are just integrable (Grad cut-off
assumption), with data between near two different Maxwellians.
•Need to adjust hydrodynamic limits for non conservative phenomena:
Hydrodynamic limits with energy dissipation lack of exact/local closure formulasmacroscopic equations may not have an accurate closed form.
Thank you very much for your attention!
For recent preprints and reprints see:
www.ma.utexas.edu/users/gamba/research and references therein
Recent work related to these problems:
Cercignani'95(inelastic BTE derivation);
Bobylev, JSP 97 (elastic,hard spheres in 3 d: propagation of L1-exponential estimates );
Bobylev, Carrillo and I.M.G., JSP'00 (inelastic Maxwell type interactions- self similarity- mean field);
Bobylev, Cercignani , and with Toscani, JSP '02 &'03 (inelastic Maxwell type interactions);
Bobylev, I.M.G, V.Panferov, C.Villani, JSP'04, CMP’04 (inelastic + heat sources);
Mischler and Mouhout, Rodriguez Ricart JSP '06 (inelastic + self-similar hard spheres);
Bobylev and I.M.G. JSP'06 (Maxwell type interactions-inelastic/elastic + thermostat),
Bobylev, Cercignani and I.M.G arXiv.org,06 (CMP’09); (generalized multi-linear Maxwell type interactionsinelastic/elastic: global energy dissipation)
I.M.G, V.Panferov, C.Villani, arXiv.org’07, ARMA’09 (elastic n-dimensional variable hard potentials Grad cut-off::
propagation of L1 and L∞-exponential estimates)
C. Mouhot, CMP’06 (elastic, VHP, bounded angular cross section: creation of L1-exponential )
R. Alonso and I.M.G., JMPA’08 (Grad cut-off, propagation of regularity bounds-elastic d-dim VHP)
I.M.G. and Harsha Tarskabhushanam JCP’09(spectral-lagrangian solvers-computation of singularities)
I.M.G. and Harsha Tarskabhushanam JCM’09 (Shock and Boundary Structure formation by Spectral-Lagrangian
methods for the Inhomogeneous Boltzmann Transport Equation)
R.Alonso, E.Carneiro (ArXiv.org08)(Young’s inequality for collisional integrals with integrable (grad cut-off) angular cross
section)
R.Alonso, E.Carneiro, I.M.G. ArXiv.org09 (weigthed Young’s inequality and Hardy Sobolev’s inequalities for collisional
integrals with integrable (grad cut-fff)angular cross section)
R. Alonso and I.M.G. ArXiv.org09, (Distributional and classical solutions to the Cauchy Boltzmann problem for soft
potentials with integrable angular cross section)
Alonso, Canizo, I.M.G.,Mischler, Mouhot, in preparation (The homogeneous Boltzmann eqaution with a cold thermostat
for variable hard potentials)
Alonso, Canizo, I.M.G., Mouhot, in preparation (sharper decay for moments creation estimates for variable hard
potentials)
?
Yes
(ARMA’09)
B. Wennberg~’98
generation of moments
estimates
generation of exponentially weighted
lower bound
(JPS’04)
Sharp Povzner estimates
Summability of moments series
(I.G V.Panferov, c. Villani; ARMA’09)
then
(JMPA’08)
VIII) Generation of exponential L1-weighted estimates (Mouhot’06) and better tails (Alonso,Canizo,IG and
Mouhot
in progress)
Sharp Povzner estimates: optimal control of weights in `average’
Angular Averaging Inequality: (A.Bobylev, I.G., V.Panferov, JSP’04)
and of γ (rate of the intramolecular potentials)
Elastic case: β=1
d-dimensions
**
(I.G., V.Panferov C.Villani; ARMA’08
with
Our result extends the Bobylev-Povzner-type estimate (JSP'97) for d=3 and γ=1, (i.e. b(σ)=C)
to d > 1
and
kernels with monotone angular dependence on its symmetric part satisfying **
Corollary 2:
In order to study the behavior of mp with p = ks/2 for a good choice of s, take
moments of evolution forced equations:
Corollary: it is possible to choose s, such that
for r and R depending on the initial mass m0, energy bound m1 and some high order moments
mp for some p∗ > 1, depending on the 'heating' force coefficient.
0
The choice of s is done by setting: (shown in the pure diffusion case and bounded angular section
γ=1 and stationary state)
So, in order to control zp+1/2 we need to divide by Γ(a(p+1/2) + b) and find a suitable
value of a such that we can get control of a corresponding
recursion inequality relation that produces a geometric growth control for zP
Then
∂tzp +
Add the time derivative to compute the
Corresponding evolution estimate
a careful choice of a = 4/3 and b < 1 cancels the coefficients for the two terms
proportional to zp-1 , and the right hand side term (from the gain term) is controlled by a
constant!!
Similarly for the other cases: diffusion with friction: s = 2.
Self-similar (homogeneous cooling) s = 1
Shear flow: at least s = 1 but anisotropy is admissible, so other direction might decay faster.
In the elastic case with no sources, for 0 < γ ≤ 1 and b(θ) integrable:
Exponential moments propagation (I.G. Panferov and Villani, arXiv’06, ARMA’09) moments of
equation
collision operator
Loss op.
Gain op.
(i.e. a=1 and s=2)
Bernoulli type eq. can also “create” moments
(Desvillates 93 B. Wennberg~’98) or “generate”
In the elastic case, for 0 < γ ≤ 1 and b(θ)integrable:
Propagation and generation estimates
(i.e. a=1 and s=2)
for r and R depending on the initial mass m0, energy bound m1 and some high order
moments mp for some p∗ > 1, but uniform in t ! 
0
• Summability of the series of moments is uniform intime 
• Propagation of exponential moments
1. Same argument holds for controlling moments of the derivatives of f(t,v) by iterative
methods (R. Alonso &I.G. JMPA’08)
2. SS solutions to Elastic collisions with a cold thermostat for the choice of a= γ /2 and s= γ
and existence (Alonso, Canizo, IG, Mischler, Mouhot, in preparation )
3. Generation of moments for a= γ /4 and s= γ /2 (Mouhot JSP’06) for initial data with only 2+ moments
4. Improvement in moments generation by taking a=a(t, γ) to
and Mouhot, in preparation )
s=γ
(Alonso, Canizo, IG,
Upper point-wise uniform bounds for large energy tails for elastic
hard spheres or γ-variable potentials in d-dimensions
(IG, Vlad Panferov, Cedric Villani, arXiv.org’06 -
ARMA’08, and R. Alonso and I.G- JMPA’08)
STRATEGY : Find a comparison theorem & construct a suitable barrier function
⇒ Compare to obtain point-wise bounds:
Comparison principle: Q is multi-linear, symmetric, conservative, and L1-contractive for its
linear restriction (Crandall & Tartar ’80, also Vandenjapin & Bobylev 75, Kaniel & Shimbrot ’84, Lions ’94 )
Remark: it also works in the space inhomogeneous case.
and
Crucial point:
In order to find the barrier probability distribution:
we need
That can be obtain by the following key estimate:
Tool: Carleman integral representation of
Remark: these propagation properties in L1 and L∞ Maxwellians weighted norms
also hold for all the derivatives if initial data have all derivatives under such control
(Ricardo Alonso and I.G.; ): we use iterative arguments.
Goal: Write the BTE in ( (v +v*)/2 ; u) =
(center of mass, relative velocity) coordinates.
Let u = v – v* the relative velocity associated to an elastic
interaction. Let P be the orthogonal plane to u.
Spherical coordinates to represent the d-space spanned by
{u; P} are {r; φ; ε1; ε2;…; εd-2}, where r = radial
coordinates, φ = polar angle, and
{ε1; ε2;…; εd-2}, the n-2 azimuthal angular variables.
σ
then
•
, θ = scattering angle
with
0 ≤ sin γ = b/a ≤ 1, with b = impact parameter, a = diameter of particle
• Assume scattering effects are symmetric with respect to θ = 0 → 0 ≤ θ ≤ π ↔ 0 ≤ γ ≤ π/2
• The unit direction σ is the specular reflection of u w.r.t. γ, that is
|u|σ = u-2(u · η) η
• Then write the BTE collisional integral with the σ-direction dη dv* → dσ dv* , η, σ in Sd-1
Set |u · η| dη = cos γ dη and u · σ = cos 2γ (since η·σ = cos γ ) . Then
|u|
|u|
dσ = 2d-2 cosd-2 2γ dγ
∫ Sd-1 |u · η|
g(|u · η| dη )dη =
|u|
|u · η| dη = |u| 21-d ( 1-cos θ )(3-d)/2 sin θ dθ.
2
1
|Sd-2|
|u| ∫0
g(z) (1-z2) (d-3)/2 dz