Extensions of the Kac N-particle model to
multi-particle interactions
Irene M. Gamba
Department of Mathematics and ICES
The University of Texas at Austin
IPAM KTW4, May 2009
Motivation: Connection between the kinetic Boltzmann eq.s and Kac probabilistic
interpretation of statistical mechanics -- Properties and Examples
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 structure of this equation follows from thr well-known probabilistic interpretation by
M. Kac: Consider stochastic dynamics of N particles with phase coordinates (velocities)
VN=vi(t) ∈ Rd, i = 1..N
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
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
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)  yields
the Boltzmann equation of Maxwell type in weak form
A general form statistical transport : The Boltzmann Transport Equation (BTE) with
external heating sources: important examples from mathematical physics and social sciences:
The term
models external heating sources:
Space homogeneous examples:
•background thermostat (linear collisions),
•thermal bath (diffusion)
•shear flow (friction),
•dynamically scaled long time limits (selfsimilar solutions).
Inelastic Collision
γ=0 Maxwell molecules
γ=1 hard spheres
u’= (1-β) u + β |u| σ , with σ the direction of elastic post-collisional relative velocity
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 equation,
2. in R1 = R should be replaced by reflections. An interesting one-dimensional system is based on the
above discussed multi-particle stochastic model with non-negative phase
variables v = R+, for which the Laplace transform
Recall self-similarity:
Back to molecular models of Maxwell type (as originally studied)
so
is also a probability distribution function in v.
We work in the space of characteristic functions associated to Probabilities:
The Fourier transformed problem:
Γ
characterized by
Bobylev operator
σ
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
N
1
Accounts for the integrability of the function b(1-2s)(s-s2)(3-N)/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)
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
< 1
λ1 := ∫(0,1) aβ(s) + bβ(s) ds
=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)
An example for multiplicatively interacting stochastic process (with Bobylev’08):
Phase variable: goods (monies or wealth)
particles: M- indistinguishable players
•
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 (n-particle collision)
is a quadratic n x n matrix with non-negative random elements, and must satisfy a condition that
ensures the model does not depend on numeration of identical particles.
Simplest example: a 2-parameter family
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 N-linear ‘game’ according to the Kac rules (Bobylev, Cercignani,I.M.G.):
Assume VM(t), n≥ M undergoes random jumps caused by interactions.
Intervals between two successive jumps have the Poisson distribution with the average ΔtM = θ /M, θ const.
Then we introduce M-particle distribution function F(VM; t) and consider a weak form as in the Kac Master eq:
• Jumps are caused by interactions of 1 ≤ n ≤ N ≤ M particles (the case N =1 is understood as a interaction with
background)
• Relative probabilities of interactions which involve 1; 2; : : : ;N particles are given respectively by non-negative real
numbers β1; β2 ; …. βN such that β1 + β2 + …+ βN = 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 making the “molecular chaos” assumption (factorization)
In the limit 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
,
N
N
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
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.
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 infite 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 2009
Testing: BTE with Thermostat
explicit solution problem of colored particles
Maxwell Molecules model
Rescaling of spectral modes
exponentially by the continuous
spectrum with λ(1)=-2/3
Testing: BTE with Thermostat
Moments calculations:
Rigorous results
(Bobylev, Cercignani, I.M.G.;.arXig.org ‘06 - CPAM 09)
Existence,
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
Stability estimate for a
weighted pointwise distance
for finite or infinite
initial energy
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)
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
ms> 0 for all s>1.
)
In the limit M
∞
So we obtain a model of the class being under discussion where self-similar asymptotics is possible:
N
,
N
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
0 and μ’(1) < 0 for
Non-Equilibrium Stationary Statistical States -γ - homogeneity of kernels vs. high energy tails for stationary states
Elastic case
Inelastic case
A.V. Bobylev, C. Cercignani and I. M. Gamba, On the self-similar asymptotics for generalized non-linear
kinetic Maxwell models, to appear CMP’09
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).
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