Extensions%20of%20the%20Kac%20N-particle%20model%20to%20multi-particle%20interactions

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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
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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
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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) VNvi(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 lt 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
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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 f(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
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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 (self-similar
  solutions).

?0 Maxwell molecules ?1 hard spheres
u (1-ß) u ß u s , with s the direction of
elastic post-collisional relative velocity
Inelastic Collision
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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 aj 0 are relative
probabilities of such interactions, where

• What properties of Q(j) are needed to make them
  consistent with the Maxwell-type interactions?
• 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
• Stv1(0),
  . . . , vM(0) v1(t), . . . , vM(t) , t
  0 ,

then St?v1(0), . .
. , ? vM(0) ?v1(t), . . . , ?vM(t)
for any constant ? gt 0 which leads to the
property
Q(j) (A? f) A? Q(j) (f), A? f(v) ?d
f(? v) , ? gt 0, (j 1, 2, .,M) Note that
the transformation A? is consistent with the
normalization of f with respect to v.
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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 ? ? k, k ? Rd
If solutions are isotropic
then
where Qj(a1, . . . , aj) can be an generalized
functions of j-non-negative variables.

• All these considerations remain valid for d 1,
  the only two differences are
• The evolving Boltzmann Eq should be considered
  as the one-dimensional Kac equation,
• 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

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Recall self-similarity
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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
G
Bobylev operator
s
characterized by
One may think of this model as the generalization
original Kac (59) probabilistic interpretation
of rules of dynamics on each time step ?t2/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 Kacs models and
Mc Kean work in the 60s Carlen, Carvalho,
Gabetta, Toscani, 80-90s For inelastic
interactions Bobylev,Carrillo, I.M.G.
00 Bobylev, Cercignani,Toscani, 03, Bobylev,
Cercignani, I.M.G06 and 08, for general
non-conservative problem
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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) ft
f G(f , f ) f(t,0)1,
f(0,k)F(f0)(k), ?(t) - f(0) Using
the linearization of G(f , f ) about the
stationary state f1, we can inferred the energy
rate of change by looking at ?1
kinetic energy is dissipated
lt 1
?1 ?(0,1) aß(s) bß(s) ds 1
kinetic energy is conserved
gt 1
kinetic energy is generated
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Examples
Existence, asymptotic behavior - self-similar
solutions and power like tails From a unified
point of energy dissipative Maxwell type models
?1 energy dissipation rate (Bobylev,
I.M.G.JSP06, Bobylev,Cercignani,I.G.
arXiv.org06- CMP08)
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• An example for multiplicatively interacting
  stochastic process (with Bobylev08)
• 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 R2 with some probability density
Bn(a,b).
The corresponding transformation is
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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 ?tM ? /M, ? 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 ß1 ß2
  . ßN such that ß1 ß2 ßN 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)

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In the limit M
Example For the choice of rules of random
interaction
With a jump process for ? 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 µ(p) is a curve with a unique minima at
p0gt1 and approaches 8 as p 0 Also
µ(1) lt 0 for
is a multi-linear algebraic equation whose
spectral properties can be well studied
and it is possible to find a second root
conjugate to µ(1) for ?lt?lt1 So a self-similar
attracting state with a power law exists
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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.
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Explicit solutions an elastic model in the
presence of a thermostat for d 2 Mixtures of
colored particles (same mass ß1 ) (Bobylev
I.M.G., JSP06)

Set ß1

, with
and set
Transforms The eq. into

1. Laplace transform of ?

and y(z) z-2 u(zq) B , B constant
2- set
Transforms The eq. into
and
3- Hence, choosing aß0 B(B-1)
with ?µ -1 -5µq and 6µq2 1
0
Painleve eq.
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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 ?(p)
and µ(p)
Case 1
Case 2
where the solutions are given by equalities
with
and Case 2
Case 1
Finite energy SS solutions
Infinity energy SS solutions
For p 1/3 and p1/2 then ?0 ? the Fourier
transf. Boltzmann eq. for one-component gas
? These exact solutions were already obtained by
Bobylev and Cercignani, JSP03
after transforming Fourier back in phase space
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, 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
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Testing BTE with Thermostat explicit solution
problem of colored particles
Maxwell Molecules model Rescaling of spectral
modes exponentially by the continuous spectrum
with ?(1)-2/3
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Testing BTE with Thermostat
Moments calculations
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Rigorous results
(Bobylev, Cercignani, I.M.G..arXig.org 06 -
CPAM 09)
Existence,
?
with 0 lt p lt 1 infinity energy, or p
1 finite energy
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(for initial data with finite energy)
Relates to the work of Toscani, Gabetta,Wennberg,
Villani,Carlen, Carvallo,..
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- I
Boltzmann Spectrum
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Stability estimate for a weighted pointwise
distance for finite or infinite initial energy
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In addition, the corresponding Fourier Transform
of the self-similar pdf admits an integral
representation by distributions Mp(v) with
kernels Rp(t) , for p µ-1(µ). They are given
by
Similarly, by means of Laplace transform
inversion, for v 0 and 0 lt p 1
with
These representations explain the connection of
self-similar solutions with stable distributions
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Theorem appearance of stable law (Kintchine
type of CLT)
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Study of the spectral function µ(p)
associated to the linearized collision operator
For any initial state f(x) 1 xp x(p?) , p
1. Decay rates in Fourier space (p?) µ(p)
- µ(p ?) For finite (p1) or infinite (plt1)
initial energy.
µ(p)
For µ(1) µ(s) , s gtp0 gt1

• For p0 gt1 and 0ltplt (p ?) lt p0

Power tails
CLT to a stable law
Self similar asymptotics for
p0
s
1
p
µ(s) µ(1)
µ(po)
Finite (p1) or infinite (plt1) initial energy
No self-similar asymptotics with finite energy
For p0lt 1 and p1
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msgt 0 for all sgt1.
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)
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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
p0gt1 and approaches 8 as p 0 and µ(1)
lt 0 for
And it is possible to find a second root
conjugate to µ(1) for ?lt?lt1 So a self-similar
attracting state with a power law exists
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Non-Equilibrium Stationary Statistical States --
? - homogeneity of kernels vs. high
energy tails for stationary states
Elastic case
Inelastic case
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A.V. Bobylev, C. Cercignani and I. M. Gamba, On
the self-similar asymptotics for generalized
 non-linear kinetic Maxwell models, to appear
CMP09 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, arXivmath-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) 20122036 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