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Evolution of statistical models of non-conservative particle interactions Irene M. Gamba Department of Mathematics and ICES The University of Texas at Austin 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 Kinetics and Statistical Methods for Complex Particle Systems Kinetics and Statistical Methods for Complex Particle Systems Lisbon, July 2009 Lisbon, July 2009

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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. probability density functions: Power tails * characterizations of their probability density functions: Power tails * Applications to agent interactions: information percolation and M-game multilinear model * 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 * Explicit self similar solutions to a non-linear equation with a cooling background thermostat

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Dissipative models for Variable hard potentials with heating sources: All moments bounded Stretched exponential high energy tails Some issues of variable hard and soft potential interactions 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

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Statistical transport from interactive/collisional kinetic models Rarefied ideal gases-elastic:classical conservativeBoltzmann Transport eq. 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 ө b or Rapid granular flow dynamics: (inelastic hard sphere interactions): homogeneous cooling states, randomly heated states, shear flows, shockwaves past wedges, etc. (Soft) condensed matter at nano scale: Bose-Einstein condensates models and charge transport in solids: current/voltage transport modeling semiconductor. Emerging applications from stochastic dynamics for multi-linear Maxwell type interactions : Multiplicatively Interactive Stochastic Processes: information percolation models, particle swarms in population dynamics, Goals: Understanding of analytical properties: large energy tails Long time asymptotics and characterization of asymptotics states Long time asymptotics and characterization of asymptotics states A unified approach for Maxwell type interactions and generalizations. A unified approach for Maxwell type interactions and generalizations. Spectral-Lagrangian solvers for dissipative interactionsSpectral-Lagrangian solvers for dissipative interactions Simulations of granular flows from UT Austin and CalTech groupsUT Austin CalTech

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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

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‘v‘v ‘v*‘v* v v*v* C = number of particle in the box a = diameter of the spheres d = space dimension η elastic inelastic u · η = u η := 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: ( 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 R d, with velocity v R d, at time t R +, 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 Part I γ θ

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i.e. enough intersitial space May be extended to multi-linear interactions ( in some special cases to see later) := statistical correlation function (sort of mean field ansatz,i.e. independent of v) = for elastic interactions (e=1) := mass density

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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 and v' * = v * + (1+e) (u. η) η 2 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 and 'v * = v * + (1+'e) ('u. η) η 2 2 e(z) + ze z (z) = θ z (z) =( z e(z) ) z J(e(z)) = In addition, the Jacobian of the transformation is then given by γ θ 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 σ

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σ 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; ε 1 ; ε 2 ;…; ε d-2 }, where r = radial coordinates, φ = polar angle, and {ε 1 ; ε 2 ;…; ε d-2 }, the n-2 azimuthal angular variables. thenwith, θ = scattering angle 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 S d-1 using the identity b(|u · σ| )dσ = |S d-2 | ∫ 0 b(z) (1-z 2 ) (d-3)/2 dz |u| 1 ∫ S d-1 In addition, since then any function b(u · σ) defined on S d-1 satisfies |u| dσ = |S d-2 | sin d-2 θ dθ,, z=cosθ So the exchange of coordinates can be performed.

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v*v* v. σ u v' * v' u' η θ v*v* v. σ.. 1-β u v' * v' β u' e 1- β +e = β η Elastic collision Inelastic collision σ = u ref /|u| is the unit vector in the direction of the relative velocity with respect to an elastic collision Interchange of velocities during a binary collision or interaction γ γ Remark: θ ≈ 0 grazing and θ ≈ π head on collisions or interactions

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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) Elastic/inelastic Both Elastic/inelastic formulations: The inelasticity shows only in the exchange of velocities. Center of mass-relative velocity coordinates for Q(f; f): β σ = u ref /|u| is the unit vector in the direction of the relative velocity with respect to an elastic collision 1-β 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. γ

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In addition, we shall use the α-growth condition which is satisfied for angular cross section functionfor α > d-1 (in 3-d is for α>2) is the angular cross section satisfies Collisional kernel or transition probability of interactions is calculated using intramolecular potential laws:

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Weak Formulation & fundamental properties of the collisional integral moments 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’

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The Boltzmann Theorem: The Boltzmann Theorem: there are only N+2 collision invariants Time irreversibility is expressed in this inequalitystability In addition:

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→yields the compressible Euler eqs → Small perturbations of Mawellians yield CNS eqs.

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Exact energy identity for a Maxwell type interaction models Then f(v,t) → δ 0 as t → ∞ to a singular concentrated measure (unless there is ‘source’) self-similarity 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!

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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 time irreversibility 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) NESS However: Existence of NESS: Non Equilibrium Statistical States (stable stationary states are non-Gaussian pdf’s)

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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). Inelastic Collision u’= (1-β) u + β |u| σ, with σ the direction of elastic post-collisional relative velocity ‘v‘v ‘v*‘v* v v*v* η

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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) The collision frequency is given by

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Next we need to recall self-similarity:

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Non-Equilibrium Stationary Statistical States Energy dissipation implies the appearance of

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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. probability density functions: Power tails * characterizations of their probability density functions: Power tails * Applications to agent interactions: information percolation and M-game multilinear model * Applications to agent interactions: information percolation and M-game multilinear model

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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 quantity Example: Then, one obtains the Energy Identity 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 → So it is possible to obtain the (expected) polynomial time decay rate for the kinetic energy Question: Is the kinetic decay rate what it matters for hydrodynamics? Not quite, also the behavior of the kinetic solution is relevant as well

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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 quantity 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 self- similar 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 L 1 k ( R d ), 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)

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Back to 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. 00 Bobylev, Cercignani,Toscani, 03, Bobylev, Cercignani, I.M.G’06 and 08, for general non-conservative problem characterized by sois 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/M of M particles associated to system of vectors randomly interchanging velocities pairwise while independently of their relative velocities. preserving momentum and local energy, independently of their relative velocities. Then: work in the space of “characteristic functions” associated to Probabilities 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” Bobylev operator Γ

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accounts for the integrability of the function b(1-2s)(s-s 2 ) (N-3)/2 λ 1 := ∫ 1 0 (a β (s) + b β (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) φ t + φ = Γ(φ, φ ) ; φ(t,0)=1, φ(0,k)= F (f 0 )(k), 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 λ 1 defined by And, for isotropic ( x = |k| 2 /2 ) or self similar solutions ( x = |k| 2 /2 e μt, μ is the energy dissipation rate, that is: Θt = - μ Θ ), by performing the operations Recall from Fourier transform: n th moments of f(., v) are n th derivatives of φ(.,k)| k=0 Θ then, the Fourier transformed collisional operator is written, with KdKd

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Existence, asymptotic behavior - self-similar solutions and power like tails: From a unified point of energy dissipative Maxwell type models: λ 1 energy dissipation rate (Bobylev, I.M.G.JSP’06, Bobylev,Cercignani,I.G. arXiv.org’06- CMP’08)Examples

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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 Note that if the initial coefficient |φ 0 |≤1, then |Ф n |≤1 for any n≥ 0. the series converges uniformly for τ [0; 1). Existence: 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. β/2 1-β/2

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Problem for (elastic) inelastic interaction (B-C-G, JSP’00) near a Dirac delta Spherical harmonic expansions For compact operators invariant under rotations

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Problem for (elastic) inelastic interaction (B-C-G, JSP’00)

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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) such that Thus, as t →∞, it recovers conservation of energy

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Problem for (elastic) inelastic interaction (B-C-G, JSP’00)

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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)

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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:

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Consider a spatially homogeneous d-dimensional ( d ≥ 2) rarefied gas of particles having a unit mass. Let f(v, t), where v ∈ R d 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 More generally () More generally ( Bobylev, Cercignani and IMG, arXiv.org’06, 09, CMP’09 ) Connection between the kinetic Boltzmann equations and Kac probabilistic Connection between the kinetic Boltzmann equations and Kac probabilistic interpretation of statistical mechanics interpretation of statistical mechanics

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The same stochastic model admits other possible generalizations 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 α j ≥ 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 S t is the evolution operator for the given N-particle system such that S t {v 1 (0),..., v M (0)} = {v 1 (t),..., v M (t)}, t ≥ 0, then S t {λv 1 (0),..., λ v M (0)} = {λv 1 (t),..., λv M (t)} for any constant λ > 0 which leads to the property Q + (j) (A λ f) = A λ Q + (j) (f), A λ f(v) = λ d f(λ v), λ > 0, (j = 1, 2,.,M) Note that the transformation A λ 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 → λ k, k ∈ R 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 ( 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 +, If solutions are isotropic then where Q j (a 1,..., a j ) can be an generalized functions of j-non-negative variables. -∞ -∞-∞

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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) V N = v i (t) ∈ Ω d, 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 (v i, v l ) →(v′ i, v′ l ) which corresponds to an interaction of two particles with ‘pre-collisional’ velocities v i and v l. Then introduce N-particle distribution function F(V N, t) and consider a weak form of the Kac Master equation ( Kac Master equation (we have assumed that V’ N j = V’ N j ( V N j, U N 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 v 1 = v) and the hierarchy reduction The corresponding “weak formulation” for f(v,t) for any test function φ(v) where the RHS has a bilinear structure from evaluating f(v i ’,t) f(v l ’, 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) 2 ΩdNΩdN Ω dN x S d-1 B BB for B= -∞ or B=0 dσ

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Existence, stability,uniqueness, (Bobylev, Cercignani, I.M.G.;.arXig.org ’06, ’09,- CMP’09) with 0 < p < 1 infinity energy, or p ≥ 1 finite energy Θ Rigorous results

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Relates to the work of Toscani, Gabetta,Wennberg, Villani,Carlen, Carvallo,….. (for initial data with finite energy)

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Boltzmann Spectrum - I

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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. (Bobylev, Cercignani, I.M.G.;.arXig.org ’06, ’09,- CMP’09)

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Existence of Self-Similar Solutions with initial conditions REMARK: The transformation, for p > 0 transforms the study of the initial value problem to u o (x) = 1+x and ||u o || ≤ 1 so it is enough to study the case p=1

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These representations explain the connection of self-similar solutions with stable distributions Similarly, by means of Laplace transform inversion, for v ≥0 and 0 < p ≤ 1 with In addition, the corresponding Fourier Transform of the self-similar pdf admits an integral representation by distributions M p (|v|) with kernels R p (τ), for p = μ −1 (μ ∗ ). They are given by:

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Theorem: appearance of stable law (Kintchine type of CLT)

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Recall the self similar problem Then,

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m s > 0 for all s>1.

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p 0 >1 Power tails CLT to a stable law Finite (p=1) or infinite (p<1) initial energy Study of the spectral function μ(p) associated to the linearized collision operator p

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)

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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 = and set 1.Laplace transform of ψ: Transforms The eq. into 2- set and y(z) =z -2 u(z q ) + B, B constant Transforms The eq. into and 3- Hence, choosing α=β=0 = B(B-1) Painleve eq. = 0 with θ=μ -1 -5μq and 6μq 2 = ± 1, with

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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 θ(p) and μ(p): Case 1: Case 2: where the solutions are given by equalities with Case 1: and Case 2: Infinity energy SS solutions 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

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Computations: spectral Lagrangian methods in collaboration with Harsha Tharkabhushaman JCP’09 and JCM’09 Also, rescaling back w.r.t. to M ^ (k) and Fourier transform back f 0 ss (|v|) = M T (v) and the similarity asymptotics holds as well. Qualitative results for Case 2 with finite energy:, both, for infinite and finite energy cases

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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 β 1 ; β 2 ; …. β M such that β 1 + β 2 + …+ β NM = 1, so it is possible to reduce the hierarchy of the system to Assume V N (t), N≥ M undergoes random jumps caused by interactions. Intervals between two successive jumps have the Poisson distribution with the average Δt M = Θ /N, Θ const. Then we introduce M-particle distribution function F(V N ; t) and consider a weak form as in the Kac Master eq: Model of N players participating in a M-linear ‘game’ according to the Kac rules (Bobylev, Cercignani,I.M.G.): 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) Applications to agent interactions: information percolation and M-game multi linear models

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In the limit N ∞ 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 Where μ(p) is a curve with a unique minima at p 0 >1 and approaches + ∞ as p 0 Also μ’(1) < 0 for 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 whose spectral function is M So is a multi-linear algebraic equation whose spectral properties can be well studied M

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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:, M M Where μ(p) is a curve with a unique minima at p 0 >1 and approaches + ∞ as p 0 and μ’(1) < 0 for 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 whose spectral function is M-game model M

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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

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Non-Equilibrium Stationary Statistical States

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Key property: Summability of series of moments of BTE solutions

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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

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These two constants C depends linearly of the expression given above for the constant of the angular averaging lemma Young’s inequality for variable hard potentials : 1 ≥ ≥ 0 Young’s inequality for variable hard potentials : 1 ≥ λ ≥ 0 Hardy-Littlewood-Sobolev type inequality for soft potentials : 0 > ≥ -n Hardy-Littlewood-Sobolev type inequality for soft potentials : 0 > λ ≥ -n

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Spectral - Lagrange solvers for collisional problems

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Collision Integral Algorithm

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~ ~ ~

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‘conserve’ algorithm Stabilization property Discrete Conservation operator

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t r = reference time = mft Δt= 0.25 mft.

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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.

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Shannon Sampling theorem

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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: N= Number of Fourier modes in each j-direction in 3D Spatial mesh size Δx = O.O1 mfp Time step Δt = r mft, mft= reference time

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Resolution of discontinuity ’near the wall’ for diffusive boundary conditions: (K.Aoki, Y. Sone, K. Nijino, H. Sugimoto, 1991) Sudden heating: Sudden heating: Constant moments initial state with a discontinuous pdf at the boundary wall, with wall kinetic temperature increased by twice its magnitude : Calculations in the next two pages: Mean free path l 0 = 1. Number of Fourier modes N = 24 3, Spatial mesh size Δx = 0.01 l 0. Time step Δt = r mft Boundary Conditions for sudden heating:

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Formation of a shock wave by an initial sudden change of wall temperature from T 0 to 2T 0. Sudden heating problem (BGK eq. with lattice Boltzmann solvers) K.Aoki, Y. Sone, K. Nijino, H. Sugimoto, 1991

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Jump in pdf Comparisons with K.Aoki, Y. Sone, K. Nijino, H. Sugimoto, 1991 (Lattice Boltzmann on BGK) Sudden heating problem

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Temperature: T 0 given at x o =0 and T 1 = 2T 0 at x 1 = 1. Knudsen Kn = 0.1, 0.5, 1, 2, 4 Heat transfer problem: Diffusive boundary conditions

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For recent preprints and reprints see: and references therein Thank you very much for your attention! 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 formulas- macroscopic equations may not have an accurate closed form.

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Recent work related to these problems: Cercignani'95(inelastic BTE derivation); Bobylev, JSP 97 (elastic,hard spheres in 3 d: propagation of L 1 -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 interactions- inelastic/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 L 1 and L ∞- exponential estimates) C. Mouhot, CMP’06 (elastic, VHP, bounded angular cross section: creation of L 1 -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)

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Yes ? ( ARMA’09)

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generation of moments estimates generation of exponentially weighted lower bound B. Wennberg~’98

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Sharp Povzner estimates Summability of moments series VIII) Generation of exponential L 1- weighted estimates (Mouhot’06) and better tails ( Alonso,Canizo,IG and Mouhot in progress ) (JMPA’08) (JPS’04) ( I.G V.Panferov, c. Villani; ARMA’09 ) then

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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 )

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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 ** (I.G., V.Panferov C.Villani; ARMA’08 Elastic case: β=1 d-dimensions **

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Corollary 2: In order to study the behavior of m p with p = ks/2 for a good choice of s, take moments of evolution forced equations:

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for r and R depending on the initial mass m 0, energy bound m 1 and some high order moments m p 0 for some p ∗ > 1, depending on the 'heating' force coefficient. The choice of s is done by setting: (shown in the pure diffusion case and bounded angular section γ=1 and stationary state) Corollary: it is possible to choose s, such that So, in order to control z p+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 z P

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Then a careful choice of a = 4/3 and b < 1 cancels the coefficients for the two terms proportional to z p-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. ∂tzp +∂tzp + Add the time derivative to compute the Corresponding evolution estimate

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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) (i.e. a=1 and s=2) moments of equation collision operator Loss op. Gain op. Bernoulli type eq. can also “create” moments (Desvillates 93 B. Wennberg~’98) or “generate”

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In the elastic case, for 0 < γ ≤ 1 and b(θ)integrable: Propagation and generation estimates 1.Same argument holds for controlling moments of the derivatives of f(t,v) by iterative methods ( R. Alonso &I.G. JMPA’08) Elastic collisions with a cold thermostat 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 ) Generation 3. Generation of moments for a= γ /4 and s= γ /2 (Mouhot JSP’06) for initial data with only 2+ moments generation 4. Improvement in moments generation by taking a=a(t, γ) to s=γ (Alonso, Canizo, IG, and Mouhot, in preparation ) (i.e. a=1 and s=2) but uniform in t ! for r and R depending on the initial mass m 0, energy bound m 1 and some high order moments m p 0 for some p ∗ > 1, but uniform in t ! Summability of the series of moments is uniform intime Summability of the series of moments is uniform intime Propagation of exponential moments Propagation of exponential moments

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( IG, Vlad Panferov, Cedric Villani, arXiv.org’06 - ARMA’08, and R. Alonso and I.G- JMPA’08 ) Upper point-wise uniform bounds for large energy tails for elastic hard spheres or γ-variable potentials in d-dimensions Comparison principle Comparison principle: Q is multi-linear, symmetric, conservative, and L 1 -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. STRATEGY : Find a comparison theorem & construct a suitable barrier function ⇒ Compare to obtain point-wise bounds: and

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Crucial point:

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barrier probability distribution In order to find the barrier probability distribution: we need key estimate That can be obtain by the following key estimate:

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Carleman integral representation Tool: Carleman integral representation of

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Remark: these propagation properties in L 1 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.

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σ 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; ε 1 ; ε 2 ;…; ε d-2 }, where r = radial coordinates, φ = polar angle, and {ε 1 ; ε 2 ;…; ε d-2 }, the n-2 azimuthal angular variables. thenwith, θ = scattering angle 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 S d-1 Set |u · η| dη = cos γ dη and u · σ = cos 2γ (since η·σ = cos γ ). Then |u| |u| dσ = 2 d-2 cos d-2 2γ dγ |u · η| dη = |u| 2 1-d ( 1-cos θ ) (3-d)/2 sin θ dθ. 2 |u · η| g(|u · η| dη )dη = |S d-2 | |u| ∫ 0 g(z) (1-z 2 ) (d-3)/2 dz |u| 1 ∫ S d-1

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