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Smoluchowski-Poisson equation – structures of kinetic mean field

equations

Takashi Suzuki Osaka Univ.

→pick up key factors

A. top down modeling

insight from experiments

→integrate formulae →simulation check →understand the evens as a system

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aggregating cellsmoving clustered cells

chemotaxis

zero-flux boundary condition

c.f. Childress-Percus 81, Nagai 95

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Random Walk

Ichikawa-Rouzimaimaiti-S. 11continuous particle distribution c.f. Othmer-Stevens 97

Einstein’s formula

B. bottom up modeling

semi-conductor physics high-molecular chemistry

friction-fluctuation approach

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Smoluchowski-Poisson equation ~ general form

Hamilton system → particle collision → time irreversible (kinetics)

kinetic statistical mechanics (Boltzmann) ～ where is it going?

isolated

closed

open

entropy

Helmholtz free energy

Gibbs free energy

energy

temperature

pressure

micro-canonical

canonical

grand-canonical

thermodynamics

statistical mechanics

bottom up top down

system consistency kinetics ensemble

dissipative (open) – entropy production (near equilibrium)

I. Prigogine

All ensembles equivalent in the range of short interaction (the state which the system should take in the sense of Gibbs)

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Hamiltonian → statistical mechanics (Gibbs)

micro-canonical statistics

H total energy

equal a priori probabilities

micro-canonical ensemble

canonical statistics

canonical ensemble

inverse temperature

N→∞ … mean field limit

equivalent in heat equilibrium

micro-canonical measure

canonical (Gibbs) measure

weight-factor

C. Static Theory ~ what should it be?

thermo-dynamical relation

ordered structure observed in negative inverse temperature

Onsager49 2D Euler equation of motionon simply-connected domain

high energy limit

particle density

stream function

Mean field equation

Hamiltonian for point vortices

dualiy

→ Ｇｉｂｂｓ ｍｅａｓｕｒｅ

Joyce-Montgomery 73

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micro-canonical measure

mean filed limit

propagation of chaos

one-point pdf

K-point pdf equal a priori probabilities

mean field equation

(factorization property)

point vortex mean field equation（stream function formulation)

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recursive hierarchy quantized blowup mechanismfield – particle duality

ordered structure observed in negative inverse temperature

point vortex system ↔ (Hamiltonian) mean field equation

variational structure

model (B) equation

total mass conservation

free energy decrease

Helmholtz’s free energy

particle density

field

mean field description of self-interaction

D. Nonlinear Spectral Dynamics

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dual variation

free energy field functionalduality

particle distribution potential density

stationary SP equation point vortex mean field

stationary state

Senba-S. 00

stationary quantization → dynamical quantization

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2D Smolchowski-Poissonequation (Childress-Percus, Jager-Luckhaus model)

quantized blowup mechanism

2. quantized blowup mechanics

1. recursive hierarchy

3. field-particle duality 4. nonlinear spectral dynamics

A. point vortex mean field equation B. Smoluchowski-Poisson equation

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1. blowup analysis E. Mathematical Structure

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scaling → critical exponent

Plasma confinement ↔ compressible self-gravitating fluid (Toland duality)

mass quantization (with or without boundary condition)

kinetic theory using Tsallis entropy …Chavanis

blowup threshold lack of monotonicity formulafiniteness of Type II blowup points

Related models

Energy Quantization

Landau-Ginzburg model H-sysytemYamabe problem harmonic heat flow

Hamiltonian control

Summary

Smoluchowski-Poisson equation is derived from the transport and kinetic theories in the contexts of condensed matter physics, high-molecular chemistry, cell biology, and astrophysics. Through the Toland duality its stationary state is equivalent to the point vortex mean field equation where quantized blowup mechanism is observed. Then we obtain the formation of collapses with the quantized mass

1. mathematical structure (3) 2. main result (1) 3. formation of collapse ( 3) 4. mass quantization (7)

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2. Main Result

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3. Formation of Collapse

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Parabolic envelopeParabolic envelopeParabolic envelopeParabolic envelope

infinitely largeinfinitely largeinfinitely largeinfinitely large parabolic regionparabolic regionparabolic regionparabolic region

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-1

0r

c(r)

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References

1. E.E. Espejo, A. Stevens and T. Suzuki, Simultaneous blowup and mass separation during collapses in an interacting system of chemotaxis, preprint 64/2011, Max-Planc Institute, http://www.mis.mpg.de/publications/preprints/2011

2. T. Suzuki and T. Senba, Applied Analysis – Mathematical Methods in Natural Science, second edition, Imperial College Press, London, 2011

3. T. Suzuki, Mean Field Theories and Dual Variation, Atlantis Press, Amsterdam-Paris, 2008

4. T. Suzuki, Free Energy and Self-Interacting Particles, Birkhauser, Boston, 2005