The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation...
Transcript of The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation...
![Page 1: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/1.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
The zero temperature limit of interactingcorpora
Peter Constantin
Department of MathematicsThe University of Chicago
IMA, July 21, 2008
![Page 2: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/2.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Thanks: N. Masmoudi, A. Zlatos.
Support: NSF
![Page 3: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/3.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Complex Fluid Models
• Landau Equilibrium models: order parameter (Director =Oseen, Zocher, Frank, Ericksen, Leslie. Tensor = deGennes.)
• Onsager Equilibrium models: (pdf of state), free energyderived from physics
• Passive Kinetic models: Doi, FENE and variants (pdf ofstate) effects of shear on dilute suspensions of rigid orextensible corpora = linear Fokker-Planck
• Tensorial models: (conformation tensors): closure ofcertain kinetic models, e.g. Oldroyd B
• Active Kinetic Models: (pdf) Onsager-Smoluchowski:Nonlinear Fokker-Planck, stochastic models
![Page 4: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/4.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Applications
• Nanoscale self-assembly
• Microfluidics
• Biomaterials
• Gels and Foams
• Soft Lattices, Jamming
• Pattern recognition
![Page 5: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/5.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Major Problems
1 Derivation of Micro-Macro Effect
2 Dissipation of Energy: Complex Fluids “Onsager”conjecture
3 PDE existence theory for coupled system
4 Modeling of interactions in the correct moduli space
![Page 6: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/6.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Major Problems
1 Derivation of Micro-Macro Effect
2 Dissipation of Energy: Complex Fluids “Onsager”conjecture
3 PDE existence theory for coupled system
4 Modeling of interactions in the correct moduli space
![Page 7: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/7.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Major Problems
1 Derivation of Micro-Macro Effect
2 Dissipation of Energy: Complex Fluids “Onsager”conjecture
3 PDE existence theory for coupled system
4 Modeling of interactions in the correct moduli space
![Page 8: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/8.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Major Problems
1 Derivation of Micro-Macro Effect
2 Dissipation of Energy: Complex Fluids “Onsager”conjecture
3 PDE existence theory for coupled system
4 Modeling of interactions in the correct moduli space
![Page 9: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/9.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
• Configuration space: M = compact, separable, metrizablespace. m ∈ M = corpus.
• Reference measure: dµ – Borel Probability on M.
• Corpora measure f (m)dµ(m) – Probabililty, AC w.r. dµ.
• Interaction kernel k : M ×M → R+, symmetric,by-Lipschitz.
• Operator (Kf ) (m) =∫M k(m, p)f (p)dµ(p)
• Potential U = −Kf = micro-micro interaction
• Free Energy
E [f ] =
∫M
f log fdµ− 1
2
∫M
(Kf ) fdµ
• Minima of Free Energy: Onsager Equation
f = Z−1eKf .
![Page 10: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/10.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
• Configuration space: M = compact, separable, metrizablespace. m ∈ M = corpus.
• Reference measure: dµ – Borel Probability on M.
• Corpora measure f (m)dµ(m) – Probabililty, AC w.r. dµ.
• Interaction kernel k : M ×M → R+, symmetric,by-Lipschitz.
• Operator (Kf ) (m) =∫M k(m, p)f (p)dµ(p)
• Potential U = −Kf = micro-micro interaction
• Free Energy
E [f ] =
∫M
f log fdµ− 1
2
∫M
(Kf ) fdµ
• Minima of Free Energy: Onsager Equation
f = Z−1eKf .
![Page 11: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/11.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
• Configuration space: M = compact, separable, metrizablespace. m ∈ M = corpus.
• Reference measure: dµ – Borel Probability on M.
• Corpora measure f (m)dµ(m) – Probabililty, AC w.r. dµ.
• Interaction kernel k : M ×M → R+, symmetric,by-Lipschitz.
• Operator (Kf ) (m) =∫M k(m, p)f (p)dµ(p)
• Potential U = −Kf = micro-micro interaction
• Free Energy
E [f ] =
∫M
f log fdµ− 1
2
∫M
(Kf ) fdµ
• Minima of Free Energy: Onsager Equation
f = Z−1eKf .
![Page 12: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/12.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
• Configuration space: M = compact, separable, metrizablespace. m ∈ M = corpus.
• Reference measure: dµ – Borel Probability on M.
• Corpora measure f (m)dµ(m) – Probabililty, AC w.r. dµ.
• Interaction kernel k : M ×M → R+,
symmetric,by-Lipschitz.
• Operator (Kf ) (m) =∫M k(m, p)f (p)dµ(p)
• Potential U = −Kf = micro-micro interaction
• Free Energy
E [f ] =
∫M
f log fdµ− 1
2
∫M
(Kf ) fdµ
• Minima of Free Energy: Onsager Equation
f = Z−1eKf .
![Page 13: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/13.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
• Configuration space: M = compact, separable, metrizablespace. m ∈ M = corpus.
• Reference measure: dµ – Borel Probability on M.
• Corpora measure f (m)dµ(m) – Probabililty, AC w.r. dµ.
• Interaction kernel k : M ×M → R+, symmetric,by-Lipschitz.
• Operator (Kf ) (m) =∫M k(m, p)f (p)dµ(p)
• Potential U = −Kf = micro-micro interaction
• Free Energy
E [f ] =
∫M
f log fdµ− 1
2
∫M
(Kf ) fdµ
• Minima of Free Energy: Onsager Equation
f = Z−1eKf .
![Page 14: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/14.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
• Configuration space: M = compact, separable, metrizablespace. m ∈ M = corpus.
• Reference measure: dµ – Borel Probability on M.
• Corpora measure f (m)dµ(m) – Probabililty, AC w.r. dµ.
• Interaction kernel k : M ×M → R+, symmetric,by-Lipschitz.
• Operator (Kf ) (m) =∫M k(m, p)f (p)dµ(p)
• Potential U = −Kf = micro-micro interaction
• Free Energy
E [f ] =
∫M
f log fdµ− 1
2
∫M
(Kf ) fdµ
• Minima of Free Energy: Onsager Equation
f = Z−1eKf .
![Page 15: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/15.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
• Configuration space: M = compact, separable, metrizablespace. m ∈ M = corpus.
• Reference measure: dµ – Borel Probability on M.
• Corpora measure f (m)dµ(m) – Probabililty, AC w.r. dµ.
• Interaction kernel k : M ×M → R+, symmetric,by-Lipschitz.
• Operator (Kf ) (m) =∫M k(m, p)f (p)dµ(p)
• Potential U = −Kf = micro-micro interaction
• Free Energy
E [f ] =
∫M
f log fdµ− 1
2
∫M
(Kf ) fdµ
• Minima of Free Energy: Onsager Equation
f = Z−1eKf .
![Page 16: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/16.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
• Configuration space: M = compact, separable, metrizablespace. m ∈ M = corpus.
• Reference measure: dµ – Borel Probability on M.
• Corpora measure f (m)dµ(m) – Probabililty, AC w.r. dµ.
• Interaction kernel k : M ×M → R+, symmetric,by-Lipschitz.
• Operator (Kf ) (m) =∫M k(m, p)f (p)dµ(p)
• Potential U = −Kf = micro-micro interaction
• Free Energy
E [f ] =
∫M
f log fdµ− 1
2
∫M
(Kf ) fdµ
• Minima of Free Energy: Onsager Equation
f = Z−1eKf .
![Page 17: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/17.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
• Configuration space: M = compact, separable, metrizablespace. m ∈ M = corpus.
• Reference measure: dµ – Borel Probability on M.
• Corpora measure f (m)dµ(m) – Probabililty, AC w.r. dµ.
• Interaction kernel k : M ×M → R+, symmetric,by-Lipschitz.
• Operator (Kf ) (m) =∫M k(m, p)f (p)dµ(p)
• Potential U = −Kf = micro-micro interaction
• Free Energy
E [f ] =
∫M
f log fdµ− 1
2
∫M
(Kf ) fdµ
• Minima of Free Energy: Onsager Equation
f = Z−1eKf .
![Page 18: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/18.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Goals of Theory:
1 Existence theory for solutions of Onsager’s equation
2 Classification of zero-temperature limits
3 Selection mechanism for zero-temperature limit
4 Stability of states
5 Physical Space Interaction
6 Dynamics
![Page 19: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/19.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Goals of Theory:
1 Existence theory for solutions of Onsager’s equation
2 Classification of zero-temperature limits
3 Selection mechanism for zero-temperature limit
4 Stability of states
5 Physical Space Interaction
6 Dynamics
![Page 20: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/20.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Goals of Theory:
1 Existence theory for solutions of Onsager’s equation
2 Classification of zero-temperature limits
3 Selection mechanism for zero-temperature limit
4 Stability of states
5 Physical Space Interaction
6 Dynamics
![Page 21: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/21.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Goals of Theory:
1 Existence theory for solutions of Onsager’s equation
2 Classification of zero-temperature limits
3 Selection mechanism for zero-temperature limit
4 Stability of states
5 Physical Space Interaction
6 Dynamics
![Page 22: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/22.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Goals of Theory:
1 Existence theory for solutions of Onsager’s equation
2 Classification of zero-temperature limits
3 Selection mechanism for zero-temperature limit
4 Stability of states
5 Physical Space Interaction
6 Dynamics
![Page 23: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/23.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Goals of Theory:
1 Existence theory for solutions of Onsager’s equation
2 Classification of zero-temperature limits
3 Selection mechanism for zero-temperature limit
4 Stability of states
5 Physical Space Interaction
6 Dynamics
![Page 24: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/24.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Example: Rods, Maier-Saupe potential
M = Sn−1, dµ = area.
Kf (p) = b
∫Sn−1
((p · q)2 − 1
n
)f (q)dµ
b = intensity, inverse temperature.
![Page 25: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/25.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Dimension Reduction, Maier-Saupe
n × n symmetric, traceless matrix S :
S 7→ Z (S)
Z (S) =
∫Sn−1
eb(S ijmimj )dµ.
fS(m) = (Z (S))−1eb(S ijmimj )
σ(S)ij =
∫Sn−1
(mimj −
δij
n
)fS(m)dµ.
TheoremOnsager’s equation with Maier-Saupe potential is equivalent to
σ(S) = S .
![Page 26: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/26.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Limit b →∞
[φ] =
∫S2
φ(m)f (m)dµ.
Isotropic:
limb→∞
[φ] =1
4π
∫S2
φ(p)dµ
Oblate:
limb→∞
[φ] =1
2π
∫ 2π
0φ(cos ϕ, sin ϕ, 0)dϕ
Prolate:lim
b→∞[φ] = φ(m), m ∈ S2.
![Page 27: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/27.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Limit b →∞
[φ] =
∫S2
φ(m)f (m)dµ.
Isotropic:
limb→∞
[φ] =1
4π
∫S2
φ(p)dµ
Oblate:
limb→∞
[φ] =1
2π
∫ 2π
0φ(cos ϕ, sin ϕ, 0)dϕ
Prolate:lim
b→∞[φ] = φ(m), m ∈ S2.
![Page 28: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/28.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Limit b →∞
[φ] =
∫S2
φ(m)f (m)dµ.
Isotropic:
limb→∞
[φ] =1
4π
∫S2
φ(p)dµ
Oblate:
limb→∞
[φ] =1
2π
∫ 2π
0φ(cos ϕ, sin ϕ, 0)dϕ
Prolate:lim
b→∞[φ] = φ(m), m ∈ S2.
![Page 29: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/29.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Limit b →∞
[φ] =
∫S2
φ(m)f (m)dµ.
Isotropic:
limb→∞
[φ] =1
4π
∫S2
φ(p)dµ
Oblate:
limb→∞
[φ] =1
2π
∫ 2π
0φ(cos ϕ, sin ϕ, 0)dϕ
Prolate:lim
b→∞[φ] = φ(m), m ∈ S2.
![Page 30: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/30.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Freely Articulated N-corpora
M = M1 × · · · ×MN , dµ = Πdµj
k(p1, q1, p2, q2, . . . ) =∑i ,j
kij(pi , qj)
Kf =N∑
i=1
Ki f , with
Ki f (pi ) =∑
j
∫eM kij(pi , qj)f (q1, . . . qN)dµ(q)
Onsager Equation f = Z−1eeKef
Z = ΠNj=1Zj , with Zj =
∫Mj
eKj fj dµj , fj = (Zj)−1eKj fj
f (p1, . . . pN) = f1(p1)f (p2) . . . fN(pN) product measure
![Page 31: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/31.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Freely Articulated N-corpora
M = M1 × · · · ×MN , dµ = Πdµj
k(p1, q1, p2, q2, . . . ) =∑i ,j
kij(pi , qj)
Kf =N∑
i=1
Ki f , with
Ki f (pi ) =∑
j
∫eM kij(pi , qj)f (q1, . . . qN)dµ(q)
Onsager Equation f = Z−1eeKef
Z = ΠNj=1Zj , with Zj =
∫Mj
eKj fj dµj , fj = (Zj)−1eKj fj
f (p1, . . . pN) = f1(p1)f (p2) . . . fN(pN) product measure
![Page 32: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/32.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Freely Articulated N-corpora
M = M1 × · · · ×MN , dµ = Πdµj
k(p1, q1, p2, q2, . . . ) =∑i ,j
kij(pi , qj)
Kf =N∑
i=1
Ki f ,
with
Ki f (pi ) =∑
j
∫eM kij(pi , qj)f (q1, . . . qN)dµ(q)
Onsager Equation f = Z−1eeKef
Z = ΠNj=1Zj , with Zj =
∫Mj
eKj fj dµj , fj = (Zj)−1eKj fj
f (p1, . . . pN) = f1(p1)f (p2) . . . fN(pN) product measure
![Page 33: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/33.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Freely Articulated N-corpora
M = M1 × · · · ×MN , dµ = Πdµj
k(p1, q1, p2, q2, . . . ) =∑i ,j
kij(pi , qj)
Kf =N∑
i=1
Ki f , with
Ki f (pi ) =∑
j
∫eM kij(pi , qj)f (q1, . . . qN)dµ(q)
Onsager Equation f = Z−1eeKef
Z = ΠNj=1Zj , with Zj =
∫Mj
eKj fj dµj , fj = (Zj)−1eKj fj
f (p1, . . . pN) = f1(p1)f (p2) . . . fN(pN) product measure
![Page 34: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/34.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Freely Articulated N-corpora
M = M1 × · · · ×MN , dµ = Πdµj
k(p1, q1, p2, q2, . . . ) =∑i ,j
kij(pi , qj)
Kf =N∑
i=1
Ki f , with
Ki f (pi ) =∑
j
∫eM kij(pi , qj)f (q1, . . . qN)dµ(q)
Onsager Equation f = Z−1eeKef
Z = ΠNj=1Zj , with Zj =
∫Mj
eKj fj dµj , fj = (Zj)−1eKj fj
f (p1, . . . pN) = f1(p1)f (p2) . . . fN(pN) product measure
![Page 35: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/35.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Freely Articulated N-corpora
M = M1 × · · · ×MN , dµ = Πdµj
k(p1, q1, p2, q2, . . . ) =∑i ,j
kij(pi , qj)
Kf =N∑
i=1
Ki f , with
Ki f (pi ) =∑
j
∫eM kij(pi , qj)f (q1, . . . qN)dµ(q)
Onsager Equation f = Z−1eeKef
Z = ΠNj=1Zj , with Zj =
∫Mj
eKj fj dµj , fj = (Zj)−1eKj fj
f (p1, . . . pN) = f1(p1)f (p2) . . . fN(pN) product measure
![Page 36: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/36.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Freely Articulated N-corpora
M = M1 × · · · ×MN , dµ = Πdµj
k(p1, q1, p2, q2, . . . ) =∑i ,j
kij(pi , qj)
Kf =N∑
i=1
Ki f , with
Ki f (pi ) =∑
j
∫eM kij(pi , qj)f (q1, . . . qN)dµ(q)
Onsager Equation f = Z−1eeKef
Z = ΠNj=1Zj , with Zj =
∫Mj
eKj fj dµj , fj = (Zj)−1eKj fj
f (p1, . . . pN) = f1(p1)f (p2) . . . fN(pN) product measure
![Page 37: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/37.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Example of Interacting Corpora
M = S1, M = S1 × S1.
Kf (p1, p2) =−b
∫T2 ‖e(p1) ∧ e(p2)− e(q1) ∧ e(q2)‖2f (q1, q2)dq1dq2
with e(p) = (cos p, sin p) if p ∈ [0, 2π].
‖e(p1)∧e(p2)−e(q1)∧e(q2)‖2 = (sin(p1 − p2)− sin(q1 − q2))2
Dimension reduction: Onsager’s equation f = Z−1eKf
reduces toa = [sin θ](a)
with [φ](a) =
∫ 2π0 φ(θ)g(θ)dθ
g(θ) = Z−1e−b(sin(θ)−a)2
Z =∫ 2π0 e−b(sin(θ)−a)2dθ
![Page 38: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/38.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Example of Interacting Corpora
M = S1, M = S1 × S1.
Kf (p1, p2) =−b
∫T2 ‖e(p1) ∧ e(p2)− e(q1) ∧ e(q2)‖2f (q1, q2)dq1dq2
with e(p) = (cos p, sin p) if p ∈ [0, 2π].
‖e(p1)∧e(p2)−e(q1)∧e(q2)‖2 = (sin(p1 − p2)− sin(q1 − q2))2
Dimension reduction: Onsager’s equation f = Z−1eKf
reduces toa = [sin θ](a)
with [φ](a) =
∫ 2π0 φ(θ)g(θ)dθ
g(θ) = Z−1e−b(sin(θ)−a)2
Z =∫ 2π0 e−b(sin(θ)−a)2dθ
![Page 39: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/39.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Example of Interacting Corpora
M = S1, M = S1 × S1.
Kf (p1, p2) =−b
∫T2 ‖e(p1) ∧ e(p2)− e(q1) ∧ e(q2)‖2f (q1, q2)dq1dq2
with e(p) = (cos p, sin p) if p ∈ [0, 2π].
‖e(p1)∧e(p2)−e(q1)∧e(q2)‖2 = (sin(p1 − p2)− sin(q1 − q2))2
Dimension reduction: Onsager’s equation f = Z−1eKf
reduces toa = [sin θ](a)
with [φ](a) =
∫ 2π0 φ(θ)g(θ)dθ
g(θ) = Z−1e−b(sin(θ)−a)2
Z =∫ 2π0 e−b(sin(θ)−a)2dθ
![Page 40: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/40.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Example of Interacting Corpora
M = S1, M = S1 × S1.
Kf (p1, p2) =−b
∫T2 ‖e(p1) ∧ e(p2)− e(q1) ∧ e(q2)‖2f (q1, q2)dq1dq2
with e(p) = (cos p, sin p) if p ∈ [0, 2π].
‖e(p1)∧e(p2)−e(q1)∧e(q2)‖2 = (sin(p1 − p2)− sin(q1 − q2))2
Dimension reduction: Onsager’s equation f = Z−1eKf
reduces toa = [sin θ](a)
with [φ](a) =
∫ 2π0 φ(θ)g(θ)dθ
g(θ) = Z−1e−b(sin(θ)−a)2
Z =∫ 2π0 e−b(sin(θ)−a)2dθ
![Page 41: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/41.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Example of Interacting Corpora
M = S1, M = S1 × S1.
Kf (p1, p2) =−b
∫T2 ‖e(p1) ∧ e(p2)− e(q1) ∧ e(q2)‖2f (q1, q2)dq1dq2
with e(p) = (cos p, sin p) if p ∈ [0, 2π].
‖e(p1)∧e(p2)−e(q1)∧e(q2)‖2 = (sin(p1 − p2)− sin(q1 − q2))2
Dimension reduction: Onsager’s equation f = Z−1eKf
reduces toa = [sin θ](a)
with [φ](a) =
∫ 2π0 φ(θ)g(θ)dθ
g(θ) = Z−1e−b(sin(θ)−a)2
Z =∫ 2π0 e−b(sin(θ)−a)2dθ
![Page 42: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/42.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Example of Interacting Corpora
M = S1, M = S1 × S1.
Kf (p1, p2) =−b
∫T2 ‖e(p1) ∧ e(p2)− e(q1) ∧ e(q2)‖2f (q1, q2)dq1dq2
with e(p) = (cos p, sin p) if p ∈ [0, 2π].
‖e(p1)∧e(p2)−e(q1)∧e(q2)‖2 = (sin(p1 − p2)− sin(q1 − q2))2
Dimension reduction: Onsager’s equation f = Z−1eKf
reduces toa = [sin θ](a)
with [φ](a) =
∫ 2π0 φ(θ)g(θ)dθ
g(θ) = Z−1e−b(sin(θ)−a)2
Z =∫ 2π0 e−b(sin(θ)−a)2dθ
![Page 43: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/43.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
The solution is f (θ1, θ2) = g(θ1 − θ2).
Let
u(θ, a) = sin θ − a,
and let
[u](b, a) =
∫ 2π0 u(θ, a)e−bu2(θ,a)dθ∫ 2π
0 e−bu2(θ,a)dθ.
The Onsager equation is equivalent to
[u](b, a) = 0.
This determines a, which in turn determines g , f .a = 0 always a solution. It yields
f0(p1, p2) = Z−1e−b sin2(p1−p2).
As b →∞ this tends to δ((p1 − p2)modπ).
![Page 44: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/44.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
The solution is f (θ1, θ2) = g(θ1 − θ2). Let
u(θ, a) = sin θ − a,
and let
[u](b, a) =
∫ 2π0 u(θ, a)e−bu2(θ,a)dθ∫ 2π
0 e−bu2(θ,a)dθ.
The Onsager equation is equivalent to
[u](b, a) = 0.
This determines a, which in turn determines g , f .a = 0 always a solution. It yields
f0(p1, p2) = Z−1e−b sin2(p1−p2).
As b →∞ this tends to δ((p1 − p2)modπ).
![Page 45: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/45.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
The solution is f (θ1, θ2) = g(θ1 − θ2). Let
u(θ, a) = sin θ − a,
and let
[u](b, a) =
∫ 2π0 u(θ, a)e−bu2(θ,a)dθ∫ 2π
0 e−bu2(θ,a)dθ.
The Onsager equation is equivalent to
[u](b, a) = 0.
This determines a, which in turn determines g , f .a = 0 always a solution. It yields
f0(p1, p2) = Z−1e−b sin2(p1−p2).
As b →∞ this tends to δ((p1 − p2)modπ).
![Page 46: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/46.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
The solution is f (θ1, θ2) = g(θ1 − θ2). Let
u(θ, a) = sin θ − a,
and let
[u](b, a) =
∫ 2π0 u(θ, a)e−bu2(θ,a)dθ∫ 2π
0 e−bu2(θ,a)dθ.
The Onsager equation is equivalent to
[u](b, a) = 0.
This determines a, which in turn determines g , f .a = 0 always a solution. It yields
f0(p1, p2) = Z−1e−b sin2(p1−p2).
As b →∞ this tends to δ((p1 − p2)modπ).
![Page 47: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/47.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
The solution is f (θ1, θ2) = g(θ1 − θ2). Let
u(θ, a) = sin θ − a,
and let
[u](b, a) =
∫ 2π0 u(θ, a)e−bu2(θ,a)dθ∫ 2π
0 e−bu2(θ,a)dθ.
The Onsager equation is equivalent to
[u](b, a) = 0.
This determines a, which in turn determines g , f .
a = 0 always a solution. It yields
f0(p1, p2) = Z−1e−b sin2(p1−p2).
As b →∞ this tends to δ((p1 − p2)modπ).
![Page 48: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/48.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
The solution is f (θ1, θ2) = g(θ1 − θ2). Let
u(θ, a) = sin θ − a,
and let
[u](b, a) =
∫ 2π0 u(θ, a)e−bu2(θ,a)dθ∫ 2π
0 e−bu2(θ,a)dθ.
The Onsager equation is equivalent to
[u](b, a) = 0.
This determines a, which in turn determines g , f .a = 0 always a solution. It yields
f0(p1, p2) = Z−1e−b sin2(p1−p2).
As b →∞ this tends to δ((p1 − p2)modπ).
![Page 49: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/49.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
The solution is f (θ1, θ2) = g(θ1 − θ2). Let
u(θ, a) = sin θ − a,
and let
[u](b, a) =
∫ 2π0 u(θ, a)e−bu2(θ,a)dθ∫ 2π
0 e−bu2(θ,a)dθ.
The Onsager equation is equivalent to
[u](b, a) = 0.
This determines a, which in turn determines g , f .a = 0 always a solution. It yields
f0(p1, p2) = Z−1e−b sin2(p1−p2).
As b →∞ this tends to δ((p1 − p2)modπ).
![Page 50: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/50.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Consider
λ(a, τ) = b12
∫ 2π
0e−b(sin θ−a)2dθ
with τ = b−1.
Note
[u] =1
2b
∂aλ
λand
∂τλ =1
4∂2
aλ
limτ→0
λ(a, τ) = 2√
π1√
1− a2, 0 < a < 1.
Increasing. But things are subtle, ∂λ∂a (1, τ) < 0.
In fact, phase transition at positive τ
∂aλ((a(τ), τ) = 0
and limit limτ→0 a(τ) = 1, and consequently
limb→∞
f (p1 − p2) = δ((
p1 − p2 −π
2
)modπ
)
![Page 51: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/51.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Consider
λ(a, τ) = b12
∫ 2π
0e−b(sin θ−a)2dθ
with τ = b−1.Note
[u] =1
2b
∂aλ
λ
and
∂τλ =1
4∂2
aλ
limτ→0
λ(a, τ) = 2√
π1√
1− a2, 0 < a < 1.
Increasing. But things are subtle, ∂λ∂a (1, τ) < 0.
In fact, phase transition at positive τ
∂aλ((a(τ), τ) = 0
and limit limτ→0 a(τ) = 1, and consequently
limb→∞
f (p1 − p2) = δ((
p1 − p2 −π
2
)modπ
)
![Page 52: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/52.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Consider
λ(a, τ) = b12
∫ 2π
0e−b(sin θ−a)2dθ
with τ = b−1.Note
[u] =1
2b
∂aλ
λand
∂τλ =1
4∂2
aλ
limτ→0
λ(a, τ) = 2√
π1√
1− a2, 0 < a < 1.
Increasing. But things are subtle, ∂λ∂a (1, τ) < 0.
In fact, phase transition at positive τ
∂aλ((a(τ), τ) = 0
and limit limτ→0 a(τ) = 1, and consequently
limb→∞
f (p1 − p2) = δ((
p1 − p2 −π
2
)modπ
)
![Page 53: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/53.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Consider
λ(a, τ) = b12
∫ 2π
0e−b(sin θ−a)2dθ
with τ = b−1.Note
[u] =1
2b
∂aλ
λand
∂τλ =1
4∂2
aλ
limτ→0
λ(a, τ) = 2√
π1√
1− a2, 0 < a < 1.
Increasing.
But things are subtle, ∂λ∂a (1, τ) < 0.
In fact, phase transition at positive τ
∂aλ((a(τ), τ) = 0
and limit limτ→0 a(τ) = 1, and consequently
limb→∞
f (p1 − p2) = δ((
p1 − p2 −π
2
)modπ
)
![Page 54: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/54.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Consider
λ(a, τ) = b12
∫ 2π
0e−b(sin θ−a)2dθ
with τ = b−1.Note
[u] =1
2b
∂aλ
λand
∂τλ =1
4∂2
aλ
limτ→0
λ(a, τ) = 2√
π1√
1− a2, 0 < a < 1.
Increasing. But things are subtle, ∂λ∂a (1, τ) < 0.
In fact, phase transition at positive τ
∂aλ((a(τ), τ) = 0
and limit limτ→0 a(τ) = 1, and consequently
limb→∞
f (p1 − p2) = δ((
p1 − p2 −π
2
)modπ
)
![Page 55: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/55.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Consider
λ(a, τ) = b12
∫ 2π
0e−b(sin θ−a)2dθ
with τ = b−1.Note
[u] =1
2b
∂aλ
λand
∂τλ =1
4∂2
aλ
limτ→0
λ(a, τ) = 2√
π1√
1− a2, 0 < a < 1.
Increasing. But things are subtle, ∂λ∂a (1, τ) < 0.
In fact, phase transition at positive τ
∂aλ((a(τ), τ) = 0
and limit limτ→0 a(τ) = 1, and consequently
limb→∞
f (p1 − p2) = δ((
p1 − p2 −π
2
)modπ
)
![Page 56: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/56.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
More degrees of freedom
M = [0, L]× [0, L]× [0, π], dµ = 1πL2 dx1dx2dθ.
U[f ](x1, x2, θ) =
∫M
(x1x2 sin(θ)− y1y2 sin(φ))2f (y1, y2, φ)dµ
The solutions of Onsager’s equation are of the form
g(x1, x2, θ) = Z−1e−b(x1x2 sin θ−a)2
with Z determined by the requirement of normalization∫M gdµ = 1, a determined by
a =
∫M
(x1x2 sin θ)g(x1, x2, θ)dµ
![Page 57: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/57.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
More degrees of freedom
M = [0, L]× [0, L]× [0, π], dµ = 1πL2 dx1dx2dθ.
U[f ](x1, x2, θ) =
∫M
(x1x2 sin(θ)− y1y2 sin(φ))2f (y1, y2, φ)dµ
The solutions of Onsager’s equation are of the form
g(x1, x2, θ) = Z−1e−b(x1x2 sin θ−a)2
with Z determined by the requirement of normalization∫M gdµ = 1, a determined by
a =
∫M
(x1x2 sin θ)g(x1, x2, θ)dµ
![Page 58: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/58.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
More degrees of freedom
M = [0, L]× [0, L]× [0, π], dµ = 1πL2 dx1dx2dθ.
U[f ](x1, x2, θ) =
∫M
(x1x2 sin(θ)− y1y2 sin(φ))2f (y1, y2, φ)dµ
The solutions of Onsager’s equation are of the form
g(x1, x2, θ) = Z−1e−b(x1x2 sin θ−a)2
with Z determined by the requirement of normalization∫M gdµ = 1, a determined by
a =
∫M
(x1x2 sin θ)g(x1, x2, θ)dµ
![Page 59: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/59.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
More degrees of freedom
M = [0, L]× [0, L]× [0, π], dµ = 1πL2 dx1dx2dθ.
U[f ](x1, x2, θ) =
∫M
(x1x2 sin(θ)− y1y2 sin(φ))2f (y1, y2, φ)dµ
The solutions of Onsager’s equation are of the form
g(x1, x2, θ) = Z−1e−b(x1x2 sin θ−a)2
with Z determined by the requirement of normalization∫M gdµ = 1,
a determined by
a =
∫M
(x1x2 sin θ)g(x1, x2, θ)dµ
![Page 60: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/60.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
More degrees of freedom
M = [0, L]× [0, L]× [0, π], dµ = 1πL2 dx1dx2dθ.
U[f ](x1, x2, θ) =
∫M
(x1x2 sin(θ)− y1y2 sin(φ))2f (y1, y2, φ)dµ
The solutions of Onsager’s equation are of the form
g(x1, x2, θ) = Z−1e−b(x1x2 sin θ−a)2
with Z determined by the requirement of normalization∫M gdµ = 1, a determined by
a =
∫M
(x1x2 sin θ)g(x1, x2, θ)dµ
![Page 61: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/61.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Letu(x1, x2, θ, a) = x1x2 sin θ − a
[u] =
∫M
ugdµ
a is determined by [u] = 0.
λ(a, τ) = τ−1/2
∫M
e−u2/τdµ
obeys the heat equation
∂τλ =1
4∂2
aλ
with τ = b−1.
[u] =1
2b∂a log λ.
a → 0, as b →∞.
![Page 62: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/62.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Letu(x1, x2, θ, a) = x1x2 sin θ − a
[u] =
∫M
ugdµ
a is determined by [u] = 0.
λ(a, τ) = τ−1/2
∫M
e−u2/τdµ
obeys the heat equation
∂τλ =1
4∂2
aλ
with τ = b−1.
[u] =1
2b∂a log λ.
a → 0, as b →∞.
![Page 63: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/63.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Letu(x1, x2, θ, a) = x1x2 sin θ − a
[u] =
∫M
ugdµ
a is determined by [u] = 0.
λ(a, τ) = τ−1/2
∫M
e−u2/τdµ
obeys the heat equation
∂τλ =1
4∂2
aλ
with τ = b−1.
[u] =1
2b∂a log λ.
a → 0, as b →∞.
![Page 64: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/64.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Letu(x1, x2, θ, a) = x1x2 sin θ − a
[u] =
∫M
ugdµ
a is determined by [u] = 0.
λ(a, τ) = τ−1/2
∫M
e−u2/τdµ
obeys the heat equation
∂τλ =1
4∂2
aλ
with τ = b−1.
[u] =1
2b∂a log λ.
a → 0, as b →∞.
![Page 65: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/65.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Letu(x1, x2, θ, a) = x1x2 sin θ − a
[u] =
∫M
ugdµ
a is determined by [u] = 0.
λ(a, τ) = τ−1/2
∫M
e−u2/τdµ
obeys the heat equation
∂τλ =1
4∂2
aλ
with τ = b−1.
[u] =1
2b∂a log λ.
a → 0, as b →∞.
![Page 66: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/66.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Letu(x1, x2, θ, a) = x1x2 sin θ − a
[u] =
∫M
ugdµ
a is determined by [u] = 0.
λ(a, τ) = τ−1/2
∫M
e−u2/τdµ
obeys the heat equation
∂τλ =1
4∂2
aλ
with τ = b−1.
[u] =1
2b∂a log λ.
a → 0, as b →∞.
![Page 67: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/67.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Letu(x1, x2, θ, a) = x1x2 sin θ − a
[u] =
∫M
ugdµ
a is determined by [u] = 0.
λ(a, τ) = τ−1/2
∫M
e−u2/τdµ
obeys the heat equation
∂τλ =1
4∂2
aλ
with τ = b−1.
[u] =1
2b∂a log λ.
a → 0, as b →∞.
![Page 68: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/68.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Even More Degrees of Freedom...
V (r) nonnegative, nonincreasing, compactly supported.p = (x1, . . . xN), xi ∈ Ω ⊂ Rn.
Packing energy:
F (p) =∑i<j
V (|xi − xj |).
M = Ω× · · · × Ω ∩ F ≤ F0.
(Kf )(p) = −∫
eM |F (p)− F (q)|2f (q)dq
Connection to the example of freely articulated 2n corpora,jamming, perhaps...
![Page 69: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/69.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Even More Degrees of Freedom...
V (r) nonnegative, nonincreasing, compactly supported.p = (x1, . . . xN), xi ∈ Ω ⊂ Rn. Packing energy:
F (p) =∑i<j
V (|xi − xj |).
M = Ω× · · · × Ω ∩ F ≤ F0.
(Kf )(p) = −∫
eM |F (p)− F (q)|2f (q)dq
Connection to the example of freely articulated 2n corpora,jamming, perhaps...
![Page 70: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/70.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Even More Degrees of Freedom...
V (r) nonnegative, nonincreasing, compactly supported.p = (x1, . . . xN), xi ∈ Ω ⊂ Rn. Packing energy:
F (p) =∑i<j
V (|xi − xj |).
M = Ω× · · · × Ω ∩ F ≤ F0.
(Kf )(p) = −∫
eM |F (p)− F (q)|2f (q)dq
Connection to the example of freely articulated 2n corpora,jamming, perhaps...
![Page 71: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/71.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Even More Degrees of Freedom...
V (r) nonnegative, nonincreasing, compactly supported.p = (x1, . . . xN), xi ∈ Ω ⊂ Rn. Packing energy:
F (p) =∑i<j
V (|xi − xj |).
M = Ω× · · · × Ω ∩ F ≤ F0.
(Kf )(p) = −∫
eM |F (p)− F (q)|2f (q)dq
Connection to the example of freely articulated 2n corpora,jamming, perhaps...
![Page 72: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/72.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Even More Degrees of Freedom...
V (r) nonnegative, nonincreasing, compactly supported.p = (x1, . . . xN), xi ∈ Ω ⊂ Rn. Packing energy:
F (p) =∑i<j
V (|xi − xj |).
M = Ω× · · · × Ω ∩ F ≤ F0.
(Kf )(p) = −∫
eM |F (p)− F (q)|2f (q)dq
Connection to the example of freely articulated 2n corpora,jamming, perhaps...
![Page 73: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/73.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
M compact metric space, d distance, µ Borel probabilitymeasure on M.
Let
−k = u : M ×M → R
• symmetric u(m, p) = u(p,m)
• bounded below u(m, n) ≥ 0
• uniformly bi-Lipschitz:
|u(m, n)− u(p, n)| ≤ Ld(m, p)
If f > 0,∫M fdµ = 1, define
E [f ] =
∫M
f log fdµ +b
2
∫M
∫M
u(p, q)f (p)dµ(p)f (q)dµ(q).
![Page 74: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/74.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
M compact metric space, d distance, µ Borel probabilitymeasure on M. Let
−k = u : M ×M → R
• symmetric u(m, p) = u(p,m)
• bounded below u(m, n) ≥ 0
• uniformly bi-Lipschitz:
|u(m, n)− u(p, n)| ≤ Ld(m, p)
If f > 0,∫M fdµ = 1, define
E [f ] =
∫M
f log fdµ +b
2
∫M
∫M
u(p, q)f (p)dµ(p)f (q)dµ(q).
![Page 75: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/75.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
M compact metric space, d distance, µ Borel probabilitymeasure on M. Let
−k = u : M ×M → R
• symmetric u(m, p) = u(p,m)
• bounded below u(m, n) ≥ 0
• uniformly bi-Lipschitz:
|u(m, n)− u(p, n)| ≤ Ld(m, p)
If f > 0,∫M fdµ = 1, define
E [f ] =
∫M
f log fdµ +b
2
∫M
∫M
u(p, q)f (p)dµ(p)f (q)dµ(q).
![Page 76: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/76.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
M compact metric space, d distance, µ Borel probabilitymeasure on M. Let
−k = u : M ×M → R
• symmetric u(m, p) = u(p,m)
• bounded below u(m, n) ≥ 0
• uniformly bi-Lipschitz:
|u(m, n)− u(p, n)| ≤ Ld(m, p)
If f > 0,∫M fdµ = 1, define
E [f ] =
∫M
f log fdµ +b
2
∫M
∫M
u(p, q)f (p)dµ(p)f (q)dµ(q).
![Page 77: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/77.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
TheoremFor any b > 0 there exists a solution g that minimizes theenergy:
E [g ] = minf≥0,
RM fdµ=1
E [f ]
The function g solves the Onsager equation
g(x) = (Z (b))−1e−bU(x)
with
Z (b) =
∫M
e−bU(x)dµ(x)
and
U(x) =
∫M
u(x , y)g(y)dµ(y).
The function g is normalized∫
gdµ = 1, strictly positive andLipschitz continuous.
![Page 78: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/78.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
TheoremFor any b > 0 there exists a solution g that minimizes theenergy:
E [g ] = minf≥0,
RM fdµ=1
E [f ]
The function g solves the Onsager equation
g(x) = (Z (b))−1e−bU(x)
with
Z (b) =
∫M
e−bU(x)dµ(x)
and
U(x) =
∫M
u(x , y)g(y)dµ(y).
The function g is normalized∫
gdµ = 1, strictly positive andLipschitz continuous.
![Page 79: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/79.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
TheoremFor any b > 0 there exists a solution g that minimizes theenergy:
E [g ] = minf≥0,
RM fdµ=1
E [f ]
The function g solves the Onsager equation
g(x) = (Z (b))−1e−bU(x)
with
Z (b) =
∫M
e−bU(x)dµ(x)
and
U(x) =
∫M
u(x , y)g(y)dµ(y).
The function g is normalized∫
gdµ = 1, strictly positive andLipschitz continuous.
![Page 80: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/80.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
TheoremFor any b > 0 there exists a solution g that minimizes theenergy:
E [g ] = minf≥0,
RM fdµ=1
E [f ]
The function g solves the Onsager equation
g(x) = (Z (b))−1e−bU(x)
with
Z (b) =
∫M
e−bU(x)dµ(x)
and
U(x) =
∫M
u(x , y)g(y)dµ(y).
The function g is normalized∫
gdµ = 1, strictly positive andLipschitz continuous.
![Page 81: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/81.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
TheoremFor any b > 0 there exists a solution g that minimizes theenergy:
E [g ] = minf≥0,
RM fdµ=1
E [f ]
The function g solves the Onsager equation
g(x) = (Z (b))−1e−bU(x)
with
Z (b) =
∫M
e−bU(x)dµ(x)
and
U(x) =
∫M
u(x , y)g(y)dµ(y).
The function g is normalized∫
gdµ = 1, strictly positive andLipschitz continuous.
![Page 82: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/82.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
TheoremFor any b > 0 there exists a solution g that minimizes theenergy:
E [g ] = minf≥0,
RM fdµ=1
E [f ]
The function g solves the Onsager equation
g(x) = (Z (b))−1e−bU(x)
with
Z (b) =
∫M
e−bU(x)dµ(x)
and
U(x) =
∫M
u(x , y)g(y)dµ(y).
The function g is normalized∫
gdµ = 1, strictly positive andLipschitz continuous.
![Page 83: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/83.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
The ur-corpus
Let M be a compact metrizable space and let u(x , y) besymmetric, bi-Lipschitz and bounded below.
In addition,assume:
u(x , x) = 0.
Theorem(C-Zlatos) Let ν be a weak limit of a sequence fndµ ofminima of the free energy E corresponding to bn →∞. Thenthere exists m ∈ M such that ν is concentrated on the level setΣ(m) = p | u(m, p) = 0.
![Page 84: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/84.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
The ur-corpus
Let M be a compact metrizable space and let u(x , y) besymmetric, bi-Lipschitz and bounded below. In addition,assume:
u(x , x) = 0.
Theorem(C-Zlatos) Let ν be a weak limit of a sequence fndµ ofminima of the free energy E corresponding to bn →∞. Thenthere exists m ∈ M such that ν is concentrated on the level setΣ(m) = p | u(m, p) = 0.
![Page 85: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/85.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
The ur-corpus
Let M be a compact metrizable space and let u(x , y) besymmetric, bi-Lipschitz and bounded below. In addition,assume:
u(x , x) = 0.
Theorem(C-Zlatos) Let ν be a weak limit of a sequence fndµ ofminima of the free energy E corresponding to bn →∞. Thenthere exists m ∈ M such that ν is concentrated on the level setΣ(m) = p | u(m, p) = 0.
![Page 86: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/86.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Idea of proof:
limb→∞
1
b
min
f >0,RM fdµ=1
E [f ]
= 0
and
ε
∫ ∫u(p,q)≥ε
f (p)dµ(p)f (q)dµ(q) ≤ 2
bE [f ].
if ε2n = 2bnE [fn], 0 < εn → 0, and
Q(p, ε) = q|u(p, q) ≤ ε,
then ∫M
fn(p)
[∫Q(p,εn)
fn(q)dµ(q)
]dµ(p) ≥ 1− εn
∃ pn,∫Q(pn,εn)
fn(q)dµ(q) ≥ 1− 2εn. Pass to subsequencepn → p.
![Page 87: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/87.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Idea of proof:
limb→∞
1
b
min
f >0,RM fdµ=1
E [f ]
= 0
and
ε
∫ ∫u(p,q)≥ε
f (p)dµ(p)f (q)dµ(q) ≤ 2
bE [f ].
if ε2n = 2bnE [fn], 0 < εn → 0, and
Q(p, ε) = q|u(p, q) ≤ ε,
then ∫M
fn(p)
[∫Q(p,εn)
fn(q)dµ(q)
]dµ(p) ≥ 1− εn
∃ pn,∫Q(pn,εn)
fn(q)dµ(q) ≥ 1− 2εn. Pass to subsequencepn → p.
![Page 88: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/88.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Idea of proof:
limb→∞
1
b
min
f >0,RM fdµ=1
E [f ]
= 0
and
ε
∫ ∫u(p,q)≥ε
f (p)dµ(p)f (q)dµ(q) ≤ 2
bE [f ].
if ε2n = 2bnE [fn], 0 < εn → 0,
and
Q(p, ε) = q|u(p, q) ≤ ε,
then ∫M
fn(p)
[∫Q(p,εn)
fn(q)dµ(q)
]dµ(p) ≥ 1− εn
∃ pn,∫Q(pn,εn)
fn(q)dµ(q) ≥ 1− 2εn. Pass to subsequencepn → p.
![Page 89: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/89.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Idea of proof:
limb→∞
1
b
min
f >0,RM fdµ=1
E [f ]
= 0
and
ε
∫ ∫u(p,q)≥ε
f (p)dµ(p)f (q)dµ(q) ≤ 2
bE [f ].
if ε2n = 2bnE [fn], 0 < εn → 0, and
Q(p, ε) = q|u(p, q) ≤ ε,
then ∫M
fn(p)
[∫Q(p,εn)
fn(q)dµ(q)
]dµ(p) ≥ 1− εn
∃ pn,∫Q(pn,εn)
fn(q)dµ(q) ≥ 1− 2εn. Pass to subsequencepn → p.
![Page 90: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/90.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Idea of proof:
limb→∞
1
b
min
f >0,RM fdµ=1
E [f ]
= 0
and
ε
∫ ∫u(p,q)≥ε
f (p)dµ(p)f (q)dµ(q) ≤ 2
bE [f ].
if ε2n = 2bnE [fn], 0 < εn → 0, and
Q(p, ε) = q|u(p, q) ≤ ε,
then ∫M
fn(p)
[∫Q(p,εn)
fn(q)dµ(q)
]dµ(p) ≥ 1− εn
∃ pn,∫Q(pn,εn)
fn(q)dµ(q) ≥ 1− 2εn. Pass to subsequencepn → p.
![Page 91: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/91.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Idea of proof:
limb→∞
1
b
min
f >0,RM fdµ=1
E [f ]
= 0
and
ε
∫ ∫u(p,q)≥ε
f (p)dµ(p)f (q)dµ(q) ≤ 2
bE [f ].
if ε2n = 2bnE [fn], 0 < εn → 0, and
Q(p, ε) = q|u(p, q) ≤ ε,
then ∫M
fn(p)
[∫Q(p,εn)
fn(q)dµ(q)
]dµ(p) ≥ 1− εn
∃ pn,∫Q(pn,εn)
fn(q)dµ(q) ≥ 1− 2εn.
Pass to subsequencepn → p.
![Page 92: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/92.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Idea of proof:
limb→∞
1
b
min
f >0,RM fdµ=1
E [f ]
= 0
and
ε
∫ ∫u(p,q)≥ε
f (p)dµ(p)f (q)dµ(q) ≤ 2
bE [f ].
if ε2n = 2bnE [fn], 0 < εn → 0, and
Q(p, ε) = q|u(p, q) ≤ ε,
then ∫M
fn(p)
[∫Q(p,εn)
fn(q)dµ(q)
]dµ(p) ≥ 1− εn
∃ pn,∫Q(pn,εn)
fn(q)dµ(q) ≥ 1− 2εn. Pass to subsequencepn → p.
![Page 93: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/93.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Principle: if a µ measure-preserving transformation Texists such that locally around p = p0,u(Tp, Tq) ≤ cu(p, q) with c < 1, then p0 cannot be anur-corpus.
If a local u-preserving transformation around p = p0 hasthe property that µ(T (B)) ≥ Cµ(B) for small balls aroundp0, with C > 1, then p0 cannot be an ur-corpus.
Example: Rhombi centered at the origin. The ur-rhombus isthe square.
![Page 94: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/94.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Principle: if a µ measure-preserving transformation Texists such that locally around p = p0,u(Tp, Tq) ≤ cu(p, q) with c < 1, then p0 cannot be anur-corpus.If a local u-preserving transformation around p = p0 hasthe property that µ(T (B)) ≥ Cµ(B) for small balls aroundp0, with C > 1, then p0 cannot be an ur-corpus.
Example: Rhombi centered at the origin. The ur-rhombus isthe square.
![Page 95: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/95.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Kinetics
M compact connected Riemannian manifold with metric g .
∂t f = divg
(f∇g
(δEδf
))δEδf
= log f −Kf
dEdt
= −∫
Mf |∇g (log f −Kf )|2 dµ(p)
Gradient system, steady solutions = Onsager equation.
∂t f = ∆g f − divg (f∇g (Kf ))
![Page 96: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/96.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Kinetics
M compact connected Riemannian manifold with metric g .
∂t f = divg
(f∇g
(δEδf
))
δEδf
= log f −Kf
dEdt
= −∫
Mf |∇g (log f −Kf )|2 dµ(p)
Gradient system, steady solutions = Onsager equation.
∂t f = ∆g f − divg (f∇g (Kf ))
![Page 97: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/97.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Kinetics
M compact connected Riemannian manifold with metric g .
∂t f = divg
(f∇g
(δEδf
))δEδf
= log f −Kf
dEdt
= −∫
Mf |∇g (log f −Kf )|2 dµ(p)
Gradient system, steady solutions = Onsager equation.
∂t f = ∆g f − divg (f∇g (Kf ))
![Page 98: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/98.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Kinetics
M compact connected Riemannian manifold with metric g .
∂t f = divg
(f∇g
(δEδf
))δEδf
= log f −Kf
dEdt
= −∫
Mf |∇g (log f −Kf )|2 dµ(p)
Gradient system, steady solutions = Onsager equation.
∂t f = ∆g f − divg (f∇g (Kf ))
![Page 99: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/99.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Embedding in Physical Space
f : Rn ×M × [0,∞) → (0,∞):
∂t f = ∆x f + divg (f∇g (log f −Kf ))
Example: n = 1, M = S1, Maier-Saupe potential:
f (x , θ, t) = 12π + 1
π
∑∞j=1 yj(x , t) cos(2jθ)
∂tyj = ∂2xyj − 4j2yj + bjy1(yj−1 − yj+1)
Boundary conditions
limx→±∞
f (x , θ, t) = g±(θ)
g±(θ) steady solutions.
Standing Waves, Traveling Waves.
![Page 100: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/100.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Embedding in Physical Space
f : Rn ×M × [0,∞) → (0,∞):
∂t f = ∆x f + divg (f∇g (log f −Kf ))
Example: n = 1, M = S1, Maier-Saupe potential:
f (x , θ, t) = 12π + 1
π
∑∞j=1 yj(x , t) cos(2jθ)
∂tyj = ∂2xyj − 4j2yj + bjy1(yj−1 − yj+1)
Boundary conditions
limx→±∞
f (x , θ, t) = g±(θ)
g±(θ) steady solutions.
Standing Waves, Traveling Waves.
![Page 101: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/101.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Embedding in Physical Space
f : Rn ×M × [0,∞) → (0,∞):
∂t f = ∆x f + divg (f∇g (log f −Kf ))
Example: n = 1, M = S1, Maier-Saupe potential:
f (x , θ, t) = 12π + 1
π
∑∞j=1 yj(x , t) cos(2jθ)
∂tyj = ∂2xyj − 4j2yj + bjy1(yj−1 − yj+1)
Boundary conditions
limx→±∞
f (x , θ, t) = g±(θ)
g±(θ) steady solutions.
Standing Waves, Traveling Waves.
![Page 102: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/102.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Embedding in Physical Space
f : Rn ×M × [0,∞) → (0,∞):
∂t f = ∆x f + divg (f∇g (log f −Kf ))
Example: n = 1, M = S1, Maier-Saupe potential:
f (x , θ, t) = 12π + 1
π
∑∞j=1 yj(x , t) cos(2jθ)
∂tyj = ∂2xyj − 4j2yj + bjy1(yj−1 − yj+1)
Boundary conditions
limx→±∞
f (x , θ, t) = g±(θ)
g±(θ) steady solutions.
Standing Waves, Traveling Waves.
![Page 103: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/103.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Embedding in Physical Space
f : Rn ×M × [0,∞) → (0,∞):
∂t f = ∆x f + divg (f∇g (log f −Kf ))
Example: n = 1, M = S1, Maier-Saupe potential:
f (x , θ, t) = 12π + 1
π
∑∞j=1 yj(x , t) cos(2jθ)
∂tyj = ∂2xyj − 4j2yj + bjy1(yj−1 − yj+1)
Boundary conditions
limx→±∞
f (x , θ, t) = g±(θ)
g±(θ) steady solutions.
Standing Waves, Traveling Waves.
![Page 104: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/104.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Passive
∂t f + v · ∇x f + divg (Wf ) = divg (f∇g (log f −Kf ))
withW (x ,m, t) =
=(∑n
i ,j=1 c ji (m)∂v i
∂x j (x , t))
c ji (m) ∈ Tm(M).
Example, rods in 3D:
W (x ,m, t) = (∇xv(x , t))m − ((∇xu(x , t))m ·m)m.
Macro-Micro Effect: from first principles, in principle...
![Page 105: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/105.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Passive
∂t f + v · ∇x f + divg (Wf ) = divg (f∇g (log f −Kf ))
withW (x ,m, t) =
=(∑n
i ,j=1 c ji (m)∂v i
∂x j (x , t))
c ji (m) ∈ Tm(M).
Example, rods in 3D:
W (x ,m, t) = (∇xv(x , t))m − ((∇xu(x , t))m ·m)m.
Macro-Micro Effect: from first principles, in principle...
![Page 106: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/106.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Passive
∂t f + v · ∇x f + divg (Wf ) = divg (f∇g (log f −Kf ))
withW (x ,m, t) =
=(∑n
i ,j=1 c ji (m)∂v i
∂x j (x , t))
c ji (m) ∈ Tm(M).
Example, rods in 3D:
W (x ,m, t) = (∇xv(x , t))m − ((∇xu(x , t))m ·m)m.
Macro-Micro Effect: from first principles, in principle...
![Page 107: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/107.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Passive
∂t f + v · ∇x f + divg (Wf ) = divg (f∇g (log f −Kf ))
withW (x ,m, t) =
=(∑n
i ,j=1 c ji (m)∂v i
∂x j (x , t))
c ji (m) ∈ Tm(M).
Example, rods in 3D:
W (x ,m, t) = (∇xv(x , t))m − ((∇xu(x , t))m ·m)m.
Macro-Micro Effect: from first principles, in principle...
![Page 108: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/108.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Passive
∂t f + v · ∇x f + divg (Wf ) = divg (f∇g (log f −Kf ))
withW (x ,m, t) =
=(∑n
i ,j=1 c ji (m)∂v i
∂x j (x , t))
c ji (m) ∈ Tm(M).
Example, rods in 3D:
W (x ,m, t) = (∇xv(x , t))m − ((∇xu(x , t))m ·m)m.
Macro-Micro Effect: from first principles, in principle...
![Page 109: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/109.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Active: Navier-Stokes
∂tv + v · ∇v +∇p = ν∆v +∇ · σ∇ · v = 0
σ = σij (x , t)
added stress tensor.
Micro-Macro Effect
σij (x) = −
∫M
(divgc i
j + c ij · ∇gKf (x ,m)
)f (x ,m)dµ(m) ∗
![Page 110: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/110.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Active: Navier-Stokes
∂tv + v · ∇v +∇p = ν∆v +∇ · σ∇ · v = 0
σ = σij (x , t)
added stress tensor.
Micro-Macro Effect
σij (x) = −
∫M
(divgc i
j + c ij · ∇gKf (x ,m)
)f (x ,m)dµ(m) ∗
![Page 111: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/111.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Active: Navier-Stokes
∂tv + v · ∇v +∇p = ν∆v +∇ · σ∇ · v = 0
σ = σij (x , t)
added stress tensor.
Micro-Macro Effect
σij (x) = −
∫M
(divgc i
j + c ij · ∇gKf (x ,m)
)f (x ,m)dµ(m) ∗
![Page 112: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/112.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Theorem3DNS + Fokker-Planck eqns with *. Then
E (t) = 12
∫|v |2dx+
+∫
f log f − 12(Kf )f
dxdµ.
is nondecreasing on solutions.
If (v , f ) is a smooth solution then
dEdt = −ν
∫|∇xv |2dx−
−∫ ∫
M
f |∇g (log f −Kf )|2 dmdx .
If the smooth solution is time independent, then v = 0 and fsolves the Onsager equation
f = Z−1eK[f ].
![Page 113: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/113.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Theorem3DNS + Fokker-Planck eqns with *. Then
E (t) = 12
∫|v |2dx+
+∫
f log f − 12(Kf )f
dxdµ.
is nondecreasing on solutions.If (v , f ) is a smooth solution then
dEdt = −ν
∫|∇xv |2dx−
−∫ ∫
M
f |∇g (log f −Kf )|2 dmdx .
If the smooth solution is time independent, then v = 0 and fsolves the Onsager equation
f = Z−1eK[f ].
![Page 114: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/114.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Theorem3DNS + Fokker-Planck eqns with *. Then
E (t) = 12
∫|v |2dx+
+∫
f log f − 12(Kf )f
dxdµ.
is nondecreasing on solutions.If (v , f ) is a smooth solution then
dEdt = −ν
∫|∇xv |2dx−
−∫ ∫
M
f |∇g (log f −Kf )|2 dmdx .
If the smooth solution is time independent, then v = 0 and fsolves the Onsager equation
f = Z−1eK[f ].
![Page 115: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/115.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
NFP + 3D time-dependent Stokes
∂t f + v · ∇x f + divg (Wf ) = divg (f∇g (log f −Kf )),∂tv − ν∆xv +∇xp = divxσ + F , ∇x · v = 0.
TheoremLet v0 divergence-free, in W 2,r (T3), r > 3, f0 positive,∫M f0(x ,m)dµ = 1,
f0 ∈ L∞(dx ; C(M)) ∩∇x f0 ∈ Lr (dx ;H−s(M)), s ≤ d2 + 1.
Then the solution exists for all time and
‖v‖Lp[(0,T );W 2,r (dx)] < ∞,
‖∇x f ‖L∞[(0,T );Lr (dx ;H−s(M))] < ∞
for any p > 2rr−3 , T > 0.
![Page 116: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/116.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
NFP + 3D time-dependent Stokes
∂t f + v · ∇x f + divg (Wf ) = divg (f∇g (log f −Kf )),∂tv − ν∆xv +∇xp = divxσ + F , ∇x · v = 0.
TheoremLet v0 divergence-free, in W 2,r (T3), r > 3, f0 positive,∫M f0(x ,m)dµ = 1,
f0 ∈ L∞(dx ; C(M)) ∩∇x f0 ∈ Lr (dx ;H−s(M)), s ≤ d2 + 1.
Then the solution exists for all time and
‖v‖Lp[(0,T );W 2,r (dx)] < ∞,
‖∇x f ‖L∞[(0,T );Lr (dx ;H−s(M))] < ∞
for any p > 2rr−3 , T > 0.
![Page 117: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/117.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
NFP + 3D time-dependent Stokes
∂t f + v · ∇x f + divg (Wf ) = divg (f∇g (log f −Kf )),∂tv − ν∆xv +∇xp = divxσ + F , ∇x · v = 0.
TheoremLet v0 divergence-free, in W 2,r (T3), r > 3, f0 positive,∫M f0(x ,m)dµ = 1,
f0 ∈ L∞(dx ; C(M)) ∩∇x f0 ∈ Lr (dx ;H−s(M)), s ≤ d2 + 1.
Then the solution exists for all time and
‖v‖Lp[(0,T );W 2,r (dx)] < ∞,
‖∇x f ‖L∞[(0,T );Lr (dx ;H−s(M))] < ∞
for any p > 2rr−3 , T > 0.
![Page 118: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/118.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
NFP + 2D time dependent Navier-Stokes
Theorem(C-Masmoudi) Let v0 ∈
(W α,r ∩ L2
)(R2), divergence-free,
f0 ∈ W 1,r (H−s(M)), with r > 2, α > 1, s ≤ d2 + 1 and f0 ≥ 0,∫
M f0dµ ∈ (L1 ∩ L∞)(R2). Then the coupled NS and nonlinearFokker-Planck system in 2D has a global solutionv ∈ L∞loc(W
1,r ) ∩ L2loc(W
2,r ) and f ∈ L∞loc(W1,r (H−s)).
Moreover, for T > T0 > 0, we have v ∈ L∞((T0,T );W 2−0,r ).
![Page 119: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/119.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
NFP + 2D time dependent Navier-Stokes
Theorem(C-Masmoudi) Let v0 ∈
(W α,r ∩ L2
)(R2), divergence-free,
f0 ∈ W 1,r (H−s(M)), with r > 2, α > 1, s ≤ d2 + 1 and f0 ≥ 0,∫
M f0dµ ∈ (L1 ∩ L∞)(R2).
Then the coupled NS and nonlinearFokker-Planck system in 2D has a global solutionv ∈ L∞loc(W
1,r ) ∩ L2loc(W
2,r ) and f ∈ L∞loc(W1,r (H−s)).
Moreover, for T > T0 > 0, we have v ∈ L∞((T0,T );W 2−0,r ).
![Page 120: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/120.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
NFP + 2D time dependent Navier-Stokes
Theorem(C-Masmoudi) Let v0 ∈
(W α,r ∩ L2
)(R2), divergence-free,
f0 ∈ W 1,r (H−s(M)), with r > 2, α > 1, s ≤ d2 + 1 and f0 ≥ 0,∫
M f0dµ ∈ (L1 ∩ L∞)(R2). Then the coupled NS and nonlinearFokker-Planck system in 2D has a global solutionv ∈ L∞loc(W
1,r ) ∩ L2loc(W
2,r ) and f ∈ L∞loc(W1,r (H−s)).
Moreover, for T > T0 > 0, we have v ∈ L∞((T0,T );W 2−0,r ).
![Page 121: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/121.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
NFP + 2D time dependent Navier-Stokes
Theorem(C-Masmoudi) Let v0 ∈
(W α,r ∩ L2
)(R2), divergence-free,
f0 ∈ W 1,r (H−s(M)), with r > 2, α > 1, s ≤ d2 + 1 and f0 ≥ 0,∫
M f0dµ ∈ (L1 ∩ L∞)(R2). Then the coupled NS and nonlinearFokker-Planck system in 2D has a global solutionv ∈ L∞loc(W
1,r ) ∩ L2loc(W
2,r ) and f ∈ L∞loc(W1,r (H−s)).
Moreover, for T > T0 > 0, we have v ∈ L∞((T0,T );W 2−0,r ).
![Page 122: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/122.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Outlook
1 n-gons, Hausdorff-Gromov distance
2 soft sphere packing, jamming
3 kinetics w/o Riemannian structure
![Page 123: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/123.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Outlook
1 n-gons, Hausdorff-Gromov distance
2 soft sphere packing, jamming
3 kinetics w/o Riemannian structure
![Page 124: The zero temperature limit of interacting corpora...Equation General Goals Examples Onsager equation for general corpora Kinetics Physical space connections Embedding in Fluid Outlook](https://reader034.fdocuments.us/reader034/viewer/2022042211/5eb2b3f1538fb41cf5498164/html5/thumbnails/124.jpg)
The zerotemperature
limit ofinteracting
corpora
PeterConstantin
Introduction
OnsagerEquation
General Goals
Examples
Onsagerequation forgeneralcorpora
Kinetics
Physical spaceconnections
Embedding inFluid
Outlook
Outlook
1 n-gons, Hausdorff-Gromov distance
2 soft sphere packing, jamming
3 kinetics w/o Riemannian structure