sticky particles with interaction
play

Sticky particles with interaction Giuseppe Savar e - PowerPoint PPT Presentation

Aug 11, 2023 •264 likes •1.33k views

Sticky particles Cumulative distribution Lagrangian representation Sticky particles with interaction Giuseppe Savar e http://www.imati.cnr.it/ savare Department of Mathematics, University of Pavia, Italy HYP2012 - Padova, June 26, 2012

  1. Sticky particles Cumulative distribution Lagrangian representation Starting point: motion of a finite number of particles. Discrete particle model N particles P i := ( m i , x i , v i ), i = 1 , . . . , N , with positive mass m i satisfying � N i =1 m i = 1 ordered positions x 1 < x 2 < . . . < x N − 1 < x N , and velocities v i . P 1 P 2 P 3 P 4 At the initial time t = 0 the particles are disjoint and start to move according to the system of ODE’s d d d tx i ( t ) = v i ( t ) , d tv i ( t ) = a i ( x ( t )) . The first collision time t = t 1 correspond to x j ( t 1 ) = x j +1 ( t 1 ) = . . . = x k ( t 1 ) for some indices j < k . The particles P j , P j +1 , . . . , P k collapse and stick in a new particle P with mass m := m j + . . . + m k and v := m j v j ( t 1 ) + m j +1 v j +1 ( t 1 ) + . . . + m k v k ( t 1 ) “barycentric” velocity m 6

  2. Sticky particles Cumulative distribution Lagrangian representation Starting point: motion of a finite number of particles. Discrete particle model N particles P i := ( m i , x i , v i ), i = 1 , . . . , N , with positive mass m i satisfying � N i =1 m i = 1 ordered positions x 1 < x 2 < . . . < x N − 1 < x N , and velocities v i . P 1 P 2 P 3 P 4 At the initial time t = 0 the particles are disjoint and start to move according to the system of ODE’s d d d tx i ( t ) = v i ( t ) , d tv i ( t ) = a i ( x ( t )) . The first collision time t = t 1 correspond to x j ( t 1 ) = x j +1 ( t 1 ) = . . . = x k ( t 1 ) for some indices j < k . The particles P j , P j +1 , . . . , P k collapse and stick in a new particle P with mass m := m j + . . . + m k and v := m j v j ( t 1 ) + m j +1 v j +1 ( t 1 ) + . . . + m k v k ( t 1 ) “barycentric” velocity m 6

  3. Sticky particles Cumulative distribution Lagrangian representation Starting point: motion of a finite number of particles. Discrete particle model N particles P i := ( m i , x i , v i ), i = 1 , . . . , N , with positive mass m i satisfying � N i =1 m i = 1 ordered positions x 1 < x 2 < . . . < x N − 1 < x N , and velocities v i . P 1 P 2 P 3 At the initial time t = 0 the particles are disjoint and start to move according to the system of ODE’s d d d tx i ( t ) = v i ( t ) , d tv i ( t ) = a i ( x ( t )) . The first collision time t = t 1 correspond to x j ( t 1 ) = x j +1 ( t 1 ) = . . . = x k ( t 1 ) for some indices j < k . The particles P j , P j +1 , . . . , P k collapse and stick in a new particle P with mass m := m j + . . . + m k and v := m j v j ( t 1 ) + m j +1 v j +1 ( t 1 ) + . . . + m k v k ( t 1 ) “barycentric” velocity m 6

  4. Sticky particles Cumulative distribution Lagrangian representation Starting point: motion of a finite number of particles. Discrete particle model N particles P i := ( m i , x i , v i ), i = 1 , . . . , N , with positive mass m i satisfying � N i =1 m i = 1 ordered positions x 1 < x 2 < . . . < x N − 1 < x N , and velocities v i . P 1 P 2 P 3 At the initial time t = 0 the particles are disjoint and start to move according to the system of ODE’s d d d tx i ( t ) = v i ( t ) , d tv i ( t ) = a i ( x ( t )) . The first collision time t = t 1 correspond to x j ( t 1 ) = x j +1 ( t 1 ) = . . . = x k ( t 1 ) for some indices j < k . The particles P j , P j +1 , . . . , P k collapse and stick in a new particle P with mass m := m j + . . . + m k and v := m j v j ( t 1 ) + m j +1 v j +1 ( t 1 ) + . . . + m k v k ( t 1 ) “barycentric” velocity m 6

  5. Sticky particles Cumulative distribution Lagrangian representation Starting point: motion of a finite number of particles. Discrete particle model N particles P i := ( m i , x i , v i ), i = 1 , . . . , N , with positive mass m i satisfying � N i =1 m i = 1 ordered positions x 1 < x 2 < . . . < x N − 1 < x N , and velocities v i . P 1 P 2 At the initial time t = 0 the particles are disjoint and start to move according to the system of ODE’s d d d tx i ( t ) = v i ( t ) , d tv i ( t ) = a i ( x ( t )) . The first collision time t = t 1 correspond to x j ( t 1 ) = x j +1 ( t 1 ) = . . . = x k ( t 1 ) for some indices j < k . The particles P j , P j +1 , . . . , P k collapse and stick in a new particle P with mass m := m j + . . . + m k and v := m j v j ( t 1 ) + m j +1 v j +1 ( t 1 ) + . . . + m k v k ( t 1 ) “barycentric” velocity m 6

  6. Sticky particles Cumulative distribution Lagrangian representation Starting point: motion of a finite number of particles. Discrete particle model N particles P i := ( m i , x i , v i ), i = 1 , . . . , N , with positive mass m i satisfying � N i =1 m i = 1 ordered positions x 1 < x 2 < . . . < x N − 1 < x N , and velocities v i . P 1 P 2 At the initial time t = 0 the particles are disjoint and start to move according to the system of ODE’s d d d tx i ( t ) = v i ( t ) , d tv i ( t ) = a i ( x ( t )) . The first collision time t = t 1 correspond to x j ( t 1 ) = x j +1 ( t 1 ) = . . . = x k ( t 1 ) for some indices j < k . The particles P j , P j +1 , . . . , P k collapse and stick in a new particle P with mass m := m j + . . . + m k and v := m j v j ( t 1 ) + m j +1 v j +1 ( t 1 ) + . . . + m k v k ( t 1 ) “barycentric” velocity m 6

  7. Sticky particles Cumulative distribution Lagrangian representation Starting point: motion of a finite number of particles. Discrete particle model N particles P i := ( m i , x i , v i ), i = 1 , . . . , N , with positive mass m i satisfying � N i =1 m i = 1 ordered positions x 1 < x 2 < . . . < x N − 1 < x N , and velocities v i . P 1 P 2 At the initial time t = 0 the particles are disjoint and start to move according to the system of ODE’s d d d tx i ( t ) = v i ( t ) , d tv i ( t ) = a i ( x ( t )) . The first collision time t = t 1 correspond to x j ( t 1 ) = x j +1 ( t 1 ) = . . . = x k ( t 1 ) for some indices j < k . The particles P j , P j +1 , . . . , P k collapse and stick in a new particle P with mass m := m j + . . . + m k and v := m j v j ( t 1 ) + m j +1 v j +1 ( t 1 ) + . . . + m k v k ( t 1 ) “barycentric” velocity m 6

  8. Sticky particles Cumulative distribution Lagrangian representation Measure-theoretic description We thus have: a (finite) sequence of collision times 0 < t 1 < t 2 < . . . in each interval [ t h , t h +1 ) a finite number N h of (suitably relabeled) particles P 1 ( t ) , · · · , P N h ( t ), P i ( t ) := ( m i , x i ( t ) , v i ( t )). We can introduce the measures N h N h � � ̺ t := m i δ x i ( t ) ∈ P ( R ) , ( ̺ v ) t := m i v i ( t ) δ x i ( t ) ∈ M ( R ) i =1 i =1 if t ∈ [ t h , t h +1 ) . N h � f [ ̺ t ] := m i a i ( t ) δ x i ( t ) ∈ M ( R ) i =1 They satisfy the 1 -dimensional pressureless Euler system in the sense of distributions � ∂ t ̺ + ∂ x ( ̺ v ) = 0 , in R × (0 , + ∞ ); ̺ | t =0 = ̺ 0 , v | t =0 = v 0 , ∂ t ( ̺ v ) + ∂ x ( ̺ v 2 ) = f [ ̺ ] , 7

  9. Sticky particles Cumulative distribution Lagrangian representation Measure-theoretic description We thus have: a (finite) sequence of collision times 0 < t 1 < t 2 < . . . in each interval [ t h , t h +1 ) a finite number N h of (suitably relabeled) particles P 1 ( t ) , · · · , P N h ( t ), P i ( t ) := ( m i , x i ( t ) , v i ( t )). We can introduce the measures N h N h � � ̺ t := m i δ x i ( t ) ∈ P ( R ) , ( ̺ v ) t := m i v i ( t ) δ x i ( t ) ∈ M ( R ) i =1 i =1 if t ∈ [ t h , t h +1 ) . N h � f [ ̺ t ] := m i a i ( t ) δ x i ( t ) ∈ M ( R ) i =1 They satisfy the 1 -dimensional pressureless Euler system in the sense of distributions � ∂ t ̺ + ∂ x ( ̺ v ) = 0 , in R × (0 , + ∞ ); ̺ | t =0 = ̺ 0 , v | t =0 = v 0 , ∂ t ( ̺ v ) + ∂ x ( ̺ v 2 ) = f [ ̺ ] , 7

  10. Sticky particles Cumulative distribution Lagrangian representation Measure-theoretic description We thus have: a (finite) sequence of collision times 0 < t 1 < t 2 < . . . in each interval [ t h , t h +1 ) a finite number N h of (suitably relabeled) particles P 1 ( t ) , · · · , P N h ( t ), P i ( t ) := ( m i , x i ( t ) , v i ( t )). We can introduce the measures N h N h � � ̺ t := m i δ x i ( t ) ∈ P ( R ) , ( ̺ v ) t := m i v i ( t ) δ x i ( t ) ∈ M ( R ) i =1 i =1 if t ∈ [ t h , t h +1 ) . N h � f [ ̺ t ] := m i a i ( t ) δ x i ( t ) ∈ M ( R ) i =1 They satisfy the 1 -dimensional pressureless Euler system in the sense of distributions � ∂ t ̺ + ∂ x ( ̺ v ) = 0 , in R × (0 , + ∞ ); ̺ | t =0 = ̺ 0 , v | t =0 = v 0 , ∂ t ( ̺ v ) + ∂ x ( ̺ v 2 ) = f [ ̺ ] , 7

  11. Sticky particles Cumulative distribution Lagrangian representation The various models Motion driven by a potential V d  d tx i = v i , � ∂ t ̺ + ∂ x ( ̺ v ) = 0 ,   � d ∂ t ( ̺ v ) + ∂ x ( ̺ v 2 ) = − ̺∂ x V d tv i = − ∂ x V ( x i )   Motion driven by an interaction potential W d  d tx i = v i ,  � ∂ t ̺ + ∂ x ( ̺ v ) = 0 ,   d � ∂ t ( ̺ v ) + ∂ x ( ̺ v 2 ) = − ̺ � � ̺ ∗ ∂ x W � d tv i = − m j ∂ x W ( x i − x j )    j � = i Non smooth interaction: the Euler-Poisson system when W ( x ) = ± λ | x | . 8

  12. Sticky particles Cumulative distribution Lagrangian representation The various models Motion driven by a potential V d  d tx i = v i , � ∂ t ̺ + ∂ x ( ̺ v ) = 0 ,   � d ∂ t ( ̺ v ) + ∂ x ( ̺ v 2 ) = − ̺∂ x V d tv i = − ∂ x V ( x i )   Motion driven by an interaction potential W d  d tx i = v i ,  � ∂ t ̺ + ∂ x ( ̺ v ) = 0 ,   d � ∂ t ( ̺ v ) + ∂ x ( ̺ v 2 ) = − ̺ � � ̺ ∗ ∂ x W � d tv i = − m j ∂ x W ( x i − x j )    j � = i Non smooth interaction: the Euler-Poisson system when W ( x ) = ± λ | x | . 8

  13. Sticky particles Cumulative distribution Lagrangian representation The various models Motion driven by a potential V d  d tx i = v i , � ∂ t ̺ + ∂ x ( ̺ v ) = 0 ,   � d ∂ t ( ̺ v ) + ∂ x ( ̺ v 2 ) = − ̺∂ x V d tv i = − ∂ x V ( x i )   Motion driven by an interaction potential W d  d tx i = v i ,  � ∂ t ̺ + ∂ x ( ̺ v ) = 0 ,   d � ∂ t ( ̺ v ) + ∂ x ( ̺ v 2 ) = − ̺ � � ̺ ∗ ∂ x W � d tv i = − m j ∂ x W ( x i − x j )    j � = i Non smooth interaction: the Euler-Poisson system when W ( x ) = ± λ | x | . 8

  14. Sticky particles Cumulative distribution Lagrangian representation Main problem: continuous limit Consider a sequence of discrete initial data µ n 0 := ( ̺ n 0 , ̺ n 0 v n 0 ) converging to µ 0 = ( ̺ 0 , ̺ 0 v 0 ) in a suitable measure-theoretic sense and let µ n t = ( ̺ n t , ̺ n t v n t ) be the (discrete) solution of SPS. Problem ◮ Prove that the limit µ t = ( ̺ t , ̺ t v t ) of the SPS µ n t = ( ̺ n t , ̺ n t v n t ) as n ↑ + ∞ exists. ◮ Find a suitable characterization of µ t ◮ Show that ( ̺ t , ̺ t v t ) solves the pressureless Euler system � ∂ t ̺ + ∂ x ( ̺ v ) = 0 , in R × (0 , + ∞ ); ̺ | t =0 = ̺ 0 , v | t =0 = v 0 , ∂ t ( ̺ v ) + ∂ x ( ̺ v 2 ) = f [ ̺ ] , 9

  15. Sticky particles Cumulative distribution Lagrangian representation Main problem: continuous limit Consider a sequence of discrete initial data µ n 0 := ( ̺ n 0 , ̺ n 0 v n 0 ) converging to µ 0 = ( ̺ 0 , ̺ 0 v 0 ) in a suitable measure-theoretic sense and let µ n t = ( ̺ n t , ̺ n t v n t ) be the (discrete) solution of SPS. Problem ◮ Prove that the limit µ t = ( ̺ t , ̺ t v t ) of the SPS µ n t = ( ̺ n t , ̺ n t v n t ) as n ↑ + ∞ exists. ◮ Find a suitable characterization of µ t ◮ Show that ( ̺ t , ̺ t v t ) solves the pressureless Euler system � ∂ t ̺ + ∂ x ( ̺ v ) = 0 , in R × (0 , + ∞ ); ̺ | t =0 = ̺ 0 , v | t =0 = v 0 , ∂ t ( ̺ v ) + ∂ x ( ̺ v 2 ) = f [ ̺ ] , 9

  16. Sticky particles Cumulative distribution Lagrangian representation Main problem: continuous limit Consider a sequence of discrete initial data µ n 0 := ( ̺ n 0 , ̺ n 0 v n 0 ) converging to µ 0 = ( ̺ 0 , ̺ 0 v 0 ) in a suitable measure-theoretic sense and let µ n t = ( ̺ n t , ̺ n t v n t ) be the (discrete) solution of SPS. Problem ◮ Prove that the limit µ t = ( ̺ t , ̺ t v t ) of the SPS µ n t = ( ̺ n t , ̺ n t v n t ) as n ↑ + ∞ exists. ◮ Find a suitable characterization of µ t ◮ Show that ( ̺ t , ̺ t v t ) solves the pressureless Euler system � ∂ t ̺ + ∂ x ( ̺ v ) = 0 , in R × (0 , + ∞ ); ̺ | t =0 = ̺ 0 , v | t =0 = v 0 , ∂ t ( ̺ v ) + ∂ x ( ̺ v 2 ) = f [ ̺ ] , 9

  17. Sticky particles Cumulative distribution Lagrangian representation Main problem: continuous limit Consider a sequence of discrete initial data µ n 0 := ( ̺ n 0 , ̺ n 0 v n 0 ) converging to µ 0 = ( ̺ 0 , ̺ 0 v 0 ) in a suitable measure-theoretic sense and let µ n t = ( ̺ n t , ̺ n t v n t ) be the (discrete) solution of SPS. Problem ◮ Prove that the limit µ t = ( ̺ t , ̺ t v t ) of the SPS µ n t = ( ̺ n t , ̺ n t v n t ) as n ↑ + ∞ exists. ◮ Find a suitable characterization of µ t ◮ Show that ( ̺ t , ̺ t v t ) solves the pressureless Euler system � ∂ t ̺ + ∂ x ( ̺ v ) = 0 , in R × (0 , + ∞ ); ̺ | t =0 = ̺ 0 , v | t =0 = v 0 , ∂ t ( ̺ v ) + ∂ x ( ̺ v 2 ) = f [ ̺ ] , 9

  18. Sticky particles Cumulative distribution Lagrangian representation Outline 1 A one-dimensional fluid flow and the sticky particle model 2 Free motion: representation formulas by the cumulative distribution function 3 Monotone rearrangement and Lagrangian representation 10

  19. Sticky particles Cumulative distribution Lagrangian representation Main contributions in the case f [ ̺ ] ≡ 0 • Existence and convergence ◮ Grenier ’95 , E-Rykov-Sinai ’96 : first existence and convergence result. ◮ Brenier-Grenier ’96 : Characterization of the limit in terms of a suitable scalar conservation law, uniqueness. ◮ Huang-Wang ’01, Nguyen-Tudorascu ’08, Moutsinga ’08 : further refinements. Gangbo-Nguyen-Tudorascu ’09 : Euler-Poisson. Basic assumptions: ̺ n 0 → ̺ 0 in the L 2 -Wasserstein distance, v n 0 = v 0 is given by a continuous function with (at most) linear growth. In particular the result cover the case when ̺ n 0 , ̺ 0 have a common compact support and ̺ n 0 → ̺ 0 weakly in the sense of distribution (or, equivalently, in the duality with continuous functions). • Pioneering ideas which lies (more or less explicitly) at the core of the papers by E-Rykov-Sinai and Brenier-Grenier have been introduced by ◮ Shnirelman ’86 and further clarified by ◮ Andrievwsky-Gurbatov-Soboelvski˘ ı ’07 in a formal way. • Different approaches and models: ◮ Bouchut-James ’95 , Poupaud-Rascle ’97 ◮ Sobolevski˘ ı ’97 , Boudin ’00 : viscous regularization. ◮ Wolansky ’07 : particles with finite size. 11

  20. Sticky particles Cumulative distribution Lagrangian representation Main contributions in the case f [ ̺ ] ≡ 0 • Existence and convergence ◮ Grenier ’95 , E-Rykov-Sinai ’96 : first existence and convergence result. ◮ Brenier-Grenier ’96 : Characterization of the limit in terms of a suitable scalar conservation law, uniqueness. ◮ Huang-Wang ’01, Nguyen-Tudorascu ’08, Moutsinga ’08 : further refinements. Gangbo-Nguyen-Tudorascu ’09 : Euler-Poisson. Basic assumptions: ̺ n 0 → ̺ 0 in the L 2 -Wasserstein distance, v n 0 = v 0 is given by a continuous function with (at most) linear growth. In particular the result cover the case when ̺ n 0 , ̺ 0 have a common compact support and ̺ n 0 → ̺ 0 weakly in the sense of distribution (or, equivalently, in the duality with continuous functions). • Pioneering ideas which lies (more or less explicitly) at the core of the papers by E-Rykov-Sinai and Brenier-Grenier have been introduced by ◮ Shnirelman ’86 and further clarified by ◮ Andrievwsky-Gurbatov-Soboelvski˘ ı ’07 in a formal way. • Different approaches and models: ◮ Bouchut-James ’95 , Poupaud-Rascle ’97 ◮ Sobolevski˘ ı ’97 , Boudin ’00 : viscous regularization. ◮ Wolansky ’07 : particles with finite size. 11

  21. Sticky particles Cumulative distribution Lagrangian representation Main contributions in the case f [ ̺ ] ≡ 0 • Existence and convergence ◮ Grenier ’95 , E-Rykov-Sinai ’96 : first existence and convergence result. ◮ Brenier-Grenier ’96 : Characterization of the limit in terms of a suitable scalar conservation law, uniqueness. ◮ Huang-Wang ’01, Nguyen-Tudorascu ’08, Moutsinga ’08 : further refinements. Gangbo-Nguyen-Tudorascu ’09 : Euler-Poisson. Basic assumptions: ̺ n 0 → ̺ 0 in the L 2 -Wasserstein distance, v n 0 = v 0 is given by a continuous function with (at most) linear growth. In particular the result cover the case when ̺ n 0 , ̺ 0 have a common compact support and ̺ n 0 → ̺ 0 weakly in the sense of distribution (or, equivalently, in the duality with continuous functions). • Pioneering ideas which lies (more or less explicitly) at the core of the papers by E-Rykov-Sinai and Brenier-Grenier have been introduced by ◮ Shnirelman ’86 and further clarified by ◮ Andrievwsky-Gurbatov-Soboelvski˘ ı ’07 in a formal way. • Different approaches and models: ◮ Bouchut-James ’95 , Poupaud-Rascle ’97 ◮ Sobolevski˘ ı ’97 , Boudin ’00 : viscous regularization. ◮ Wolansky ’07 : particles with finite size. 11

  22. Sticky particles Cumulative distribution Lagrangian representation Main contributions in the case f [ ̺ ] ≡ 0 • Existence and convergence ◮ Grenier ’95 , E-Rykov-Sinai ’96 : first existence and convergence result. ◮ Brenier-Grenier ’96 : Characterization of the limit in terms of a suitable scalar conservation law, uniqueness. ◮ Huang-Wang ’01, Nguyen-Tudorascu ’08, Moutsinga ’08 : further refinements. Gangbo-Nguyen-Tudorascu ’09 : Euler-Poisson. Basic assumptions: ̺ n 0 → ̺ 0 in the L 2 -Wasserstein distance, v n 0 = v 0 is given by a continuous function with (at most) linear growth. In particular the result cover the case when ̺ n 0 , ̺ 0 have a common compact support and ̺ n 0 → ̺ 0 weakly in the sense of distribution (or, equivalently, in the duality with continuous functions). • Pioneering ideas which lies (more or less explicitly) at the core of the papers by E-Rykov-Sinai and Brenier-Grenier have been introduced by ◮ Shnirelman ’86 and further clarified by ◮ Andrievwsky-Gurbatov-Soboelvski˘ ı ’07 in a formal way. • Different approaches and models: ◮ Bouchut-James ’95 , Poupaud-Rascle ’97 ◮ Sobolevski˘ ı ’97 , Boudin ’00 : viscous regularization. ◮ Wolansky ’07 : particles with finite size. 11

  23. Sticky particles Cumulative distribution Lagrangian representation Main contributions in the case f [ ̺ ] ≡ 0 • Existence and convergence ◮ Grenier ’95 , E-Rykov-Sinai ’96 : first existence and convergence result. ◮ Brenier-Grenier ’96 : Characterization of the limit in terms of a suitable scalar conservation law, uniqueness. ◮ Huang-Wang ’01, Nguyen-Tudorascu ’08, Moutsinga ’08 : further refinements. Gangbo-Nguyen-Tudorascu ’09 : Euler-Poisson. Basic assumptions: ̺ n 0 → ̺ 0 in the L 2 -Wasserstein distance, v n 0 = v 0 is given by a continuous function with (at most) linear growth. In particular the result cover the case when ̺ n 0 , ̺ 0 have a common compact support and ̺ n 0 → ̺ 0 weakly in the sense of distribution (or, equivalently, in the duality with continuous functions). • Pioneering ideas which lies (more or less explicitly) at the core of the papers by E-Rykov-Sinai and Brenier-Grenier have been introduced by ◮ Shnirelman ’86 and further clarified by ◮ Andrievwsky-Gurbatov-Soboelvski˘ ı ’07 in a formal way. • Different approaches and models: ◮ Bouchut-James ’95 , Poupaud-Rascle ’97 ◮ Sobolevski˘ ı ’97 , Boudin ’00 : viscous regularization. ◮ Wolansky ’07 : particles with finite size. 11

  24. Sticky particles Cumulative distribution Lagrangian representation The cumulative distribution function For every probability measure ̺ ∈ P ( R ) 4 � m 2 A discrete measure ρ = m j δ x j j =1 m 3 m 1 m 4 x 1 x 2 x 3 x 4 x 12

  25. Sticky particles Cumulative distribution Lagrangian representation The cumulative distribution function For every probability measure ̺ ∈ P ( R ) 4 � m 2 A discrete measure ρ = m j δ x j j =1 m 3 m 1 m 4 x 1 x 2 x 3 x 4 x � � we introduce the distribution function M ρ ( x ) := ρ ( −∞ , x ] , Total mass = 1 Its distribution function M ρ x 2 x 3 x 4 x x 1 12

  26. Sticky particles Cumulative distribution Lagrangian representation The Brenier-Grenier formulation in the absence of force in D ′ ( R ) . � � M ̺ ( x ) := ̺ ( −∞ , x ] , x ∈ R , so that ̺ = ∂ x M ̺ Main idea: study the evolution of M t := M ̺ t , where ̺ t is the solution of the SPS. Theorem (Brenier-Grenier ’96) M is the unique entropy solution of the scalar conservation law ∂ t M + ∂ x A ( M ) = 0 in R × (0 , + ∞ ) where A : [0 , 1] → R is a continuous flux function depending only on the initial data ̺ 0 and v 0 . It is characterized by A ′ ( M 0 ( x )) = v 0 ( x ) . 13

  27. Sticky particles Cumulative distribution Lagrangian representation The Brenier-Grenier formulation in the absence of force in D ′ ( R ) . � � M ̺ ( x ) := ̺ ( −∞ , x ] , x ∈ R , so that ̺ = ∂ x M ̺ Main idea: study the evolution of M t := M ̺ t , where ̺ t is the solution of the SPS. Theorem (Brenier-Grenier ’96) M is the unique entropy solution of the scalar conservation law ∂ t M + ∂ x A ( M ) = 0 in R × (0 , + ∞ ) where A : [0 , 1] → R is a continuous flux function depending only on the initial data ̺ 0 and v 0 . It is characterized by A ′ ( M 0 ( x )) = v 0 ( x ) . 13

  28. Sticky particles Cumulative distribution Lagrangian representation The Brenier-Grenier formulation in the absence of force in D ′ ( R ) . � � M ̺ ( x ) := ̺ ( −∞ , x ] , x ∈ R , so that ̺ = ∂ x M ̺ Main idea: study the evolution of M t := M ̺ t , where ̺ t is the solution of the SPS. Theorem (Brenier-Grenier ’96) M is the unique entropy solution of the scalar conservation law ∂ t M + ∂ x A ( M ) = 0 in R × (0 , + ∞ ) where A : [0 , 1] → R is a continuous flux function depending only on the initial data ̺ 0 and v 0 . It is characterized by A ′ ( M 0 ( x )) = v 0 ( x ) . 13

  29. Sticky particles Cumulative distribution Lagrangian representation The Brenier-Grenier formulation in the absence of force in D ′ ( R ) . � � M ̺ ( x ) := ̺ ( −∞ , x ] , x ∈ R , so that ̺ = ∂ x M ̺ Main idea: study the evolution of M t := M ̺ t , where ̺ t is the solution of the SPS. Theorem (Brenier-Grenier ’96) M is the unique entropy solution of the scalar conservation law ∂ t M + ∂ x A ( M ) = 0 in R × (0 , + ∞ ) where A : [0 , 1] → R is a continuous flux function depending only on the initial data ̺ 0 and v 0 . It is characterized by A ′ ( M 0 ( x )) = v 0 ( x ) . 13

  30. Sticky particles Cumulative distribution Lagrangian representation Outline 1 A one-dimensional fluid flow and the sticky particle model 2 Free motion: representation formulas by the cumulative distribution function 3 Monotone rearrangement and Lagrangian representation 14

  31. Sticky particles Cumulative distribution Lagrangian representation Monotone rearrangement ρ m 2 m 3 m 1 m 4 x x 1 x 2 x 3 x 4 15

  32. Sticky particles Cumulative distribution Lagrangian representation Monotone rearrangement ρ m 2 m 3 m 1 m 4 x x 1 x 2 x 3 x 4 The distribution function M ρ Mass = 1 w 3 w 2 w 1 x 1 x 2 x 3 x 4 x 15

  33. Sticky particles Cumulative distribution Lagrangian representation Monotone rearrangement The monotone rearrangement ρ m 2 − 1 X ρ = M ρ m 3 m 1 m 4 x x 1 x 2 x 3 x 4 x 4 x 3 The distribution function M ρ Mass = 1 x 2 w 3 w 1 w 2 w 3 1 w 2 x 1 w 1 = m 1 w 1 w 2 = m 1 + m 2 w 3 = m 1 + m 2 + m 3 x 1 x 2 x 3 x 4 x 15

  34. Sticky particles Cumulative distribution Lagrangian representation Optimal transport and monotone rearrangement Point of view of 1 -dimensional optimal transport: instead of using � � the cumulative distribution function M ̺ ( x ) = ̺ ( −∞ , x ] , we represent each probability measure ̺ by its monotone rearrangement X ̺ : (0 , 1) → R : X ̺ ( w ) is the position x such that ̺ (( −∞ , x ]) = w . � � X ̺ ( w ) := inf x ∈ R : M ̺ ( x ) > w w ∈ (0 , 1) which is the so-called pseudo-inverse of M ̺ . The map X ̺ is nondecreasing and right-continuous . It pushes the Lebesgue measure λ := L 1 | (0 , 1) on (0 , 1) onto ̺ , i.e. ( X ̺ ) # L 1 L 1 ( X − 1 | (0 , 1) = ̺, ̺ ( B )) = ̺ ( B ) for every Borel set B ⊂ R It satisfies the change of variable formula � 1 � φ ( x ) d ̺ ( x ) = φ ( X ̺ ( w )) d w 0 R for every nonnegative/bounded Borel function φ : R → R . In particular, � 1 � | x | 2 d ̺ ( x ) = � 2 d w = � 2 � � � � m 2 ( ̺ ) := � X ̺ ( w ) � X ̺ L 2 (0 , 1) 0 R 16

  35. Sticky particles Cumulative distribution Lagrangian representation Optimal transport and monotone rearrangement Point of view of 1 -dimensional optimal transport: instead of using � � the cumulative distribution function M ̺ ( x ) = ̺ ( −∞ , x ] , we represent each probability measure ̺ by its monotone rearrangement X ̺ : (0 , 1) → R : X ̺ ( w ) is the position x such that ̺ (( −∞ , x ]) = w . � � X ̺ ( w ) := inf x ∈ R : M ̺ ( x ) > w w ∈ (0 , 1) which is the so-called pseudo-inverse of M ̺ . The map X ̺ is nondecreasing and right-continuous . It pushes the Lebesgue measure λ := L 1 | (0 , 1) on (0 , 1) onto ̺ , i.e. ( X ̺ ) # L 1 L 1 ( X − 1 | (0 , 1) = ̺, ̺ ( B )) = ̺ ( B ) for every Borel set B ⊂ R It satisfies the change of variable formula � 1 � φ ( x ) d ̺ ( x ) = φ ( X ̺ ( w )) d w 0 R for every nonnegative/bounded Borel function φ : R → R . In particular, � 1 � | x | 2 d ̺ ( x ) = � 2 d w = � 2 � � � � m 2 ( ̺ ) := � X ̺ ( w ) � X ̺ L 2 (0 , 1) 0 R 16

  36. Sticky particles Cumulative distribution Lagrangian representation Optimal transport and monotone rearrangement Point of view of 1 -dimensional optimal transport: instead of using � � the cumulative distribution function M ̺ ( x ) = ̺ ( −∞ , x ] , we represent each probability measure ̺ by its monotone rearrangement X ̺ : (0 , 1) → R : X ̺ ( w ) is the position x such that ̺ (( −∞ , x ]) = w . � � X ̺ ( w ) := inf x ∈ R : M ̺ ( x ) > w w ∈ (0 , 1) which is the so-called pseudo-inverse of M ̺ . The map X ̺ is nondecreasing and right-continuous . It pushes the Lebesgue measure λ := L 1 | (0 , 1) on (0 , 1) onto ̺ , i.e. ( X ̺ ) # L 1 L 1 ( X − 1 | (0 , 1) = ̺, ̺ ( B )) = ̺ ( B ) for every Borel set B ⊂ R It satisfies the change of variable formula � 1 � φ ( x ) d ̺ ( x ) = φ ( X ̺ ( w )) d w 0 R for every nonnegative/bounded Borel function φ : R → R . In particular, � 1 � | x | 2 d ̺ ( x ) = � 2 d w = � 2 � � � � m 2 ( ̺ ) := � X ̺ ( w ) � X ̺ L 2 (0 , 1) 0 R 16

  37. Sticky particles Cumulative distribution Lagrangian representation Optimal transport and monotone rearrangement Point of view of 1 -dimensional optimal transport: instead of using � � the cumulative distribution function M ̺ ( x ) = ̺ ( −∞ , x ] , we represent each probability measure ̺ by its monotone rearrangement X ̺ : (0 , 1) → R : X ̺ ( w ) is the position x such that ̺ (( −∞ , x ]) = w . � � X ̺ ( w ) := inf x ∈ R : M ̺ ( x ) > w w ∈ (0 , 1) which is the so-called pseudo-inverse of M ̺ . The map X ̺ is nondecreasing and right-continuous . It pushes the Lebesgue measure λ := L 1 | (0 , 1) on (0 , 1) onto ̺ , i.e. ( X ̺ ) # L 1 L 1 ( X − 1 | (0 , 1) = ̺, ̺ ( B )) = ̺ ( B ) for every Borel set B ⊂ R It satisfies the change of variable formula � 1 � φ ( x ) d ̺ ( x ) = φ ( X ̺ ( w )) d w 0 R for every nonnegative/bounded Borel function φ : R → R . In particular, � 1 � | x | 2 d ̺ ( x ) = � 2 d w = � 2 � � � � m 2 ( ̺ ) := � X ̺ ( w ) � X ̺ L 2 (0 , 1) 0 R 16

  38. Sticky particles Cumulative distribution Lagrangian representation Optimal transport and monotone rearrangement Point of view of 1 -dimensional optimal transport: instead of using � � the cumulative distribution function M ̺ ( x ) = ̺ ( −∞ , x ] , we represent each probability measure ̺ by its monotone rearrangement X ̺ : (0 , 1) → R : X ̺ ( w ) is the position x such that ̺ (( −∞ , x ]) = w . � � X ̺ ( w ) := inf x ∈ R : M ̺ ( x ) > w w ∈ (0 , 1) which is the so-called pseudo-inverse of M ̺ . The map X ̺ is nondecreasing and right-continuous . It pushes the Lebesgue measure λ := L 1 | (0 , 1) on (0 , 1) onto ̺ , i.e. ( X ̺ ) # L 1 L 1 ( X − 1 | (0 , 1) = ̺, ̺ ( B )) = ̺ ( B ) for every Borel set B ⊂ R It satisfies the change of variable formula � 1 � φ ( x ) d ̺ ( x ) = φ ( X ̺ ( w )) d w 0 R for every nonnegative/bounded Borel function φ : R → R . In particular, � 1 � | x | 2 d ̺ ( x ) = � 2 d w = � 2 � � � � m 2 ( ̺ ) := � X ̺ ( w ) � X ̺ L 2 (0 , 1) 0 R 16

  39. Sticky particles Cumulative distribution Lagrangian representation Wasserstein distance and the L 2 isometry The map ̺ �→ X ̺ is a one-to-one correspondence between the space P 2 ( R ) of probability measures with finite quadratic moment R | x | 2 d ̺ ( x ) < + ∞ � m 2 ( ̺ ) = and the closed convex cone K of all the nondecreasing function in L 2 (0 , 1) (among which we can always choose the right-continuous representative). L 2 -Wasserstein distance W 2 ( ̺ 1 , ̺ 2 ) between ̺ 1 , ̺ 2 ∈ P 2 ( R ): � 1 � 2 d w = � 2 W 2 2 ( ̺ 1 , ̺ 2 ) := � � � � � X ̺ 1 ( w ) − X ̺ 2 ( w ) � X ̺ 1 − X ̺ 2 L 2 (0 , 1) 0 In this way ̺ ↔ X ̺ is an isometry between ( P 2 ( R ) , W 2 ) and ( K , � · � L 2 (0 , 1) ). 17

  40. Sticky particles Cumulative distribution Lagrangian representation Wasserstein distance and the L 2 isometry The map ̺ �→ X ̺ is a one-to-one correspondence between the space P 2 ( R ) of probability measures with finite quadratic moment R | x | 2 d ̺ ( x ) < + ∞ � m 2 ( ̺ ) = and the closed convex cone K of all the nondecreasing function in L 2 (0 , 1) (among which we can always choose the right-continuous representative). L 2 -Wasserstein distance W 2 ( ̺ 1 , ̺ 2 ) between ̺ 1 , ̺ 2 ∈ P 2 ( R ): � 1 � 2 d w = � 2 W 2 2 ( ̺ 1 , ̺ 2 ) := � � � � � X ̺ 1 ( w ) − X ̺ 2 ( w ) � X ̺ 1 − X ̺ 2 L 2 (0 , 1) 0 In this way ̺ ↔ X ̺ is an isometry between ( P 2 ( R ) , W 2 ) and ( K , � · � L 2 (0 , 1) ). 17

  41. Sticky particles Cumulative distribution Lagrangian representation Wasserstein distance and the L 2 isometry The map ̺ �→ X ̺ is a one-to-one correspondence between the space P 2 ( R ) of probability measures with finite quadratic moment R | x | 2 d ̺ ( x ) < + ∞ � m 2 ( ̺ ) = and the closed convex cone K of all the nondecreasing function in L 2 (0 , 1) (among which we can always choose the right-continuous representative). L 2 -Wasserstein distance W 2 ( ̺ 1 , ̺ 2 ) between ̺ 1 , ̺ 2 ∈ P 2 ( R ): � 1 � 2 d w = � 2 W 2 2 ( ̺ 1 , ̺ 2 ) := � � � � � X ̺ 1 ( w ) − X ̺ 2 ( w ) � X ̺ 1 − X ̺ 2 L 2 (0 , 1) 0 In this way ̺ ↔ X ̺ is an isometry between ( P 2 ( R ) , W 2 ) and ( K , � · � L 2 (0 , 1) ). 17

  42. Sticky particles Cumulative distribution Lagrangian representation Wasserstein distance and the L 2 isometry The map ̺ �→ X ̺ is a one-to-one correspondence between the space P 2 ( R ) of probability measures with finite quadratic moment R | x | 2 d ̺ ( x ) < + ∞ � m 2 ( ̺ ) = and the closed convex cone K of all the nondecreasing function in L 2 (0 , 1) (among which we can always choose the right-continuous representative). L 2 -Wasserstein distance W 2 ( ̺ 1 , ̺ 2 ) between ̺ 1 , ̺ 2 ∈ P 2 ( R ): � 1 � 2 d w = � 2 W 2 2 ( ̺ 1 , ̺ 2 ) := � � � � � X ̺ 1 ( w ) − X ̺ 2 ( w ) � X ̺ 1 − X ̺ 2 L 2 (0 , 1) 0 In this way ̺ ↔ X ̺ is an isometry between ( P 2 ( R ) , W 2 ) and ( K , � · � L 2 (0 , 1) ). 17

  43. Sticky particles Cumulative distribution Lagrangian representation Lagrangian parametrizations To the (discrete) data µ t = ( ̺ t , ̺ t v t ) we associate the functions ( X t , V t ) ∈ K × L 2 (0 , 1) by X t := X ̺ t , V t := v t ◦ X t . Notice that the second component of ( X t , V t ) do not span the whole space L 2 ( R ) in general, but it is contained in the closed subspace � � V = v ◦ X t for some Borel map v ∈ L 2 H X t := ̺ t ( R ). The reduced cone of the discrete particle system with masses m = ( m 1 , m 2 , · · · , m N ) K m := � X ∈ K : X | [ w i − 1 ,w i ) ≡ x i � is constant 18

  44. Sticky particles Cumulative distribution Lagrangian representation Lagrangian parametrizations To the (discrete) data µ t = ( ̺ t , ̺ t v t ) we associate the functions ( X t , V t ) ∈ K × L 2 (0 , 1) by X t := X ̺ t , V t := v t ◦ X t . Notice that the second component of ( X t , V t ) do not span the whole space L 2 ( R ) in general, but it is contained in the closed subspace � � V = v ◦ X t for some Borel map v ∈ L 2 H X t := ̺ t ( R ). The reduced cone of the discrete particle system with masses m = ( m 1 , m 2 , · · · , m N ) K m := � X ∈ K : X | [ w i − 1 ,w i ) ≡ x i � is constant 18

  45. Sticky particles Cumulative distribution Lagrangian representation Lagrangian parametrizations To the (discrete) data µ t = ( ̺ t , ̺ t v t ) we associate the functions ( X t , V t ) ∈ K × L 2 (0 , 1) by X t := X ̺ t , V t := v t ◦ X t . Notice that the second component of ( X t , V t ) do not span the whole space L 2 ( R ) in general, but it is contained in the closed subspace � � V = v ◦ X t for some Borel map v ∈ L 2 H X t := ̺ t ( R ). The monotone rearrangement − 1 X ρ = M ρ The reduced cone of the discrete particle system with masses m = ( m 1 , m 2 , · · · , m N ) x 4 x 3 K m := � X ∈ K : X | [ w i − 1 ,w i ) ≡ x i � x 2 is constant w 1 w 2 w 3 1 x 1 w 1 = m 1 , w 2 = m 1 + m 2 , · · · 18

  46. Sticky particles Cumulative distribution Lagrangian representation The normal cone N X K Ξ ∈ N X K if and only if X ∈ K and one of the following equivalent conditions holds (Ξ | Z − X ) ≤ 0 for every Z ∈ K , K is the L 2 projection on K P K ( X + Ξ) = X, P The normal cone N X K coincides with the subdifferential ∂I K ( X ) of the indi- � 0 if X ∈ K , cator (convex, lower semicontinuous) I K ( X ) = + ∞ otherwise function of K in L 2 (0 , 1) 19

  47. Sticky particles Cumulative distribution Lagrangian representation The normal cone N X K Ξ ∈ N X K if and only if X ∈ K and one of the following equivalent conditions holds (Ξ | Z − X ) ≤ 0 for every Z ∈ K , K is the L 2 projection on K P K ( X + Ξ) = X, P The normal cone N X K coincides with the subdifferential ∂I K ( X ) of the indi- � 0 if X ∈ K , cator (convex, lower semicontinuous) I K ( X ) = + ∞ otherwise function of K in L 2 (0 , 1) 19

  48. Sticky particles Cumulative distribution Lagrangian representation The normal cone N X K Ξ ∈ N X K if and only if X ∈ K and one of the following equivalent conditions holds (Ξ | Z − X ) ≤ 0 for every Z ∈ K , K is the L 2 projection on K P K ( X + Ξ) = X, P The normal cone N X K coincides with the subdifferential ∂I K ( X ) of the indi- � 0 if X ∈ K , cator (convex, lower semicontinuous) I K ( X ) = + ∞ otherwise function of K in L 2 (0 , 1) 19

  49. Sticky particles Cumulative distribution Lagrangian representation The normal cone N X K Ξ ∈ N X K if and only if X ∈ K and one of the following equivalent conditions holds (Ξ | Z − X ) ≤ 0 for every Z ∈ K , K is the L 2 projection on K P K ( X + Ξ) = X, P The normal cone N X K coincides with the subdifferential ∂I K ( X ) of the indi- � 0 if X ∈ K , cator (convex, lower semicontinuous) I K ( X ) = + ∞ otherwise function of K in L 2 (0 , 1) N x 2 K T x 1 K T x 2 K N x 1 K K x 1 x 2 Figure: Normal and tangent cones. 19

  50. Sticky particles Cumulative distribution Lagrangian representation Second order differential equations and jump conditions Force field: f [ ̺ ] � F [ X ̺ ]( w ) such that � 1 � for every test function ζ ∈ C 0 ζ ( x ) d f [ ̺ ] = ζ ( X ̺ ) F d w b ( R ). R 0 Differential equation for the collision free motion in Lagrangian coordinates: d 2 X ′ d t 2 X t = F [ X t ] , X 0 = X ̺ 0 , 0 = V 0 = v 0 ◦ X 0 . 20

  51. Sticky particles Cumulative distribution Lagrangian representation Second order differential equations and jump conditions Force field: f [ ̺ ] � F [ X ̺ ]( w ) such that � 1 � for every test function ζ ∈ C 0 ζ ( x ) d f [ ̺ ] = ζ ( X ̺ ) F d w b ( R ). R 0 Differential equation for the collision free motion in Lagrangian coordinates: d 2 X ′ d t 2 X t = F [ X t ] , X 0 = X ̺ 0 , 0 = V 0 = v 0 ◦ X 0 . 20

  52. Sticky particles Cumulative distribution Lagrangian representation Second order differential equations and jump conditions Force field: f [ ̺ ] � F [ X ̺ ]( w ) such that � 1 � for every test function ζ ∈ C 0 ζ ( x ) d f [ ̺ ] = ζ ( X ̺ ) F d w b ( R ). R 0 Differential equation for the collision free motion in Lagrangian coordinates: d 2 X ′ d t 2 X t = F [ X t ] , X 0 = X ̺ 0 , 0 = V 0 = v 0 ◦ X 0 . After a collision at a time ¯ t : V ¯ t + = P t ) K ( V ¯ t − ) i.e. V ¯ t + + N X (¯ t ) K ∋ V ¯ T X (¯ t − particle velocity after collision K particle trajectory Figure: Projection of velocities onto the tangent cone. 20

  53. Sticky particles Cumulative distribution Lagrangian representation Second order differential inclusions: the role of the sticky condition “Formal” differential inclusion for the SPS: d 2 d t 2 X t + ∂ I K ( X t ) ∋ F [ X t ] ( ⋆ ) The discrete case still satisfies ( ⋆ ) if F satisfies the “ non splitting ” condition F [ X ] ∈ H X i.e. F [ X ] depends on X or, more generally, F [ X ] − P H X F [ X ] ∈ N X K as for the Euler-Poisson system in attractive regime. The theory of second order differential inclusion ( Schatzman ’78, Moreau ’83 ) only covers the finite-dimensional case, lacks uniqueness, and stability. Sticky condition: ⇒ X t ∈ H X s i.e. X t “depends on” X s . s < t By the monotonicity property of ∂ I K we have s < t ⇒ ∂ I K ( X s ) ⊂ ∂ I K ( X t ) 21

  54. Sticky particles Cumulative distribution Lagrangian representation Second order differential inclusions: the role of the sticky condition “Formal” differential inclusion for the SPS: d 2 d t 2 X t + ∂ I K ( X t ) ∋ F [ X t ] ( ⋆ ) The discrete case still satisfies ( ⋆ ) if F satisfies the “ non splitting ” condition F [ X ] ∈ H X i.e. F [ X ] depends on X or, more generally, F [ X ] − P H X F [ X ] ∈ N X K as for the Euler-Poisson system in attractive regime. The theory of second order differential inclusion ( Schatzman ’78, Moreau ’83 ) only covers the finite-dimensional case, lacks uniqueness, and stability. Sticky condition: ⇒ X t ∈ H X s i.e. X t “depends on” X s . s < t By the monotonicity property of ∂ I K we have s < t ⇒ ∂ I K ( X s ) ⊂ ∂ I K ( X t ) 21

  55. Sticky particles Cumulative distribution Lagrangian representation Second order differential inclusions: the role of the sticky condition “Formal” differential inclusion for the SPS: d 2 d t 2 X t + ∂ I K ( X t ) ∋ F [ X t ] ( ⋆ ) The discrete case still satisfies ( ⋆ ) if F satisfies the “ non splitting ” condition F [ X ] ∈ H X i.e. F [ X ] depends on X or, more generally, F [ X ] − P H X F [ X ] ∈ N X K as for the Euler-Poisson system in attractive regime. The theory of second order differential inclusion ( Schatzman ’78, Moreau ’83 ) only covers the finite-dimensional case, lacks uniqueness, and stability. Sticky condition: ⇒ X t ∈ H X s i.e. X t “depends on” X s . s < t By the monotonicity property of ∂ I K we have s < t ⇒ ∂ I K ( X s ) ⊂ ∂ I K ( X t ) 21

  56. Sticky particles Cumulative distribution Lagrangian representation Second order differential inclusions: the role of the sticky condition “Formal” differential inclusion for the SPS: d 2 d t 2 X t + ∂ I K ( X t ) ∋ F [ X t ] ( ⋆ ) The discrete case still satisfies ( ⋆ ) if F satisfies the “ non splitting ” condition F [ X ] ∈ H X i.e. F [ X ] depends on X or, more generally, F [ X ] − P H X F [ X ] ∈ N X K as for the Euler-Poisson system in attractive regime. The theory of second order differential inclusion ( Schatzman ’78, Moreau ’83 ) only covers the finite-dimensional case, lacks uniqueness, and stability. Sticky condition: ⇒ X t ∈ H X s i.e. X t “depends on” X s . s < t By the monotonicity property of ∂ I K we have s < t ⇒ ∂ I K ( X s ) ⊂ ∂ I K ( X t ) 21

  57. Sticky particles Cumulative distribution Lagrangian representation Second order differential inclusions: the role of the sticky condition “Formal” differential inclusion for the SPS: d 2 d t 2 X t + ∂ I K ( X t ) ∋ F [ X t ] ( ⋆ ) The discrete case still satisfies ( ⋆ ) if F satisfies the “ non splitting ” condition F [ X ] ∈ H X i.e. F [ X ] depends on X or, more generally, F [ X ] − P H X F [ X ] ∈ N X K as for the Euler-Poisson system in attractive regime. The theory of second order differential inclusion ( Schatzman ’78, Moreau ’83 ) only covers the finite-dimensional case, lacks uniqueness, and stability. Sticky condition: ⇒ X t ∈ H X s i.e. X t “depends on” X s . s < t By the monotonicity property of ∂ I K we have s < t ⇒ ∂ I K ( X s ) ⊂ ∂ I K ( X t ) 21

  58. Sticky particles Cumulative distribution Lagrangian representation Second order differential inclusions: the role of the sticky condition “Formal” differential inclusion for the SPS: d 2 d t 2 X t + ∂ I K ( X t ) ∋ F [ X t ] ( ⋆ ) The discrete case still satisfies ( ⋆ ) if F satisfies the “ non splitting ” condition F [ X ] ∈ H X i.e. F [ X ] depends on X or, more generally, F [ X ] − P H X F [ X ] ∈ N X K as for the Euler-Poisson system in attractive regime. The theory of second order differential inclusion ( Schatzman ’78, Moreau ’83 ) only covers the finite-dimensional case, lacks uniqueness, and stability. Sticky condition: ⇒ X t ∈ H X s i.e. X t “depends on” X s . s < t By the monotonicity property of ∂ I K we have s < t ⇒ ∂ I K ( X s ) ⊂ ∂ I K ( X t ) 21

  59. Sticky particles Cumulative distribution Lagrangian representation Reduction to first-order differential inclusions d 2 d t 2 X t + ∂ I K ( X t ) ∋ F [ X t ] , ⇒ ∂ I K ( X s ) ⊂ ∂ I K ( X t ) ( ⋆ ) s < t We can then integrate ( ⋆ ) with respect to time from 0 to a final time t : since � t ∂ I K ( X s ) d s ∈ ∂ I K ( X t ) . 0 � t d d tX t − V 0 + ∂ I K ( X t ) ∋ F [ X s ] d s ( ⋆⋆ ) 0 Integrating again we get � t X t − ( X 0 + tV 0 ) + ∂ I K ( X t ) ∋ ( t − s ) F [ X s ] d s 0 i.e. � t � � X t = P X 0 + tV 0 + ( t − s ) F [ X s ] d s ( ⋆ ⋆ ⋆ ) K 0 22

  60. Sticky particles Cumulative distribution Lagrangian representation Reduction to first-order differential inclusions d 2 d t 2 X t + ∂ I K ( X t ) ∋ F [ X t ] , ⇒ ∂ I K ( X s ) ⊂ ∂ I K ( X t ) ( ⋆ ) s < t We can then integrate ( ⋆ ) with respect to time from 0 to a final time t : since � t ∂ I K ( X s ) d s ∈ ∂ I K ( X t ) . 0 � t d d tX t − V 0 + ∂ I K ( X t ) ∋ F [ X s ] d s ( ⋆⋆ ) 0 Integrating again we get � t X t − ( X 0 + tV 0 ) + ∂ I K ( X t ) ∋ ( t − s ) F [ X s ] d s 0 i.e. � t � � X t = P X 0 + tV 0 + ( t − s ) F [ X s ] d s ( ⋆ ⋆ ⋆ ) K 0 22

  61. Sticky particles Cumulative distribution Lagrangian representation Reduction to first-order differential inclusions d 2 d t 2 X t + ∂ I K ( X t ) ∋ F [ X t ] , ⇒ ∂ I K ( X s ) ⊂ ∂ I K ( X t ) ( ⋆ ) s < t We can then integrate ( ⋆ ) with respect to time from 0 to a final time t : since � t ∂ I K ( X s ) d s ∈ ∂ I K ( X t ) . 0 � t d d tX t − V 0 + ∂ I K ( X t ) ∋ F [ X s ] d s ( ⋆⋆ ) 0 Integrating again we get � t X t − ( X 0 + tV 0 ) + ∂ I K ( X t ) ∋ ( t − s ) F [ X s ] d s 0 i.e. � t � � X t = P X 0 + tV 0 + ( t − s ) F [ X s ] d s ( ⋆ ⋆ ⋆ ) K 0 22

  62. Sticky particles Cumulative distribution Lagrangian representation A description of the evolution by a differential inclusion Recall that to the (discrete) data µ t = ( ̺ t , ̺ t v t ) we associate the functions ( X t , V t ) ∈ K × L 2 (0 , 1) by X t := X ̺ t , V t := v t ◦ X t . Theorem (Lagrangian representation) A family µ t = ( ̺ t , ̺ t v t ) is a solution of the (discrete) SPS if and only if X is the unique strong solution of the differential inclusion � t d d tX t + ∂ I K ( X t ) ∋ V 0 + F [ X s ] d s, lim t ↓ 0 X t = X 0 . 0 � t Equivalently, setting Y t := V 0 + 0 F [ X s ] d s we get the well posed system d  d tX t + ∂ I K ( X t ) ∋ Y t   d  d tY t = F [ X t ]  23

  63. Sticky particles Cumulative distribution Lagrangian representation A description of the evolution by a differential inclusion Recall that to the (discrete) data µ t = ( ̺ t , ̺ t v t ) we associate the functions ( X t , V t ) ∈ K × L 2 (0 , 1) by X t := X ̺ t , V t := v t ◦ X t . Theorem (Lagrangian representation) A family µ t = ( ̺ t , ̺ t v t ) is a solution of the (discrete) SPS if and only if X is the unique strong solution of the differential inclusion � t d d tX t + ∂ I K ( X t ) ∋ V 0 + F [ X s ] d s, lim t ↓ 0 X t = X 0 . 0 � t Equivalently, setting Y t := V 0 + 0 F [ X s ] d s we get the well posed system d  d tX t + ∂ I K ( X t ) ∋ Y t   d  d tY t = F [ X t ]  23

  64. Sticky particles Cumulative distribution Lagrangian representation A description of the evolution by a differential inclusion Recall that to the (discrete) data µ t = ( ̺ t , ̺ t v t ) we associate the functions ( X t , V t ) ∈ K × L 2 (0 , 1) by X t := X ̺ t , V t := v t ◦ X t . Theorem (Lagrangian representation) A family µ t = ( ̺ t , ̺ t v t ) is a solution of the (discrete) SPS if and only if X is the unique strong solution of the differential inclusion � t d d tX t + ∂ I K ( X t ) ∋ V 0 + F [ X s ] d s, lim t ↓ 0 X t = X 0 . 0 � t Equivalently, setting Y t := V 0 + 0 F [ X s ] d s we get the well posed system d  d tX t + ∂ I K ( X t ) ∋ Y t   d  d tY t = F [ X t ]  23

  65. Sticky particles Cumulative distribution Lagrangian representation Stability properties Suppose that F is Lipschitz in L 2 (0 , 1). By general results on solution of differential inclusion of the type Z ′ t + ∂φ ( Z t ) ∋ G t , φ convex Theorem If X 1 t , X 2 t are the Lagrangian representation of two (discrete) solutions ̺ 1 t , ̺ 2 t of the SPS we have � � � X 1 t − X 2 � X 1 0 − X 2 0 � L 2 + � V 1 0 − V 2 sup t � L 2 ≤ C T 0 � L 2 . t ∈ [0 ,T ] X is right-differentiable in each point and the velocity field v t can be recovered by the formula V t = d + d t X t = v t ◦ X t ∈ H X t . One gets [S. ’96] the following integral estimate for the velocity component: � T � � �� � � V 1 r − V 2 r � 2 � X ℓ 0 � + � V ℓ � X 1 0 − X 2 0 � + � V 1 0 − V 2 L 2 d r ≤ C T 0 � 0 � . 0 ℓ 24

  66. Sticky particles Cumulative distribution Lagrangian representation Stability properties Suppose that F is Lipschitz in L 2 (0 , 1). By general results on solution of differential inclusion of the type Z ′ t + ∂φ ( Z t ) ∋ G t , φ convex Theorem If X 1 t , X 2 t are the Lagrangian representation of two (discrete) solutions ̺ 1 t , ̺ 2 t of the SPS we have � � � X 1 t − X 2 � X 1 0 − X 2 0 � L 2 + � V 1 0 − V 2 sup t � L 2 ≤ C T 0 � L 2 . t ∈ [0 ,T ] X is right-differentiable in each point and the velocity field v t can be recovered by the formula V t = d + d t X t = v t ◦ X t ∈ H X t . One gets [S. ’96] the following integral estimate for the velocity component: � T � � �� � � V 1 r − V 2 r � 2 � X ℓ 0 � + � V ℓ � X 1 0 − X 2 0 � + � V 1 0 − V 2 L 2 d r ≤ C T 0 � 0 � . 0 ℓ 24

  67. Sticky particles Cumulative distribution Lagrangian representation Stability properties Suppose that F is Lipschitz in L 2 (0 , 1). By general results on solution of differential inclusion of the type Z ′ t + ∂φ ( Z t ) ∋ G t , φ convex Theorem If X 1 t , X 2 t are the Lagrangian representation of two (discrete) solutions ̺ 1 t , ̺ 2 t of the SPS we have � � � X 1 t − X 2 � X 1 0 − X 2 0 � L 2 + � V 1 0 − V 2 sup t � L 2 ≤ C T 0 � L 2 . t ∈ [0 ,T ] X is right-differentiable in each point and the velocity field v t can be recovered by the formula V t = d + d t X t = v t ◦ X t ∈ H X t . One gets [S. ’96] the following integral estimate for the velocity component: � T � � �� � � V 1 r − V 2 r � 2 � X ℓ 0 � + � V ℓ � X 1 0 − X 2 0 � + � V 1 0 − V 2 L 2 d r ≤ C T 0 � 0 � . 0 ℓ 24

  68. Sticky particles Cumulative distribution Lagrangian representation Convergence of discrete solutions Theorem Let µ n := ( ̺ n , ̺ n v n ) be a sequence of discrete sticky-particle solutions with ̺ n 0 ⇀ ̺ 0 in P 2 ( R ) and ̺ n 0 v n 0 ⇀ ̺ 0 v 0 weakly in the space of signed measures with � � 0 | 2 d ρ n | v 0 | 2 d ̺ 0 | v n 0 → Then ̺ n t ⇀ ̺ t in P 2 ( R ) for every t ≥ 0 and ̺ n t v n t ⇀ ̺ t v t weakly in the space of signed measures with � � t | 2 d ρ n | v t | 2 d ̺ t | v n t → and ( ρ t , ρ t v t ) is a solution of � ∂ t ̺ + ∂ x ( ̺ v ) = 0 , ̺ | t =0 = ̺ 0 , v | t =0 = v 0 , ∂ t ( ̺ v ) + ∂ x ( ̺ v 2 ) = f [ ̺ ] , 25

  69. Sticky particles Cumulative distribution Lagrangian representation Convergence of discrete solutions Theorem Let µ n := ( ̺ n , ̺ n v n ) be a sequence of discrete sticky-particle solutions with ̺ n 0 ⇀ ̺ 0 in P 2 ( R ) and ̺ n 0 v n 0 ⇀ ̺ 0 v 0 weakly in the space of signed measures with � � 0 | 2 d ρ n | v 0 | 2 d ̺ 0 | v n 0 → Then ̺ n t ⇀ ̺ t in P 2 ( R ) for every t ≥ 0 and ̺ n t v n t ⇀ ̺ t v t weakly in the space of signed measures with � � t | 2 d ρ n | v t | 2 d ̺ t | v n t → and ( ρ t , ρ t v t ) is a solution of � ∂ t ̺ + ∂ x ( ̺ v ) = 0 , ̺ | t =0 = ̺ 0 , v | t =0 = v 0 , ∂ t ( ̺ v ) + ∂ x ( ̺ v 2 ) = f [ ̺ ] , 25

  70. Sticky particles Cumulative distribution Lagrangian representation Convergence of discrete solutions Theorem Let µ n := ( ̺ n , ̺ n v n ) be a sequence of discrete sticky-particle solutions with ̺ n 0 ⇀ ̺ 0 in P 2 ( R ) and ̺ n 0 v n 0 ⇀ ̺ 0 v 0 weakly in the space of signed measures with � � 0 | 2 d ρ n | v 0 | 2 d ̺ 0 | v n 0 → Then ̺ n t ⇀ ̺ t in P 2 ( R ) for every t ≥ 0 and ̺ n t v n t ⇀ ̺ t v t weakly in the space of signed measures with � � t | 2 d ρ n | v t | 2 d ̺ t | v n t → and ( ρ t , ρ t v t ) is a solution of � ∂ t ̺ + ∂ x ( ̺ v ) = 0 , ̺ | t =0 = ̺ 0 , v | t =0 = v 0 , ∂ t ( ̺ v ) + ∂ x ( ̺ v 2 ) = f [ ̺ ] , 25

  71. Sticky particles Cumulative distribution Lagrangian representation Convergence of discrete solutions Theorem Let µ n := ( ̺ n , ̺ n v n ) be a sequence of discrete sticky-particle solutions with ̺ n 0 ⇀ ̺ 0 in P 2 ( R ) and ̺ n 0 v n 0 ⇀ ̺ 0 v 0 weakly in the space of signed measures with � � 0 | 2 d ρ n | v 0 | 2 d ̺ 0 | v n 0 → Then ̺ n t ⇀ ̺ t in P 2 ( R ) for every t ≥ 0 and ̺ n t v n t ⇀ ̺ t v t weakly in the space of signed measures with � � t | 2 d ρ n | v t | 2 d ̺ t | v n t → and ( ρ t , ρ t v t ) is a solution of � ∂ t ̺ + ∂ x ( ̺ v ) = 0 , ̺ | t =0 = ̺ 0 , v | t =0 = v 0 , ∂ t ( ̺ v ) + ∂ x ( ̺ v 2 ) = f [ ̺ ] , 25

  72. Sticky particles Cumulative distribution Lagrangian representation Convergence of discrete solutions Theorem Let µ n := ( ̺ n , ̺ n v n ) be a sequence of discrete sticky-particle solutions with ̺ n 0 ⇀ ̺ 0 in P 2 ( R ) and ̺ n 0 v n 0 ⇀ ̺ 0 v 0 weakly in the space of signed measures with � � 0 | 2 d ρ n | v 0 | 2 d ̺ 0 | v n 0 → Then ̺ n t ⇀ ̺ t in P 2 ( R ) for every t ≥ 0 and ̺ n t v n t ⇀ ̺ t v t weakly in the space of signed measures with � � t | 2 d ρ n | v t | 2 d ̺ t | v n t → and ( ρ t , ρ t v t ) is a solution of � ∂ t ̺ + ∂ x ( ̺ v ) = 0 , ̺ | t =0 = ̺ 0 , v | t =0 = v 0 , ∂ t ( ̺ v ) + ∂ x ( ̺ v 2 ) = f [ ̺ ] , 25

  73. Sticky particles Cumulative distribution Lagrangian representation An explicit representation formula for the Euler-Poisson system [ Tadmor-Wei] − ∂ 2 f [ ̺ ] = − ̺ ∂ x q ̺ with q ̺ solution of xx q ̺ = ̺ F [ X ] = 1 − 2 w � X 0 + tV 0 + (1 − 2 w ) t 2 � X t = P . K 26

  74. Sticky particles Cumulative distribution Lagrangian representation Extensions and open problems Extensions: ◮ Uniformly continuous force fields ◮ The behaviour of the Euler-Poisson system in the repulsive regime: W ( x ) = −| x | . Open problems: ◮ Adding a pressure term. 27

  75. Sticky particles Cumulative distribution Lagrangian representation Extensions and open problems Extensions: ◮ Uniformly continuous force fields ◮ The behaviour of the Euler-Poisson system in the repulsive regime: W ( x ) = −| x | . Open problems: ◮ Adding a pressure term. 27

  76. Sticky particles Cumulative distribution Lagrangian representation Extensions and open problems Extensions: ◮ Uniformly continuous force fields ◮ The behaviour of the Euler-Poisson system in the repulsive regime: W ( x ) = −| x | . Open problems: ◮ Adding a pressure term. 27

  77. Sticky particles Cumulative distribution Lagrangian representation Extensions and open problems Extensions: ◮ Uniformly continuous force fields ◮ The behaviour of the Euler-Poisson system in the repulsive regime: W ( x ) = −| x | . Open problems: ◮ Adding a pressure term. 27

  78. Sticky particles Cumulative distribution Lagrangian representation The L 2 -projection on K Theorem � w If X ∈ L 2 (0 , 1) and X ( w ) = 0 X ( s ) d s is its primitive then d where X ∗∗ is the convex envelope of X . d w X ∗∗ P K ( X ) = 28

  79. Sticky particles Cumulative distribution Lagrangian representation The L 2 -projection on K Theorem � w If X ∈ L 2 (0 , 1) and X ( w ) = 0 X ( s ) d s is its primitive then d where X ∗∗ is the convex envelope of X . d w X ∗∗ P K ( X ) = X 1 28

  80. Sticky particles Cumulative distribution Lagrangian representation The L 2 -projection on K Theorem � w If X ∈ L 2 (0 , 1) and X ( w ) = 0 X ( s ) d s is its primitive then d where X ∗∗ is the convex envelope of X . d w X ∗∗ P K ( X ) = � w X X ( w ) = 0 X ( s ) ds 1 1 28

  81. Sticky particles Cumulative distribution Lagrangian representation The L 2 -projection on K Theorem � w If X ∈ L 2 (0 , 1) and X ( w ) = 0 X ( s ) d s is its primitive then d where X ∗∗ is the convex envelope of X . d w X ∗∗ P K ( X ) = � w X X ( w ) = 0 X ( s ) ds 1 1 X ∗∗ 1 28

  82. Sticky particles Cumulative distribution Lagrangian representation The L 2 -projection on K Theorem � w If X ∈ L 2 (0 , 1) and X ( w ) = 0 X ( s ) d s is its primitive then d where X ∗∗ is the convex envelope of X . d w X ∗∗ P K ( X ) = � w X X ( w ) = 0 X ( s ) ds 1 1 X ∗∗ P K ( X ) 1 1 28

  83. Sticky particles Cumulative distribution Lagrangian representation The subdifferential of I K If X ∈ K we consider the open set Ω X where X is (essentially) constant: � � Ω X := w ∈ (0 , 1) : X is essentially constant in a neighborhood of w , and the cone Y ∈ C 0 ([0 , 1]) : Y ≥ 0 , � � N X := Y = 0 in [0 , 1] \ Ω X . Theorem � w Let X ∈ K and Y ∈ L 2 (0 , 1) with Y ( w ) := 0 Y ( s ) d s . Then Y ∈ ∂ I K ( X ) ⇔ Y ∈ N X . Notice that if Z = f ( X ) ∈ H X depends on X then Ω X ⊂ Ω Z , N X ⊂ N Z Corollary (Monotonicity property of ∂ I K ) If Z = f ( X ) ∈ H X depends on X then ∂ I K ( X ) ⊂ ∂ I K ( Z ) . 29

  84. Sticky particles Cumulative distribution Lagrangian representation The subdifferential of I K If X ∈ K we consider the open set Ω X where X is (essentially) constant: � � Ω X := w ∈ (0 , 1) : X is essentially constant in a neighborhood of w , and the cone Y ∈ C 0 ([0 , 1]) : Y ≥ 0 , � � N X := Y = 0 in [0 , 1] \ Ω X . Theorem � w Let X ∈ K and Y ∈ L 2 (0 , 1) with Y ( w ) := 0 Y ( s ) d s . Then Y ∈ ∂ I K ( X ) ⇔ Y ∈ N X . Notice that if Z = f ( X ) ∈ H X depends on X then Ω X ⊂ Ω Z , N X ⊂ N Z Corollary (Monotonicity property of ∂ I K ) If Z = f ( X ) ∈ H X depends on X then ∂ I K ( X ) ⊂ ∂ I K ( Z ) . 29

  85. Sticky particles Cumulative distribution Lagrangian representation The subdifferential of I K If X ∈ K we consider the open set Ω X where X is (essentially) constant: � � Ω X := w ∈ (0 , 1) : X is essentially constant in a neighborhood of w , and the cone Y ∈ C 0 ([0 , 1]) : Y ≥ 0 , � � N X := Y = 0 in [0 , 1] \ Ω X . Theorem � w Let X ∈ K and Y ∈ L 2 (0 , 1) with Y ( w ) := 0 Y ( s ) d s . Then Y ∈ ∂ I K ( X ) ⇔ Y ∈ N X . Notice that if Z = f ( X ) ∈ H X depends on X then Ω X ⊂ Ω Z , N X ⊂ N Z Corollary (Monotonicity property of ∂ I K ) If Z = f ( X ) ∈ H X depends on X then ∂ I K ( X ) ⊂ ∂ I K ( Z ) . 29

  86. Sticky particles Cumulative distribution Lagrangian representation The subdifferential of I K If X ∈ K we consider the open set Ω X where X is (essentially) constant: � � Ω X := w ∈ (0 , 1) : X is essentially constant in a neighborhood of w , and the cone Y ∈ C 0 ([0 , 1]) : Y ≥ 0 , � � N X := Y = 0 in [0 , 1] \ Ω X . Theorem � w Let X ∈ K and Y ∈ L 2 (0 , 1) with Y ( w ) := 0 Y ( s ) d s . Then Y ∈ ∂ I K ( X ) ⇔ Y ∈ N X . Notice that if Z = f ( X ) ∈ H X depends on X then Ω X ⊂ Ω Z , N X ⊂ N Z Corollary (Monotonicity property of ∂ I K ) If Z = f ( X ) ∈ H X depends on X then ∂ I K ( X ) ⊂ ∂ I K ( Z ) . 29

Recommend

Particles approximation for Vlasov equation with singular interaction M. Hauray, in collaboration

Particles approximation for Vlasov equation with singular interaction M. Hauray, in collaboration

Particles approximation for Vlasov equation with singular interaction M. Hauray, in collaboration with P.-E. Jabin. Universit e dAix-Marseille Oberwolfach Worshop, December 2013 M. Hauray (UAM) Particles systems towards Vlasov

711 views • 50 slides
protoDUNE Wenqiang Gu Brookhaven National Laboratory 1 Sticky Code The 6 LSBs in ADC ASIC

protoDUNE Wenqiang Gu Brookhaven National Laboratory 1 Sticky Code The 6 LSBs in ADC ASIC

Sticky Code Mitigation for protoDUNE Wenqiang Gu Brookhaven National Laboratory 1 Sticky Code The 6 LSBs in ADC ASIC was found to be sticky around 000000 (0x00) or 111111 (0x3F) So called sticky code, or stuck bit Digital Out

506 views • 16 slides
Interaction of particles with matter (lecture 2) 25/58 Johann Collot collot@in2p3.fr

Interaction of particles with matter (lecture 2) 25/58 Johann Collot collot@in2p3.fr

Interaction of particles with matter (lecture 2) 25/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG Bremsstrahlung : electromagnetic radiative energy loss A decelerated or accelerated charged particle radiates

663 views • 33 slides
Improving Social Mobility Removing the Sticky Floors and Sticky Ceilings Karen Iles

Improving Social Mobility Removing the Sticky Floors and Sticky Ceilings Karen Iles

Improving Social Mobility Removing the Sticky Floors and Sticky Ceilings Karen Iles National Director Achievement for All Jan Gouveia Deputy Regional Lead South East A broken social elevator? (OECD, 2018) The picture in the UK

693 views • 54 slides
Interaction of particles with matter (lecture 1) 1/58 Johann Collot collot@in2p3.fr

Interaction of particles with matter (lecture 1) 1/58 Johann Collot collot@in2p3.fr

Interaction of particles with matter (lecture 1) 1/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG A brief review of a few typical situations is going to greatly simplify the subject. Mean free path of a

811 views • 24 slides
Calorimetry Peter Krian Basic principles Interaction of charged particles and

Calorimetry Peter Krian Basic principles Interaction of charged particles and

Calorimetry Peter Krian Basic principles Interaction of charged particles and photons Electromagnetic cascades Nuclear interactions Hadronic cascades Homogeneous calorimeters Sampling calorimeters

605 views • 36 slides
Bottom Up In Bloomsbury - open, local and sticky! open, local and sticky! Bloomsbury Village

Bottom Up In Bloomsbury - open, local and sticky! open, local and sticky! Bloomsbury Village

Planning Aid for London 9 th May 2012 Bottom Up In Bloomsbury - open, local and sticky! open, local and sticky! Bloomsbury Village Neighbourhood Plan Jim Murray Bloomsbury Association www.bloomsburyvillage.org Bloomsbury Village 25

290 views • 16 slides
Elementary Particles The particles we will be looking at in this course are either elementary ,

Elementary Particles The particles we will be looking at in this course are either elementary ,

Heidi Schellman Northwestern Elementary Particles The particles we will be looking at in this course are either elementary , like the electron , or composite , like the pion and proton . The elementary particles can be characterized by their

526 views • 38 slides
STICKY HEELZ COMPANY PRESENTATION OVERVIEW OF BUSINESS Company LMB Enterprises Limited t/a

STICKY HEELZ COMPANY PRESENTATION OVERVIEW OF BUSINESS Company LMB Enterprises Limited t/a

STICKY HEELZ COMPANY PRESENTATION OVERVIEW OF BUSINESS Company LMB Enterprises Limited t/a Sticky Heelz Mission: To improve the fit and comfort of shoes Patent pending Two Part Anti-slip heel Product pads Challenger brand

248 views • 12 slides
A heat conduction model with localized billiard balls and weak interaction forces Imre P eter

A heat conduction model with localized billiard balls and weak interaction forces Imre P eter

Preliminary remarks The model Programme two particles finitely many particles heat conduction A heat conduction model with localized billiard balls and weak interaction forces Imre P eter T oth University of Helsinki, Budapest

494 views • 18 slides
Sticky Information in General Equilibrium N. Gregory Mankiw and Ricardo Reis Harvard

Sticky Information in General Equilibrium N. Gregory Mankiw and Ricardo Reis Harvard

Sticky Information in General Equilibrium N. Gregory Mankiw and Ricardo Reis Harvard University and Princeton University August 2006 Abstract This paper develops and analyzes a general-equilibrium model with sticky informa- tion. The only

672 views • 28 slides
Elem entary Particles Fundam ental Forces &amp; Forces of Nature Four forces responsible for

Elem entary Particles Fundam ental Forces &amp; Forces of Nature Four forces responsible for

Elem entary Particles Fundam ental Forces &amp; Forces of Nature Four forces responsible for all phenomena Gravitational force ( 1 0 - 4 5 ) interaction between masses (all particles) most familiar to us W eak force ( 1 0 -8 )

734 views • 24 slides
Plan for 3 lectures Introduction. The need for new physics. Portals to new Physics.

Plan for 3 lectures Introduction. The need for new physics. Portals to new Physics.

Plan for 3 lectures Introduction. The need for new physics. Portals to new Physics. Strong CP problem and axion solutions. Searches for axions and axion-like particles. Weakly interacting massive particles. WIMP interaction with

558 views • 37 slides
Dynamics of non-neutrally buoyant particles Particles with r vanishing intertia r r d X r

Dynamics of non-neutrally buoyant particles Particles with r vanishing intertia r r d X r

Dynamics of non-neutrally buoyant particles Particles with r vanishing intertia r r d X r V u ( X , t ) = = (fluid elements) dt Impurities: r r r 2 particles with d X d V F = = 2 finite inertia dt dt m and/or

513 views • 39 slides
Sun Safe Presentation for Sticky Fingers Staff. Sun Safe Nursery Accreditation Each year at

Sun Safe Presentation for Sticky Fingers Staff. Sun Safe Nursery Accreditation Each year at

Sun Safe Presentation for Sticky Fingers Staff. Sun Safe Nursery Accreditation Each year at Sticky Fingers Pre-School we will complete the requirements of the Sun Safe Accreditation. The following slides provide information for

544 views • 15 slides
the interaction physical characteristics of interaction interaction styles the

the interaction physical characteristics of interaction interaction styles the

The Interaction interaction models chapter 3 translations between user and system ergonomics the interaction physical characteristics of interaction interaction styles the nature of user/ system dialog context

506 views • 12 slides
the interaction The Interaction interaction models translations between user and system

the interaction The Interaction interaction models translations between user and system

chapter 3 the interaction The Interaction interaction models translations between user and system ergonomics physical characteristics of interaction interaction styles the nature of user/ system dialog context

733 views • 24 slides
1 Introduction Individual consumer and producer prices change every six months to one year. 1 In

1 Introduction Individual consumer and producer prices change every six months to one year. 1 In

Sticky Information and Sticky Prices Peter J. Klenow and Jonathan L. Willis June 2007 Abstract In the U.S. and Europe, prices change at least once a year. Yet nominal macro shocks seem to have real effects lasting well beyond a year.

611 views • 42 slides
Urban Interaction Design Luoning Zhang Yueting Ji Manoj Bonnke Quick Activity: Balloon Tower

Urban Interaction Design Luoning Zhang Yueting Ji Manoj Bonnke Quick Activity: Balloon Tower

Urban Interaction Design Luoning Zhang Yueting Ji Manoj Bonnke Quick Activity: Balloon Tower Form 2 groups of around 5 people each. Make a balloon tower as tall as possible by using just balloons and some sticky tape within 5 mins .

596 views • 40 slides
1 Creating more particles What have we learned? All that is needed to create particles is

1 Creating more particles What have we learned? All that is needed to create particles is

Particles as fields Final Exam : Sat. Dec. 19, 2:45-4:45 pm, 2103 Cham. Exam is cumulative, covering all material Electromagnetic field spread out over space. Review Chap. 18: Particle Physics Stronger near the the source of the

481 views • 8 slides
Sweet, Sticky, and Healthy using metabolomics to develop a green procotol for

Sweet, Sticky, and Healthy using metabolomics to develop a green procotol for

Sweet, Sticky, and Healthy using metabolomics to develop a green procotol for polyphenolics extraction from wine grape pomace Andrew Valente Shehab Selim (BSc) Joao Fonseca Shannon McGowan Lucas Maddalena (MSc) MSc student

490 views • 48 slides
MA Macroeconomics 9. Sticky Prices and the Phillips Curve Karl Whelan School of Economics, UCD

MA Macroeconomics 9. Sticky Prices and the Phillips Curve Karl Whelan School of Economics, UCD

MA Macroeconomics 9. Sticky Prices and the Phillips Curve Karl Whelan School of Economics, UCD Autumn 2014 Karl Whelan (UCD) Sticky Prices and the Phillips Curve Autumn 2014 1 / 19 Back to Price Stickiness One of the themes of the first

362 views • 19 slides
The project INTERACTION Driver INTERACTION with in-vehicle technologies EU 7 th framework

The project INTERACTION Driver INTERACTION with in-vehicle technologies EU 7 th framework

Interaction with IVT-systems Results from driving behaviour observations from the EU-project INTERACTION The project INTERACTION Driver INTERACTION with in-vehicle technologies EU 7 th framework programme 2008 to 2012 Partners

272 views • 15 slides
Dark photon searches at MAGIX U(1) Gauge boson Carrier of an unknown interaction between

Dark photon searches at MAGIX U(1) Gauge boson Carrier of an unknown interaction between

Dark photon searches at MAGIX U(1) Gauge boson Carrier of an unknown interaction between unknown particles Massive (mass unknown) No direct coupling with any SM field Kinetic mixing Same quantum numbers of the SM photon

327 views • 13 slides

More recommend