text: 20 Liping Xu | | ≤ | − | | | ≤ | − | − | − | Due to (1.8) and (cid:90)(1.5)(cid:90), c(v, v∗, z, ϕ) v v∗ , c(v, v∗, z, ϕ) (1 + z/ v v∗ γ) 1/ν v v∗ , whence (cid:12) (cid:12) (cid:12) (cid:12) ∞ (cid:90)2π |(cid:90) | − | | v + c(cid:0)(v, v∗, z, ϕ) q v q d(cid:1)ϕdz 0 0 ∞ ≤ (cid:0) 2π | | − (cid:1) | | − | − | − | − | C 1 + v q 1 + v∗ q 1 (1 + z/ v v∗ γ) 1/ν v v∗ dϕdz q(cid:0) (cid:1) 0 | 0| − | | − | − | = C 1 + v q 1 + v∗ q 1 v v∗ 1+γ q ≤ | | | | C 1 + v q + v∗ q , (4.3) q because 0 < 1 +(cid:2)γ < 1. It th(cid:3)en easily follows (cid:90)from(cid:90) the(cid:104)Itˆo formula and L (W(cid:105)∗) = f = L(W1) that α t t t E sup |W1|q ≤ E[|V 1|q] + C (cid:90) t (cid:16) 1 E 1 + |W1|q +(cid:17)|W∗(α)|q dαds s 0 q s s [0,t] 0 0 ≤ E | | t E | | [ V 1 q] + C 1 + [sup W1 q] ds. 0 q u 0 [0,s] We thus conclude (4.2) by the Gr¨onwall lemma. Next, let us recall [13, Lemma 4.3] below in order to construct our coupling. (cid:80) ∗ ≥ Lemma 4.2. Consider (f ) ≥ and (W ) ≥ introduced in Lemma 4.1 and ﬁx N 1. For v = (v , v , ..., v ) ∈ (R3)N, wetitntr0oduce thetemt p0irical measure µN := N−1 N δ . Th(cid:80)en for all t ≥ 0, a(cid:82)ll1 v 2∈ (R3N)N and all w ∈ (R3)N• , with (R3)N• := {w ∈ (R3)Nv: w (cid:54)= w ∀ii=(cid:54)=1 vji}, there are α-random ∗ ∗ ∗ · i ∗ j · − variables Z (w, α) and V (v, w, α) such that the α-law of (Z (w, ), V (v, w, )) is N 1 N δ and | ∗ −t ∗ | t W t t i=1 (wi,vi) 1 W (α) Z (w, α) 2dα = 2(f , µN). 0 t t 2 t w L ∗ · L ∗ · Observe that (Z (w, )) = µN and (V (v, w, )) = µN. Owing to technical reasons, we need to α t w α t v introduce some more notations. Notation 4.3. We consider an α-random variable Y with uniform distribution on B(0, 1) (indepen- ∈ ≥ ∈ ∈ R ∈ R dent of everything else) and, for (cid:15) (0, 1), t 0, α [0, 1], v ( 3)N and w ( 3)N• , we set ∗ ∗ ∗ ∗ L ∗ ∗ W ,(cid:15)(α) = W (α) + (cid:15)Y (α) and V ,(cid:15)(v, w, α) = V (v, w, α) + (cid:15)Y (α). It holds that (W ,(cid:15)) = f ψ t L ∗ t · ∗ t t α t t (cid:15) and (V ,(cid:15)(v, w, )) = µN ψ , where ψ (x) = (3/(4π(cid:15)3))1{| |≤ }. α t v (cid:15) (cid:15) x (cid:15) At last, we built a suitable realisation for the particle system. Lemma 4.4. Consider all the objects introduced in Lemmas 4.1-4.2 and Notation 4.3. Set W = R ≥ ≥ s (W1, ..., WN), which a.s. belongs to ( 3)N• (because f has a density for all s 0). Fix K 1 and ∈s s(cid:90) (cid:90) (cid:90) (cid:90) s (cid:15) (0, 1). There is a unique strong solution (V ) ≥ = (V 1, ..., V N) ≥ to t t 0 t t t 0 ∞ t 1 2π ∗ V i = V i+ c (V i−, V (V −, W −, α), z, ϕ+ϕ )M (ds, dα, dz, dϕ), i = 1, ..., N, (4.4) t 0 K s s s s i,α,s i 0 0 0 0 (cid:0) (cid:1) where ϕ := ϕ1 + ϕ2 + ϕ3 with i,α,s i,α,s i,α,s i,α,s (cid:0) (cid:1) − ∗ − ∗ ϕ1i,α,s =ϕ0(cid:0)Wsi− Ws (α), Wsi− Ws ,(cid:15)(α) , (cid:1) − ∗ − ∗ ϕ2 =ϕ Wi− W ,(cid:15)(α), V i− V ,(cid:15)(V −, W −, α) , i,α,s 0 s s s s s s − ∗ − ∗ ϕ3 =ϕ V i− V ,(cid:15)(V −, W −, α), V i− V (V −, W −, α) . (cid:2) (cid:3) i,α,s 0 s s s s s s s s L ∈ P R ≥ Moreover, (V ) ≥ is a Markov process with generator . If f ( 3) for some q 2, then E | t| t ≤0 N,K 0 ∈q sup V 1 q C (this last constant not depending on N, K nor (cid:15) (0, 1)). [0,T] t T,q ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Liping

text: Xu

text: Due

text: to

text: and

text: c(v,

text: v∗,

text: z,

text: ϕ)

text: v

text: v∗

text: c(v,

text: v∗,

text: z,

text: ϕ)

text: z/

text: v

text: v∗

text: γ)

text: v

text: v∗

text: whence

text: v

text: v∗,

text: z,

text: ϕ)

text: q

text: v

text: q

text: 2π

text: C

text: v

text: q

text: v∗

text: q

text: z/

text: v

text: v∗

text: γ)

text: v

text: v∗

text: dϕdz

text: C

text: v

text: q

text: v∗

text: q

text: v

text: v∗

text: q

text: C

text: v

text: q

text: v∗

text: q

text: q

text: because

text: It

text: easily

text: follows

text: formula

text: and

text: L

text: f

text: that

text: α

text: t

text: t

text: t

text: E

text: sup

text: E[|V

text: C

text: t

text: E

text: dαds

text: s

text: q

text: s

text: s

text: E

text: t

text: E

text: V

text: q]

text: C

text: [sup

text: q]

text: ds.

text: q

text: u

text: We

text: thus

text: conclude

text: by

text: the

text: Gr¨onwall

text: lemma.

text: Next,

text: let

text: us

text: recall

text: Lemma

text: below

text: in

text: order

text: to

text: construct

text: our

text: coupling.

text: Lemma

text: Consider

text: (f

text: and

text: (W

text: introduced

text: in

text: Lemma

text: and

text: ﬁx

text: N

text: For

text: v

text: (v

text: v

text: v

text: thetemt

text: measure

text: µN

text: N

text: δ

text: for

text: all

text: t

text: v

text: 2∈

text: and

text: all

text: w

text: with

text: {w

text: w

text: w

text: vji},

text: there

text: are

text: α-random

text: i

text: j

text: variables

text: Z

text: (w,

text: α)

text: and

text: V

text: (v,

text: w,

text: α)

text: such

text: that

text: the

text: α-law

text: of

text: (Z

text: (w,

text: V

text: (v,

text: w,

text: is

text: N

text: N

text: δ

text: and

text: −t

text: t

text: W

text: t

text: t

text: (wi,vi)

text: W

text: (α)

text: Z

text: (w,

text: α)

text: µN).

text: t

text: t

text: t

text: w

text: L

text: L

text: Observe

text: that

text: (Z

text: (w,

text: µN

text: and

text: (V

text: (v,

text: w,

text: µN.

text: Owing

text: to

text: technical

text: reasons,

text: we

text: need

text: to

text: α

text: t

text: w

text: α

text: t

text: v

text: introduce

text: some

text: more

text: notations.

text: Notation

text: We

text: consider

text: an

text: α-random

text: variable

text: Y

text: with

text: uniform

text: distribution

text: on

text: (indepen-

text: R

text: R

text: dent

text: of

text: everything

text: else)

text: and,

text: for

text: t

text: α

text: v

text: and

text: w

text: we

text: set

text: L

text: W

text: W

text: (α)

text: (α)

text: and

text: V

text: w,

text: α)

text: V

text: (v,

text: w,

text: α)

text: (α).

text: It

text: holds

text: that

text: (W

text: f

text: ψ

text: t

text: L

text: t

text: t

text: t

text: α

text: t

text: t

text: and

text: (V

text: w,

text: µN

text: ψ

text: where

text: ψ

text: (x)

text: α

text: t

text: v

text: x

text: At

text: last,

text: we

text: built

text: a

text: suitable

text: realisation

text: for

text: the

text: particle

text: system.

text: Lemma

text: Consider

text: all

text: the

text: objects

text: introduced

text: in

text: Lemmas

text: and

text: Notation

text: Set

text: W

text: R

text: s

text: WN),

text: which

text: a.s.

text: belongs

text: to

text: (because

text: f

text: has

text: a

text: density

text: for

text: all

text: s

text: Fix

text: K

text: and

text: ∈s

text: s

text: There

text: is

text: a

text: unique

text: strong

text: solution

text: (V

text: (V

text: V

text: N)

text: to

text: t

text: t

text: t

text: t

text: t

text: t

text: 2π

text: V

text: i

text: V

text: i+

text: c

text: (V

text: i−,

text: V

text: (V

text: W

text: α),

text: z,

text: ϕ+ϕ

text: )M

text: (ds,

text: dα,

text: dz,

text: dϕ),

text: i

text: N,

text: t

text: K

text: s

text: s

text: s

text: s

text: i,α,s

text: i

text: where

text: ϕ

text: ϕ1

text: ϕ2

text: ϕ3

text: with

text: i,α,s

text: i,α,s

text: i,α,s

text: i,α,s

text: Ws

text: (α),

text: Wsi−

text: Ws

text: ϕ2

text: =ϕ

text: Wi−

text: W

text: V

text: i−

text: V

text: W

text: α)

text: i,α,s

text: s

text: s

text: s

text: s

text: s

text: s

text: ϕ3

text: =ϕ

text: V

text: i−

text: V

text: W

text: α),

text: V

text: i−

text: V

text: (V

text: W

text: α)

text: i,α,s

text: s

text: s

text: s

text: s

text: s

text: s

text: s

text: s

text: L

text: P

text: R

text: Moreover,

text: (V

text: is

text: a

text: Markov

text: process

text: with

text: generator

text: If

text: f

text: for

text: some

text: q

text: then

text: E

text: t|

text: t

text: ≤0

text: N,K

text: ∈q

text: sup

text: V

text: q

text: C

text: (this

text: last

text: constant

text: not

text: depending

text: on

text: N,

text: K

text: nor

text: t

text: T,q

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Uniqueness and The Nanbu Particle System for Soft Potentials 23 (cid:0) (cid:1) − ≥ Z(1(k+1)N−2)∧τN Z(1kN(cid:88)−2)∧τN(cid:104)(cid:16) (cid:15)N/3. Hence, (cid:17) (cid:105) (cid:98) (cid:99) N2T p˜ ≤ (cid:88)P Z(1(k+1)N(cid:104) −2)∧(cid:110)τN − Z(1(cid:16)kN−2)∧τN ≥ (cid:15)3N (cid:17)(cid:111)(cid:105) k=0 (cid:98) (cid:99) N2T ≤ e−Nδ/3E exp 3N1/3 Z1 − ∧ − Z1 − ∧ (cid:88) ((k+1)N 2) τN (kN 2) τN k=0 (cid:98) (cid:99) N2T − =: e Nδ/3I . k k=0 (cid:104) (cid:110) (cid:16) (cid:17)(cid:111)(cid:105) Step 2. We now prove that I is (uniformly) bounded, which will complete the proof. We put k E − Jk(t) =: exp 3N1/3 Z(1t+kN−2)∧τN Z(1kN−2)∧τN . (cid:20) (cid:90) − (cid:90) (cid:90) (cid:110) (cid:16) (cid:17)(cid:111) It is obvious that I = J (N 2). Applying the Itˆo formula, we ﬁnd k k − ∧ ∞ (cid:16) E (cid:0)(t+kN 2) τN (cid:1)1 (cid:17) − (cid:21) Jk(t) = 1 + 2π − ∧ exp 3N1/3 Zs1(cid:0) Z(1kN−2)∧τN(cid:1) (kN 2) τN 0 0 (cid:110) (cid:111) × | − ∗ | | − ∗ | − e(cid:0)3N1/3G z/ Ws1 Ws (α(cid:1)) γ Ws1 Ws (α) 1 1 | − ∗ | | − ∗ | ≤ − dzdαds . G z/ W1 W (α) γ W1 W (α) /4 N 1/3 s s s s (cid:0) (cid:1) | − ∗ | | − ∗ | ≤ (cid:0) (cid:1) Since 3N1/3G z/ W1 W (α) γ W1 W (α) 12 (thanks to the indicator function), we have s s s s | − ∗ | | − ∗ | − ≤ | − ∗ | | − ∗ | e3N1/3G z/ Ws1 Ws (α) γ Ws1 Ws (α) 1 CN1/3G z/ W1 W (α) γ W1 W (α) s s s s (cid:0) (cid:1) for a positive constant C. Then using (1.5), we see that (cid:110) (cid:111) ≤ { } 1 G z/|Ws1−Ws∗(α)|γ |Ws1−Ws∗(α)|/4≤N−1/3 1 z≥CNν/3|Ws1−Ws∗(α)|γ+ν−|Ws1−Ws∗(α)|γ . (cid:20) (cid:90) (cid:90) (cid:90) (cid:110) (cid:16) (cid:17)(cid:111) Hence, − ∧ ∞ ≤ (cid:16) E (t+kN 2) (cid:17)τN 1 − (cid:21) Jk(t) 1 + CN1/3 − ∧ exp 3N1/3 Zs1 Z(1kN−2)∧τN (kN 2) τN 0 0 − × | − ∗ | 1/ν| − ∗ | 1 + z/ W1 W (α) γ W1 W (α) 1{ ≥ | − ∗ | −| − ∗ | }dzdαds . s s s s z CNν/3 W1 W (α) γ+ν W1 W (α) γ s s s s (cid:90) (cid:16) (cid:17) But, we have ∞ − | − ∗ | | − ∗ | 1/ν W1 W (α) 1 + z/ W1 W (α) γ 1{ ≥ | − ∗ | −| − ∗ | }dz s s s s z CNν/3 W1 W (α) γ+ν W1 W (α) γ 0 s s s s − − | − ∗ | =CN (1 ν)/3 W1 W (α) ν+γ s s ≤ − − | | | ∗(cid:82) | CN (1 ν)/3(1 + W1 2 + W (α) 2) s s ∈ | ∗ | | | ≤ ≤ since γ + ν (0, 1). Using now that 1 W (α) 2dα = m (f ) and(cid:90)that W1 Nδ/3 for all s τ , we 0 s 2 0 s N conclude that ≤ (cid:90) t J (t) 1 + CNν/3(1 + m (f ) + N2δ/3) J (s)ds k 2 0 k 0 ≤ t 1 + CN(ν+2δ)/3 J (s)ds. k 0 ≤ − It follows from the Gr¨onwall lemma that J (t) exp (CN(ν+2δ)/3t), and thus that I = J (N 2) is k ∈ ∈ k k uniformly bounded, because (ν + 2δ)/3 < 2 (recall that ν (0, 1) and δ (0, 1)). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Uniqueness

label: paragraph

text: and

text: The

text: Nanbu

text: Particle

text: System

text: for

text: Soft

text: Potentials

text: Hence,

text: p˜

text: exp

text: τN

text: (kN

text: τN

text: e

text: k

text: Step

text: We

text: now

text: prove

text: that

text: I

text: is

text: (uniformly)

text: bounded,

text: which

text: will

text: complete

text: the

text: proof.

text: We

text: put

text: k

text: E

text: Jk(t)

text: exp

text: It

text: is

text: obvious

text: that

text: I

text: J

text: (N

text: Applying

text: the

text: Itˆo

text: formula,

text: we

text: ﬁnd

text: k

text: k

text: E

text: τN

text: Jk(t)

text: 2π

text: exp

text: (kN

text: τN

text: z/

text: Ws

text: γ

text: Ws

text: (α)

text: dzdαds

text: G

text: z/

text: W

text: (α)

text: γ

text: W

text: (α)

text: N

text: s

text: s

text: s

text: s

text: Since

text: z/

text: W

text: (α)

text: γ

text: W

text: (α)

text: (thanks

text: to

text: the

text: indicator

text: function),

text: we

text: have

text: s

text: s

text: s

text: s

text: z/

text: Ws

text: (α)

text: γ

text: Ws

text: (α)

text: z/

text: W

text: (α)

text: γ

text: W

text: (α)

text: s

text: s

text: s

text: s

text: for

text: a

text: positive

text: constant

text: C.

text: Then

text: using

text: we

text: see

text: that

text: G

text: Hence,

text: E

text: (t+kN

text: Jk(t)

text: exp

text: (kN

text: τN

text: z/

text: W

text: (α)

text: γ

text: W

text: (α)

text: }dzdαds

text: s

text: s

text: s

text: s

text: z

text: W

text: (α)

text: γ+ν

text: W

text: (α)

text: γ

text: s

text: s

text: s

text: s

text: But,

text: we

text: have

text: W

text: (α)

text: z/

text: W

text: (α)

text: γ

text: }dz

text: s

text: s

text: s

text: s

text: z

text: W

text: (α)

text: γ+ν

text: W

text: (α)

text: γ

text: s

text: s

text: s

text: s

text: =CN

text: W

text: (α)

text: ν+γ

text: s

text: s

text: CN

text: W

text: (α)

text: s

text: s

text: since

text: γ

text: ν

text: Using

text: now

text: that

text: W

text: (α)

text: m

text: (f

text: for

text: all

text: s

text: τ

text: we

text: s

text: s

text: N

text: conclude

text: that

text: t

text: J

text: (t)

text: m

text: (f

text: J

text: (s)ds

text: k

text: k

text: t

text: J

text: (s)ds.

text: k

text: It

text: follows

text: from

text: the

text: Gr¨onwall

text: lemma

text: that

text: J

text: (t)

text: exp

text: and

text: thus

text: that

text: I

text: J

text: (N

text: is

text: k

text: k

text: k

text: uniformly

text: bounded,

text: because

text: (ν

text: (recall

text: that

text: ν

text: and

text: δ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 34 Liping Xu [7] N. Fournier. Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition. Ann. Appl. Probab., 25(2):860–897, 2015. [8] N. Fournier and H. Gu´erin. On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity. J. Stat. Phys., 131(4):749–781, 2008. [9] N. Fournier and A. Guillin. On the rate of convergence in Wasserstein distance of the empirical measure. Probab. Theory Related Fields, 162(3-4):707–738, 2015. [10] N. Fournier and M. Hauray. Propagation of chaos for the Landau equation with moderately soft potentials. To appear in Ann. Probab. arXiv:1512.01802, 2015. [11] N. Fournier, M. Hauray, and S. Mischler. Propagation of chaos for the 2D viscous vortex model. J. Eur. Math. Soc. (JEMS), 16(7):1423–1466, 2014. [12] N. Fournier and S. M´el´eard. A stochastic particle numerical method for 3D Boltzmann equations without cutoﬀ. Math. Comp., 71(238):583–604 (electronic), 2002. [13] N. Fournier and S. Mischler. Rate of convergence of the Nanbu particle system for hard potentials and Maxwell molecules. Ann. Probab., 44(1):589–627, 2016. [14] N. Fournier and C. Mouhot. On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity. Comm. Math. Phys., 289(3):803–824, 2009. [15] D. Godinho and C. Quin˜inao. Propagation of chaos for a subcritical Keller-Segel model. Ann. Inst. Henri Poincar´e Probab. Stat., 51(3):965–992, 2015. [16] C. Graham and S. M´el´eard. Stochastic particle approximations for generalized Boltzmann models and convergence estimates. Ann. Probab., 25(1):115–132, 1997. [17] F. A. Gru¨nbaum. Propagation of chaos for the Boltzmann equation. Arch. Rational Mech. Anal., 42:323–345, 1971. [18] M. Hauray and P.-E. Jabin. Particle approximation of Vlasov equations with singular forces: prop- agation of chaos. Ann. Sci. E´c. Norm. Sup´er. (4), 48(4):891–940, 2015. [19] J. Horowitz and R. L. Karandikar. Martingale problems associated with the Boltzmann equation. In Seminar on Stochastic Processes, 1989 (San Diego, CA, 1989), volume 18 of Progr. Probab., pages 75–122. Birkh¨auser Boston, Boston, MA, 1990. [20] J. Jacod and A. N. Shiryaev. Limit theorems for stochastic processes, volume 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer- Verlag, Berlin, second edition, 2003. [21] M. Kac. Foundations of kinetic theory. In Proceedings of the Third Berkeley Symposium on Mathe- matical Statistics and Probability, 1954–1955, vol. III, pages 171–197. University of California Press, Berkeley and Los Angeles, 1956. [22] X. Lu and C. Mouhot. On measure solutions of the Boltzmann equation, part I: moment production and stability estimates. J. Diﬀerential Equations, 252(4):3305–3363, 2012. [23] H. P. McKean, Jr. An exponential formula for solving Boltmann’s equation for a Maxwellian gas. J. Combinatorial Theory, 2:358–382, 1967. [24] S. Mischler and C. Mouhot. Kac’s program in kinetic theory. Invent. Math., 193(1):1–147, 2013. [25] S. Mischler and B. Wennberg. On the spatially homogeneous Boltzmann equation. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 16(4):467–501, 1999. [26] K. Nanbu. Interrelations between various direct simulation methods for solving the Boltzmann equation. Journal of the Physical Society of Japan, 52(10):3382–3388, 1983.

text: Liping

text: Xu

label: reference

text: N.

text: Fournier.

text: Finiteness

text: of

text: entropy

text: for

text: the

text: homogeneous

text: Boltzmann

text: equation

text: with

text: measure

text: initial

text: condition.

text: Ann.

text: Appl.

text: Probab.,

text: N.

text: Fournier

text: and

text: H.

text: Gu´erin.

text: On

text: the

text: uniqueness

text: for

text: the

text: spatially

text: homogeneous

text: Boltzmann

text: equation

text: with

text: a

text: strong

text: angular

text: singularity.

text: J.

text: Stat.

text: Phys.,

text: N.

text: Fournier

text: and

text: A.

text: Guillin.

text: On

text: the

text: rate

text: of

text: convergence

text: in

text: Wasserstein

text: distance

text: of

text: the

text: empirical

text: measure.

text: Probab.

text: Theory

text: Related

text: Fields,

text: N.

text: Fournier

text: and

text: M.

text: Hauray.

text: Propagation

text: of

text: chaos

text: for

text: the

text: Landau

text: equation

text: with

text: moderately

text: soft

text: potentials.

text: To

text: appear

text: in

text: Ann.

text: Probab.

text: N.

text: Fournier,

text: M.

text: Hauray,

text: and

text: S.

text: Mischler.

text: Propagation

text: of

text: chaos

text: for

text: the

text: viscous

text: vortex

text: model.

text: J.

text: Eur.

text: Math.

text: Soc.

text: (JEMS),

text: N.

text: Fournier

text: and

text: S.

text: M´el´eard.

text: A

text: stochastic

text: particle

text: numerical

text: method

text: for

text: Boltzmann

text: equations

text: without

text: cutoﬀ.

text: Math.

text: Comp.,

text: (electronic),

text: N.

text: Fournier

text: and

text: S.

text: Mischler.

text: Rate

text: of

text: convergence

text: of

text: the

text: Nanbu

text: particle

text: system

text: for

text: hard

text: potentials

text: and

text: Maxwell

text: molecules.

text: Ann.

text: Probab.,

text: N.

text: Fournier

text: and

text: C.

text: Mouhot.

text: On

text: the

text: well-posedness

text: of

text: the

text: spatially

text: homogeneous

text: Boltzmann

text: equation

text: with

text: a

text: moderate

text: angular

text: singularity.

text: Comm.

text: Math.

text: Phys.,

text: D.

text: Godinho

text: and

text: C.

text: Quin˜inao.

text: Propagation

text: of

text: chaos

text: for

text: a

text: subcritical

text: Keller-Segel

text: model.

text: Ann.

text: Inst.

text: Henri

text: Poincar´e

text: Probab.

text: Stat.,

text: C.

text: Graham

text: and

text: S.

text: M´el´eard.

text: Stochastic

text: particle

text: approximations

text: for

text: generalized

text: Boltzmann

text: models

text: and

text: convergence

text: estimates.

text: Ann.

text: Probab.,

text: F.

text: A.

text: Gru¨nbaum.

text: Propagation

text: of

text: chaos

text: for

text: the

text: Boltzmann

text: equation.

text: Arch.

text: Rational

text: Mech.

text: Anal.,

text: M.

text: Hauray

text: and

text: P.-E.

text: Jabin.

text: Particle

text: approximation

text: of

text: Vlasov

text: equations

text: with

text: singular

text: forces:

text: prop-

text: agation

text: of

text: chaos.

text: Ann.

text: Sci.

text: E´c.

text: Norm.

text: Sup´er.

text: J.

text: Horowitz

text: and

text: R.

text: L.

text: Karandikar.

text: Martingale

text: problems

text: associated

text: with

text: the

text: Boltzmann

text: equation.

text: In

text: Seminar

text: on

text: Stochastic

text: Processes,

text: (San

text: Diego,

text: CA,

text: volume

text: of

text: Progr.

text: Probab.,

text: pages

text: Birkh¨auser

text: Boston,

text: Boston,

text: MA,

text: J.

text: Jacod

text: and

text: A.

text: N.

text: Shiryaev.

text: Limit

text: theorems

text: for

text: stochastic

text: processes,

text: volume

text: of

text: Grundlehren

text: der

text: Mathematischen

text: Wissenschaften

text: [Fundamental

text: Principles

text: of

text: Mathematical

text: Sciences].

text: Springer-

text: Verlag,

text: Berlin,

text: second

text: edition,

text: M.

text: Kac.

text: Foundations

text: of

text: kinetic

text: theory.

text: In

text: Proceedings

text: of

text: the

text: Third

text: Berkeley

text: Symposium

text: on

text: Mathe-

text: matical

text: Statistics

text: and

text: Probability,

text: vol.

text: III,

text: pages

text: University

text: of

text: California

text: Press,

text: Berkeley

text: and

text: Los

text: Angeles,

text: X.

text: Lu

text: and

text: C.

text: Mouhot.

text: On

text: measure

text: solutions

text: of

text: the

text: Boltzmann

text: equation,

text: part

text: I:

text: moment

text: production

text: and

text: stability

text: estimates.

text: J.

text: Diﬀerential

text: Equations,

text: H.

text: P.

text: McKean,

text: Jr.

text: An

text: exponential

text: formula

text: for

text: solving

text: Boltmann’s

text: equation

text: for

text: a

text: Maxwellian

text: gas.

text: J.

text: Combinatorial

text: Theory,

text: S.

text: Mischler

text: and

text: C.

text: Mouhot.

text: Kac’s

text: program

text: in

text: kinetic

text: theory.

text: Invent.

text: Math.,

text: S.

text: Mischler

text: and

text: B.

text: Wennberg.

text: On

text: the

text: spatially

text: homogeneous

text: Boltzmann

text: equation.

text: Ann.

text: Inst.

text: H.

text: Poincar´e

text: Anal.

text: Non

text: Lin´eaire,

text: K.

text: Nanbu.

text: Interrelations

text: between

text: various

text: direct

text: simulation

text: methods

text: for

text: solving

text: the

text: Boltzmann

text: equation.

text: Journal

text: of

text: the

text: Physical

text: Society

text: of

text: Japan,


text: Uniqueness and The Nanbu Particle System for Soft Potentials 9 situation, see also [6, Theorem 2.6] (both concerning diﬀusion-type PDEs and Brownian SDEs). We will (cid:0) (cid:1) prove it later. ∈ − ∈ Proposition 3.1. Assume (1.3) for some γ ( 1, 0), ν (0, 1) with γ + ν > 0. Consider any weak solution (f˜) ≥ ∈ L∞ [0, ∞), P (R3) to (1.1). Then there exists, on some probability space, a random t t 0 2 ∞ × × ∞ × variable X with law f˜ , independent of a Poisson measure M(ds, dα, dz, dϕ) on [0, ) [0, 1] [0, ) 0 0 ∗ [0, 2π) with intensity dsdαdzdϕ, a measurable family (X ) ≥ of α-random variables and a c`adl`ag adapted (cid:90) (cid:90) (cid:90) (cid:90) t t 0 process (X ) ≥ solving (cid:0) (cid:1) t t 0 ∞ t 1 2π ∗ X = X + c X −, X (α), z, ϕ M(ds, dα, dz, dϕ) (3.1) t 0 s s 0 0 0 0 ≥ L L ∗ and such that for all t 0, (X ) = (X ) = f˜. t α t t The second one is the following proposition. (cid:0) (cid:1) (cid:0) ∈ − (cid:1) ∈ ∈ P R Proposition 3.2. Assume (1.3) for some γ ( 1, 0), ν (0, 1) with γ + ν > 0, that f ( 3) for ≥ ∈ 0 q some q 2 such that q > γ2/(γ + ν) and that f has a ﬁnite entropy. Fix p (3/(3 + γ), p (γ, ν, q)). Let ∈ ∞ ∞ P R ∩ ∞ 0R (cid:0) (cid:1)0 (f ) ≥ L [0, ), ( 3) L1 [0, ), Lp( 3) be the corresponding unique weak solution to (1.1) t t 0 2 loc ∗ given by Theorem 1.2. Consider also the Poisson measure M, the process (X ) ≥ and the family (X ) ≥ built in Proposition 3.1 (associated to another weak solution (f˜) ≥ ∈ L∞ [0t,t∞0), P (R3) . Let W t∼tf0 (independent of M) be such that E[|W − X |2] = W2(f , f˜ ) atndt, 0for each t ≥ 0, a2n α-random va0riable0 W∗ such that L (W∗) = f(cid:90) an(cid:90)d E(cid:90) [|W(cid:90)0∗ − X0∗|2] = W2 2(f0 , f0˜). Then for K ≥ 1, the equation t α t t α t t 2 t t ∞ t 1 2π ∗ WK = W + c (WK−, W (α), z, ϕ + ϕ )M(ds, dα, dz, dϕ), (3.2) t 0 K s s s,α,K 0 0 0 0 − ∗ − ∗ L with ϕ = ϕ (X − X (α), WK− W (α)), has a unique solution. Moreover, setting fK = (WK) s,α,K≥ 0 s s s s t t for each t 0, it holds that for all T > 0, W lim sup 2(fK, f ) = 0. (3.3) →∞ 2 t t K [0,T] ≥ Proof. For any K 1, the Poisson measure involved in (3.2) is actually ﬁnite (because c = c1{ ≤ }), K z K so the existence and uniqueness for this equation is obvious. It only remains to prove (3.3), which has already been done in [8, Lemma 4.2], where the formulation of the equation is slightly diﬀerent. But one easily checks that (WK) ≥ is a (time-inhomogeneous) Markov process with the same generator as the (cid:90) (cid:90) (cid:90) t(cid:104) t 0 R (cid:55)→ R (cid:105) ≥ one deﬁned by [8, Eq. (4.1)], because for all bounded measurable function φ : 3 and all t 0, a.s., ∞ (cid:90) 1 (cid:90) (cid:90) 2π (cid:104) ∗ (cid:105) − ∗ − ∗ − φ(w + c (w, W (α), z, ϕ + ϕ (X − X (α), w W (α))) φ(w) dϕdzdα K t 0 t t t 0 0 0 ∞ 1 2π − = φ(w + c (w, v, z, ϕ)) φ(w) dϕdzf (dv) K t 0 0 0 L ∗ by the 2π-periodicity of c (in ϕ) and since (W ) = f . K α t t Now, we use these coupled processes to conclude the (cid:0) (cid:1) Proof of Theorem 1.3. We consider a weak solution (f˜) ≥ to (1.1), with which we associate the ob- (cid:0) (cid:1) ∗ t t 0 jects M, (X ) ≥ , (X ) ≥ as in Proposition 3.1. We then consider f satisfying the assumptions of t t 0 t t 0 0 ∞ ∞ P R ∩ Theorem 1.2 and the corresponding unique weak solution (f ) ≥ belonging to L [0, ), ( 3) ∞ R ∈ t t 0 ∗ 2 L1 [0, ), Lp( 3) (with p (3/(3+γ), p (γ, ν, q))) and we consider (WK) ≥ , (W ) ≥ built in Propo- sitloicon 3.2 for any K ≥ 1. We know that W20(f , f˜ ) = E[|W −X |2] and thtat Wt 02(f , ft˜)t=0 E [|W∗−X∗|2] ≥ ∼ 2 ∼0 0 0≥ 0 2 t t α t ≥t for all t 0. Using that WK fK and X f˜ for each t 0, we deduce from (3.3) that for all t 0, t t t t W2(f , f˜) ≤ lim sup E[|WK − X |2] =: J . (3.4) 2 t t →∞ t t t K ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Uniqueness

text: and

text: The

text: Nanbu

text: Particle

text: System

text: for

text: Soft

text: Potentials

text: situation,

text: see

text: also

text: Theorem

text: (both

text: concerning

text: diﬀusion-type

text: PDEs

text: and

text: Brownian

text: SDEs).

text: We

text: will

text: prove

text: it

text: later.

text: Proposition

text: Assume

text: for

text: some

text: γ

text: ν

text: with

text: γ

text: ν

text: Consider

text: any

text: weak

text: solution

text: (f˜)

text: L∞

text: P

text: to

text: Then

text: there

text: exists,

text: on

text: some

text: probability

text: space,

text: a

text: random

text: t

text: t

text: variable

text: X

text: with

text: law

text: f˜

text: independent

text: of

text: a

text: Poisson

text: measure

text: M(ds,

text: dα,

text: dz,

text: dϕ)

text: on

text: with

text: intensity

text: dsdαdzdϕ,

text: a

text: measurable

text: family

text: (X

text: of

text: α-random

text: variables

text: and

text: a

text: c`adl`ag

text: adapted

text: t

text: t

text: process

text: (X

text: solving

text: t

text: t

text: t

text: 2π

text: X

text: X

text: c

text: X

text: X

text: (α),

text: z,

text: ϕ

text: M(ds,

text: dα,

text: dz,

text: dϕ)

text: t

text: s

text: s

text: L

text: L

text: and

text: such

text: that

text: for

text: all

text: t

text: (X

text: (X

text: f˜.

text: t

text: α

text: t

text: t

text: The

text: second

text: one

text: is

text: the

text: following

text: proposition.

text: P

text: R

text: Proposition

text: Assume

text: for

text: some

text: γ

text: ν

text: with

text: γ

text: ν

text: that

text: f

text: for

text: q

text: some

text: q

text: such

text: that

text: q

text: ν)

text: and

text: that

text: f

text: has

text: a

text: ﬁnite

text: entropy.

text: Fix

text: p

text: γ),

text: p

text: (γ,

text: ν,

text: q)).

text: Let

text: P

text: R

text: (f

text: L

text: Lp(

text: be

text: the

text: corresponding

text: unique

text: weak

text: solution

text: to

text: t

text: t

text: loc

text: given

text: by

text: Theorem

text: Consider

text: also

text: the

text: Poisson

text: measure

text: M,

text: the

text: process

text: (X

text: and

text: the

text: family

text: (X

text: built

text: in

text: Proposition

text: (associated

text: to

text: another

text: weak

text: solution

text: (f˜)

text: L∞

text: P

text: Let

text: W

text: (independent

text: of

text: M)

text: be

text: such

text: that

text: E[|W

text: X

text: f˜

text: atndt,

text: each

text: t

text: α-random

text: W∗

text: such

text: that

text: L

text: (W∗)

text: Then

text: for

text: K

text: the

text: equation

text: t

text: α

text: t

text: t

text: α

text: t

text: t

text: t

text: t

text: t

text: 2π

text: WK

text: W

text: c

text: (WK−,

text: W

text: (α),

text: z,

text: ϕ

text: ϕ

text: )M(ds,

text: dα,

text: dz,

text: dϕ),

text: t

text: K

text: s

text: s

text: s,α,K

text: L

text: with

text: ϕ

text: ϕ

text: (X

text: X

text: (α),

text: WK−

text: W

text: (α)),

text: has

text: a

text: unique

text: solution.

text: Moreover,

text: setting

text: fK

text: (WK)

text: s,α,K≥

text: s

text: s

text: s

text: s

text: t

text: t

text: for

text: each

text: t

text: it

text: holds

text: that

text: for

text: all

text: T

text: W

text: lim

text: sup

text: f

text: t

text: t

text: K

text: Proof.

text: For

text: any

text: K

text: the

text: Poisson

text: measure

text: involved

text: in

text: is

text: actually

text: ﬁnite

text: (because

text: c

text: K

text: z

text: K

text: so

text: the

text: existence

text: and

text: uniqueness

text: for

text: this

text: equation

text: is

text: obvious.

text: It

text: only

text: remains

text: to

text: prove

text: which

text: has

text: already

text: been

text: done

text: in

text: Lemma

text: where

text: the

text: formulation

text: of

text: the

text: equation

text: is

text: slightly

text: diﬀerent.

text: But

text: one

text: easily

text: checks

text: that

text: (WK)

text: is

text: a

text: (time-inhomogeneous)

text: Markov

text: process

text: with

text: the

text: same

text: generator

text: as

text: the

text: t

text: R

text: R

text: one

text: deﬁned

text: by

text: Eq.

text: because

text: for

text: all

text: bounded

text: measurable

text: function

text: φ

text: and

text: all

text: t

text: a.s.,

text: 2π

text: φ(w

text: c

text: (w,

text: W

text: (α),

text: z,

text: ϕ

text: ϕ

text: (X

text: X

text: (α),

text: w

text: W

text: (α)))

text: φ(w)

text: dϕdzdα

text: K

text: t

text: t

text: t

text: t

text: 2π

text: φ(w

text: c

text: (w,

text: v,

text: z,

text: ϕ))

text: φ(w)

text: dϕdzf

text: (dv)

text: K

text: t

text: L

text: by

text: the

text: of

text: c

text: (in

text: ϕ)

text: and

text: since

text: (W

text: f

text: K

text: α

text: t

text: t

text: Now,

text: we

text: use

text: these

text: coupled

text: processes

text: to

text: conclude

text: the

text: Proof

text: of

text: Theorem

text: We

text: consider

text: a

text: weak

text: solution

text: (f˜)

text: to

text: with

text: which

text: we

text: associate

text: the

text: ob-

text: t

text: t

text: jects

text: M,

text: (X

text: (X

text: as

text: in

text: Proposition

text: We

text: then

text: consider

text: f

text: satisfying

text: the

text: assumptions

text: of

text: t

text: t

text: t

text: t

text: P

text: R

text: Theorem

text: and

text: the

text: corresponding

text: unique

text: weak

text: solution

text: (f

text: belonging

text: to

text: L

text: R

text: t

text: t

text: Lp(

text: (with

text: p

text: p

text: (γ,

text: ν,

text: q)))

text: and

text: we

text: consider

text: (WK)

text: (W

text: built

text: in

text: Propo-

text: sitloicon

text: for

text: any

text: K

text: We

text: know

text: that

text: f˜

text: E[|W

text: −X

text: and

text: thtat

text: Wt

text: E

text: ∼0

text: 0≥

text: t

text: t

text: α

text: t

text: ≥t

text: for

text: all

text: t

text: Using

text: that

text: WK

text: fK

text: and

text: X

text: f˜

text: for

text: each

text: t

text: we

text: deduce

text: from

text: that

text: for

text: all

text: t

text: t

text: t

text: t

text: t

text: f˜)

text: lim

text: sup

text: E[|WK

text: X

text: J

text: t

text: t

text: t

text: t

text: t

text: K

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 (a) E E (b) y inc refl θK (a) (b) 5 7 T 5 B E tot j e3 4.5 4.5 B z 6 T j E x h 4 4 tot j e1 j y e2 5 T e 3.5 3.5 1 e 3 x 3 3 e2 j e1+ j e2+ j e3 = + 4 T 2.5 2.5 FIG. 1. (a) Schematic representation of the interaction of 3 T charge carriers in bismuth with light. The Fermi surface of 2 2 bismuth is sketched in the interior of the sample, with a hole pocket (red) oriented along the trigonal (z) axis (inspired by 1.5 2 T 1.5 Ref. 18). Three electron pockets (green) are oriented almost ◦ perpendicularly to the trigonal axis (with a 6 tilt), rotated ◦ 1 1 by 120 with respect to each other. A magnetic ﬁeld is applied 1 T along the z-axis. Radiation with linear polarisation is prop- agating almost parallel to the ﬁeld. Close to the intraband 0.5 0.5 resonances the light reﬂects with an elliptical polarisation and 0 T is rotated by the Kerr angle. (b) Top view on the trigonal 0 0 5 10 15 5 10 15 plane. The eﬀective electric ﬁeld E in the crystal generates tot Energy (meV) Energy (meV) currents (both in real space) in the diﬀerent pockets (in re- ciprocal space) according to j = σE . The current in the tot hole pocket results in complete magnetic circular dichroism. FIG. 2. (a,b) The real part of the measured conductivity The sum of the current in the electron pockets shows strongly for right and left circularly polarised light, σ+ and σ−, as reduced dichroism due to the large eﬀective mass anisotropy a function of the photon energy. The magnetic ﬁeld steps (bottom). are 0.5 T and the same colour coding applies to both panels. Curves are displaced for clarity. The symbols at each peak correspond to Figure 3c,d. a freshly cleaved, untreated surface, which shows only atomic height variations over large parts of the surface and θ (ω), resulting in model-independent σ (ω) and K + (see Supplemental Material [33]). The measurements σ−(ω), which are shown in Fig. 2a and b for the energy were performed at 5 K with radiation at nearly normal ◦ range where intraband Landau-level transitions occur. angle incidence (8 ). The magnetic ﬁeld was applied The transition energies, which are marked on the peak along the trigonal axis and thus almost parallel to the positions in Fig. 2a,b, are plotted with the same sym- propagation of light (Faraday geometry). bols in Fig. 3c,d as a function of magnetic ﬁeld. Before The reﬂectivity R(ω, B) and Kerr angle θK(ω, B) were we can address the magnetic circular dichroism, we ﬁrst measured in a single measurement run using two polaris- need to identify the various transitions that occur. ers, one before and one after the sample. The absolute re- The hole band of the Brillouin zone in Bi is usually ﬂectivity was measured with a double reference method. considered to be parabolic and therefore the hole Lan- At every magnetic ﬁeld the spectrum on the sample and dau levels are equidistant in energy. Due to the inﬂu- on a gold reference mirror was taken. The measurements ence of outside bands and a strong spin-orbit coupling, were repeated after in-situ evaporation of a gold layer a Zeeman-like term needs to be added, resulting in the on the sample. The Kerr angle was measured using the Landau level energies for the hole band E as a function h fast protocol explained in Ref. 34. More details on the of magnetic ﬁeld B at k =(cid:18)0 give(cid:19)n by [18] z experiment are given in the Supplemental Material [33], together with plots of the measured R(ω) and θ (ω). K − 1 ¯heB We employ a generalised magneto-optical Kramers- E (n, s, B) = E + ∆ n + + sGµ B, (1) Kronig analysis (MOKKA) to calculate σ− and σ+ from h 0 2 Mc B the measurements [34]. R(ω) and θ (ω) are initially K ± ﬁtted using a sum of Lorentzians, which relies on com- where n = 0, 1, ... is the Landau level index, s = 1 2 plementary ellipsometry data for the high frequencies the spin quantum number, E is the energy separation 0 (not shown). Subsequently two independent Kramers- between the top of the hole valence band and the bottom Kronig-consistent variational functions are added to the of the electron conduction band (E = 38.5 meV [23]), e 0 initial ﬁt, and adjusted to simultaneously reproduce R(ω) the elementary charge, M the cyclotron mass, h¯ Planck’s c )1-mcS 4 01( σ + )1-mcS 4 01( σ - ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (a)

font: default

text: E

text: E

text: (b)

text: y

text: inc

text: refl

text: θK

text: (a)

text: (b)

text: T

text: B

text: E

text: tot

text: j

text: B

text: z

text: T

text: j

text: E

text: x

text: h

text: tot

text: j

text: j

text: y

text: T

text: e

text: e

text: x

text: j

text: j

text: j

text: T

text: FIG.

text: (a)

text: Schematic

text: representation

text: of

text: the

text: interaction

text: of

text: T

text: charge

text: carriers

text: in

text: bismuth

text: with

text: light.

text: The

text: Fermi

text: surface

text: of

text: bismuth

text: is

text: sketched

text: in

text: the

text: interior

text: of

text: the

text: sample,

text: with

text: a

text: hole

text: pocket

text: (red)

text: oriented

text: along

text: the

text: trigonal

text: (z)

text: axis

text: (inspired

text: by

text: T

text: Ref.

text: Three

text: electron

text: pockets

text: (green)

text: are

text: oriented

text: almost

text: perpendicularly

text: to

text: the

text: trigonal

text: axis

text: (with

text: a

text: tilt),

text: rotated

text: by

text: with

text: respect

text: to

text: each

text: other.

text: A

text: magnetic

text: ﬁeld

text: is

text: applied

text: T

text: along

text: the

text: z-axis.

text: Radiation

text: with

text: linear

text: polarisation

text: is

text: prop-

text: agating

text: almost

text: parallel

text: to

text: the

text: ﬁeld.

text: Close

text: to

text: the

text: intraband

text: resonances

text: the

text: light

text: reﬂects

text: with

text: an

text: elliptical

text: polarisation

text: and

text: T

text: is

text: rotated

text: by

text: the

text: Kerr

text: angle.

text: (b)

text: Top

text: view

text: on

text: the

text: trigonal

text: plane.

text: The

text: eﬀective

text: electric

text: ﬁeld

text: E

text: in

text: the

text: crystal

text: generates

text: tot

text: Energy

text: (meV)

text: Energy

text: (meV)

text: currents

text: (both

text: in

text: real

text: space)

text: in

text: the

text: diﬀerent

text: pockets

text: (in

text: re-

text: ciprocal

text: space)

text: according

text: to

text: j

text: σE

text: The

text: current

text: in

text: the

text: tot

text: hole

text: pocket

text: results

text: in

text: complete

text: magnetic

text: circular

text: dichroism.

text: FIG.

text: (a,b)

text: The

text: real

text: part

text: of

text: the

text: measured

text: conductivity

text: The

text: sum

text: of

text: the

text: current

text: in

text: the

text: electron

text: pockets

text: shows

text: strongly

text: for

text: right

text: and

text: left

text: circularly

text: polarised

text: light,

text: σ+

text: and

text: σ−,

text: as

text: reduced

text: dichroism

text: due

text: to

text: the

text: large

text: eﬀective

text: mass

text: anisotropy

text: a

text: function

text: of

text: the

text: photon

text: energy.

text: The

text: magnetic

text: ﬁeld

text: steps

text: (bottom).

text: are

text: T

text: and

text: the

text: same

text: colour

text: coding

text: applies

text: to

text: both

text: panels.

text: Curves

text: are

text: displaced

text: for

text: clarity.

text: The

text: symbols

text: at

text: each

text: peak

text: correspond

text: to

text: Figure

text: a

text: freshly

text: cleaved,

text: untreated

text: surface,

text: which

text: shows

text: only

text: atomic

text: height

text: variations

text: over

text: large

text: parts

text: of

text: the

text: surface

text: and

text: θ

text: (ω),

text: resulting

text: in

text: model-independent

text: σ

text: (ω)

text: and

text: K

text: (see

text: Supplemental

text: Material

text: The

text: measurements

text: σ−(ω),

text: which

text: are

text: shown

text: in

text: Fig.

text: and

text: b

text: for

text: the

text: energy

text: were

text: performed

text: at

text: K

text: with

text: radiation

text: at

text: nearly

text: normal

text: range

text: where

text: intraband

text: Landau-level

text: transitions

text: occur.

text: angle

text: incidence

text: The

text: magnetic

text: ﬁeld

text: was

text: applied

text: The

text: transition

text: energies,

text: which

text: are

text: marked

text: on

text: the

text: peak

text: along

text: the

text: trigonal

text: axis

text: and

text: thus

text: almost

text: parallel

text: to

text: the

text: positions

text: in

text: Fig.

text: are

text: plotted

text: with

text: the

text: same

text: sym-

text: propagation

text: of

text: light

text: (Faraday

text: geometry).

text: bols

text: in

text: Fig.

text: as

text: a

text: function

text: of

text: magnetic

text: ﬁeld.

text: Before

text: The

text: reﬂectivity

text: R(ω,

text: B)

text: and

text: Kerr

text: angle

text: θK(ω,

text: B)

text: were

text: we

text: can

text: address

text: the

text: magnetic

text: circular

text: dichroism,

text: we

text: ﬁrst

text: measured

text: in

text: a

text: single

text: measurement

text: run

text: using

text: two

text: polaris-

text: need

text: to

text: identify

text: the

text: various

text: transitions

text: that

text: occur.

text: ers,

text: one

text: before

text: and

text: one

text: after

text: the

text: sample.

text: The

text: absolute

text: re-

text: The

text: hole

text: band

text: of

text: the

text: Brillouin

text: zone

text: in

text: Bi

text: is

text: usually

text: ﬂectivity

text: was

text: measured

text: with

text: a

text: double

text: reference

text: method.

text: considered

text: to

text: be

text: parabolic

text: and

text: therefore

text: the

text: hole

text: Lan-

text: At

text: every

text: magnetic

text: ﬁeld

text: the

text: spectrum

text: on

text: the

text: sample

text: and

text: dau

text: levels

text: are

text: equidistant

text: in

text: energy.

text: Due

text: to

text: the

text: inﬂu-

text: on

text: a

text: gold

text: reference

text: mirror

text: was

text: taken.

text: The

text: measurements

text: ence

text: of

text: outside

text: bands

text: and

text: a

text: strong

text: spin-orbit

text: coupling,

text: were

text: repeated

text: after

text: in-situ

text: evaporation

text: of

text: a

text: gold

text: layer

text: a

text: Zeeman-like

text: term

text: needs

text: to

text: be

text: added,

text: resulting

text: in

text: the

text: on

text: the

text: sample.

text: The

text: Kerr

text: angle

text: was

text: measured

text: using

text: the

text: Landau

text: level

text: energies

text: for

text: the

text: hole

text: band

text: E

text: as

text: a

text: function

text: h

text: fast

text: protocol

text: explained

text: in

text: Ref.

text: More

text: details

text: on

text: the

text: of

text: magnetic

text: ﬁeld

text: B

text: at

text: k

text: by

text: z

text: experiment

text: are

text: given

text: in

text: the

text: Supplemental

text: Material

text: together

text: with

text: plots

text: of

text: the

text: measured

text: R(ω)

text: and

text: θ

text: (ω).

text: K

text: ¯heB

text: We

text: employ

text: a

text: generalised

text: magneto-optical

text: Kramers-

text: E

text: (n,

text: s,

text: B)

text: E

text: n

text: sGµ

text: B,

text: Kronig

text: analysis

text: (MOKKA)

text: to

text: calculate

text: σ−

text: and

text: σ+

text: from

text: h

text: Mc

text: B

text: the

text: measurements

text: R(ω)

text: and

text: θ

text: (ω)

text: are

text: initially

text: K

text: ﬁtted

text: using

text: a

text: sum

text: of

text: Lorentzians,

text: which

text: relies

text: on

text: com-

text: where

text: n

text: is

text: the

text: Landau

text: level

text: index,

text: s

text: plementary

text: ellipsometry

text: data

text: for

text: the

text: high

text: frequencies

text: the

text: spin

text: quantum

text: number,

text: E

text: is

text: the

text: energy

text: separation

text: (not

text: shown).

text: Subsequently

text: two

text: independent

text: Kramers-

text: between

text: the

text: top

text: of

text: the

text: hole

text: valence

text: band

text: and

text: the

text: bottom

text: Kronig-consistent

text: variational

text: functions

text: are

text: added

text: to

text: the

text: of

text: the

text: electron

text: conduction

text: band

text: (E

text: meV

text: e

text: initial

text: ﬁt,

text: and

text: adjusted

text: to

text: simultaneously

text: reproduce

text: R(ω)

text: the

text: elementary

text: charge,

text: M

text: the

text: cyclotron

text: mass,

text: h¯

text: Planck’s

text: c

text: σ

text: σ

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: End of a Dark Age? 6 [6] Nobel Prize in Physics citation 2011 http://www.nobelprize.org/ [7] Planck Collaboration: Planck 2013 results. XVI. Cosmological parameters. Astronomy & Astrophysics 571, DOI: 10.1051/0004-6361/201321591 (2014) http://arxiv.org/abs/1303.5076 [8] Hu, W.: Wandering in the Background: A Cosmic Microwave Background Explorer. PhD Thesis (1995) http://arxiv.org/abs/astro-ph/9508126. [9] Eisenstein, D., Seo, H., and White, M.: On the Robustness of the Acoustic Scale in the Low-Redshift Clustering of Matter. The Astrophysical Journal 664, 660-674 (2007) http://arxiv.org/abs/astro-ph/0604361v1 [10] Gentile, G., Burkert, A., Salucci, P., Klein, U., and Walter, F.: The Dwarf Galaxy DDO 47 as a Dark Matter Laboratory: Testing Cusps Hiding in Triaxial Halos. The Astrophysical Journal Letters 634, doi:10.1086/498939 (2005) http://arxiv.org/pdf/astro-ph/0506538.pdf [11] Carroll, S.: Why Is There Dark Matter? (July 2015) http://www.preposterousuniverse.com/blog/2015/07/07/why-is-there-dark-matter/ [12] Carroll, S.: The Cosmological Constant. http://arxiv.org/abs/astro-ph/0004075 (section 4) [13] Weinberg, S.: The Cosmological Constant Problems. http://arxiv.org/abs/astro-ph/0005265 [14] Bianchi, E., Rovelli, C., and Kolb, R.: Is dark energy really a mystery? Nature 466, 321-322 (July 2010). [15] Bull, P., et al.: Beyond ΛCDM: Problems, solutions, and the road ahead. http://arxiv.org/abs/1512.05356, p 67 [16] Garﬁnkle, D.: Inhomogeneous spacetimes as a dark energy model. Classical and Quantum Gravity 23, 4811-4818 (2006). http://arxiv.org/abs/gr-qc/0605088 [17] Paranjape, A., and Singh, T.P.: The Possibility of Cosmic Acceleration via Spatial Averaging in Lemaˆıtre-Tolman-Bondi Models. Classical and Quantum Gravity 23, 6955-6969 (2006). http://arxiv.org/abs/astro-ph/0605195 [18] Tanimoto, M., and Nambu, Y.: Luminosity distance-redshift relation for the LTB solution near the center. Classical and Quantum Gravity 24, 3843-3857 (2007). http://arxiv.org/abs/gr- qc/0703012 [19] Clarkson, C., and Maartens, R.: Inhomogeneity and the foundations of concordance cosmology. Classical and Quantum Gravity 27, 124008 (2010). http://arxiv.org/abs/1005.2165 [20] Milgrom, M.: A modiﬁcation of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. The Astrophysical Journal 270, 365-370 (1983). [21] Milgrom, M: MOND theory. Canadian Journal of Physics 93, 107-118 (2015) http://arxiv.org/abs/1404.7661 [22] Sanders, R.H., and McGaugh, S.: Modiﬁed Newtonian Dynamics as an Alternative to Dark Matter. Annual Reviews of Astronomy & Astrophysics 40, 263-317 (2002) http://arxiv.org/abs/astro- ph/0204521 [23] Bekenstein, J.D.: Relativistic gravitation theory for the modiﬁed Newtonian dynamics paradigm. Physical Review D 70, 083509 (2004). [24] Sanders, R.H.: A tensor-vector-scalar framework for modiﬁed dynamics and cosmic dark matter. Monthly Notices of the Royal Astronomical Society 363, 459-468 (2005). [25] Zlosnik, T., Ferreira, P., and Starkman, G.: Modifying gravity with the aether: An alternative to dark matter. Physical Review D 75, 044017 (2007) http://arxiv.org/abs/astro-ph/0607411 [26] Zhao, H.S., and Li, B.: Dark Fluid: A Uniﬁed framework for Modiﬁed Newtonian Dynamics, Dark Matter, and Dark Engery. The Astrophysical Journal 712, doi:10.1088/0004-637X/712/1/130 (2010). [27] Blanchet, L., and Le Tiec, A.: Model of dark matter and dark energy based on gravitational polarization. Physical Review D 78, 024031 (2008). [28] Brownstein, J.R., and Moﬀat, J.W.: Galaxy Rotation Curves Without Non-Baryonic Dark Matter. The Astrophysical Journal 636, 721-741 (2006). [29] Brownstein, J.R., and Moﬀat, J.W.: Galaxy Cluster Masses Without Non-Baryonic Dark Matter. Monthly Notices of the Royal Astronomical Society 367, 527-540 (2006)

text: End

label: title

text: of

text: a

text: Dark

text: Age?

text: Nobel

text: Prize

text: in

text: Physics

text: citation

text: Planck

text: Collaboration:

text: Planck

text: results.

text: XVI.

text: Cosmological

text: parameters.

text: Astronomy

text: Astrophysics

text: DOI:

text: Hu,

text: W.:

text: Wandering

text: in

text: the

text: Background:

text: A

text: Cosmic

text: Microwave

text: Background

text: Explorer.

text: PhD

text: Thesis

text: Eisenstein,

text: D.,

text: Seo,

text: H.,

text: and

text: White,

text: M.:

text: On

text: the

text: Robustness

text: of

text: the

text: Acoustic

text: Scale

text: in

text: the

text: Low-Redshift

text: Clustering

text: of

text: Matter.

text: The

text: Astrophysical

text: Journal

text: Gentile,

text: G.,

text: Burkert,

text: A.,

text: Salucci,

text: P.,

text: Klein,

text: U.,

text: and

text: Walter,

text: F.:

text: The

text: Dwarf

text: Galaxy

text: DDO

text: as

text: a

text: Dark

text: Matter

text: Laboratory:

text: Testing

text: Cusps

text: Hiding

text: in

text: Triaxial

text: Halos.

text: The

text: Astrophysical

text: Journal

text: Letters

text: Carroll,

text: S.:

text: Why

text: Is

text: There

text: Dark

text: Matter?

text: (July

text: Carroll,

text: S.:

text: The

text: Cosmological

text: Constant.

text: (section

text: Weinberg,

text: S.:

text: The

text: Cosmological

text: Constant

text: Problems.

text: Bianchi,

text: E.,

text: Rovelli,

text: C.,

text: and

text: Kolb,

text: R.:

text: Is

text: dark

text: energy

text: really

text: a

text: mystery?

text: Nature

text: (July

text: Bull,

text: P.,

text: et

text: al.:

text: Beyond

text: ΛCDM:

text: Problems,

text: solutions,

text: and

text: the

text: road

text: ahead.

text: p

text: Garﬁnkle,

text: D.:

text: Inhomogeneous

text: spacetimes

text: as

text: a

text: dark

text: energy

text: model.

text: Classical

text: and

text: Quantum

text: Gravity

text: Paranjape,

text: A.,

text: and

text: Singh,

text: T.P.:

text: The

text: Possibility

text: of

text: Cosmic

text: Acceleration

text: via

text: Spatial

text: Averaging

text: in

text: Lemaˆıtre-Tolman-Bondi

text: Models.

text: Classical

text: and

text: Quantum

text: Gravity

text: Tanimoto,

text: M.,

text: and

text: Nambu,

text: Y.:

text: Luminosity

text: distance-redshift

text: relation

text: for

text: the

text: LTB

text: solution

text: near

text: the

text: center.

text: Classical

text: and

text: Quantum

text: Gravity

text: Clarkson,

text: C.,

text: and

text: Maartens,

text: R.:

text: Inhomogeneity

text: and

text: the

text: foundations

text: of

text: concordance

text: cosmology.

text: Classical

text: and

text: Quantum

text: Gravity

text: Milgrom,

text: M.:

text: A

text: modiﬁcation

text: of

text: the

text: Newtonian

text: dynamics

text: as

text: a

text: possible

text: alternative

text: to

text: the

text: hidden

text: mass

text: hypothesis.

text: The

text: Astrophysical

text: Journal

text: Milgrom,

text: M:

text: MOND

text: theory.

text: Canadian

text: Journal

text: of

text: Physics

text: Sanders,

text: R.H.,

text: and

text: McGaugh,

text: S.:

text: Modiﬁed

text: Newtonian

text: Dynamics

text: as

text: an

text: Alternative

text: to

text: Dark

text: Matter.

text: Annual

text: Reviews

text: of

text: Astronomy

text: Astrophysics

text: Bekenstein,

text: J.D.:

text: Relativistic

text: gravitation

text: theory

text: for

text: the

text: modiﬁed

text: Newtonian

text: dynamics

text: paradigm.

text: Physical

text: Review

text: D

text: Sanders,

text: R.H.:

text: A

text: tensor-vector-scalar

text: framework

text: for

text: modiﬁed

text: dynamics

text: and

text: cosmic

text: dark

text: matter.

text: Monthly

text: Notices

text: of

text: the

text: Royal

text: Astronomical

text: Society

text: Zlosnik,

text: T.,

text: Ferreira,

text: P.,

text: and

text: Starkman,

text: G.:

text: Modifying

text: gravity

text: with

text: the

text: aether:

text: An

text: alternative

text: to

text: dark

text: matter.

text: Physical

text: Review

text: D

text: Zhao,

text: H.S.,

text: and

text: Li,

text: B.:

text: Dark

text: Fluid:

text: A

text: Uniﬁed

text: framework

text: for

text: Modiﬁed

text: Newtonian

text: Dynamics,

text: Dark

text: Matter,

text: and

text: Dark

text: Engery.

text: The

text: Astrophysical

text: Journal

text: Blanchet,

text: L.,

text: and

text: Le

text: Tiec,

text: A.:

text: Model

text: of

text: dark

text: matter

text: and

text: dark

text: energy

text: based

text: on

text: gravitational

text: polarization.

text: Physical

text: Review

text: D

text: Brownstein,

text: J.R.,

text: and

text: Moﬀat,

text: J.W.:

text: Galaxy

text: Rotation

text: Curves

text: Without

text: Non-Baryonic

text: Dark

text: Matter.

text: The

text: Astrophysical

text: Journal

text: Brownstein,

text: J.R.,

text: and

text: Moﬀat,

text: J.W.:

text: Galaxy

text: Cluster

text: Masses

text: Without

text: Non-Baryonic

text: Dark

text: Matter.

text: Monthly

text: Notices

text: of

text: the

text: Royal

text: Astronomical

text: Society


text: 13 [4] Eghiaian F, Rigato A and Scheuring S 2015 Structural, mechanical, and dynamical variability of the actin cortex So far we have investigated a model for intracellular in living cells Biophys. J. 108 1330-1340 search strategies in two-dimensional cells. However, [5] Salbreux G, Charras G and Paluch E 2012 Actin cortex a generalization of our approach to three-dimensional mechanics and cellular morphogenesis Trends Cell Biol. spherical cells is straightforward and the former prin- 22 536-545 ciples stay valid. If we assume a cell radius of 5 µm, [6] Rebhun J I 1967 Structural aspects of saltatory particle which is consistent with the typical size of a CTL, the movement J. Gen. Physiol. 50 223-239 opening angle of α =0.02 (α =0.2) leads to an [7] B´alint Sˇ, Vilanova I V, A´lvarez A´ S and Lakadamyali M ×target target 2013 Correlative live-cell and superresolution microscopy arc length of 0.02 5 µm =100 nm (1 µm). For instance reveals cargo transport dynamics at microtubule intersec- the diameter of an immunological synapse is of the order tions P. Natl. Acad. Sci. USA 110 3375-3380 of microns [23–26]. We further assume that motors move [8] Hancock W O 2014 Bidirectional cargo transport: mov- processively at intersections with probability p=0.5 and ing beyond tug of war Nat. R 15 615-628 their velocity typically is about 1 µm/s. A transition [9] Vershinin M, Carter B C, Razafsky D S, King S J and rate to the waiting state of 10/s thus leads to a mesh size Gross S P 2007 Multiple-motor based transport and its of 100 nm, which is biologically reasonable [4, 5]. Under regulation by tau P. Natl. Acad. Sci. USA 104 87-92 [10] Welte M A 2004 Bidirectional transport along micro- these conditions, ﬁgure 9 reveals an optimal width of the tubules Curr. Biol., 14 R525-R537 actin cortex which varies from approximately δ = 0.1 opt [11] Gross S P, Tuma M C, Deacon S W, Serpinskaya A S, µm to δ = 1.2 µm in 2D and δ = 1 µm in 3D. This is opt opt Reilin A R and Gelfand V I 2002 Interactions and regu- in good qualitative agreement to biological data [4, 5, 42]. lation of molecular motors in Xenopus melanophores J. Cell. Biol. 156 855-865 In summary, our model indicates that the spatial orga- [12] Rodionov V, Yi J, Kashina A, Oladipo A and Gross S P 2003 Switching between microtubule- and actin- nization of the cytoskeleton of spherical cells with a cen- based transport systems in melanophores is controlled trosome minimizes the characteristic time necessary to by cAMP levels Curr. Biol. 13 1837-1847 detect small targets on the cell membrane by random [13] Mallik R and Gross S P 2004 Molecular motors: strate- intermittent search processes along the cytoskeletal ﬁla- gies to get along Curr. Biol. 14 R971-R982 ments (see also [43, 44] for similar ﬁndings in a model [14] Suomalainen M, Nakano M Y, Boucke K, Keller S and with intermittent diﬀusive search). The minimization is Greber U F 2001 Adenovirus-activated PKA and achieved by a small width of the actin cortical layer and p38/MAPK pathways boost microtubule-mediated nu- clear targeting of virus EMBO J. 20 1310-1319 by regulation of the motor activity and behavior at net- [15] Gross S P, Guo Y, Martinez J E and Welte M A 2003 A work intersections. Remarkably a thin actin cortex is also determinant for directionality of organelle transport in more economic than distributing cytoskeletal ﬁlaments drosophila embryos Curr. Biol. 13 1660-1668 in all directions over the cell body. Thus, our work out- [16] Bressloﬀ P C and Newby J M 2013 Stochastic models of lines that a thin conﬁnement of the actin cortex, besides intracellular transport Rev. Mod. Phys. 85 135-196 its advantages concerning cell stability or motility, also [17] Ali M Y, Krementsova E B, Kennedy G G, Mahaﬀy R, serves as an eﬃcient key to intracellular narrow escape Pollard T D, Trybus K M and Warshaw D M 2007 Myosin Va maneuvers through actin intersections and diﬀuses problems. along microtubules P. Natl. Acad. Sci. USA 104 4332- Acknowledgements 4336 [18] Ross J L, Ali M Y and Warshaw D M 2008 Cargo trans- port: molecular motors navigate a complex cytoskeleton This work was funded by the German Research Foun- Curr. Opin. Cell Biol. 20 41-47 dation (DFG) within the Collaborative Research Center [19] Ross J L, Shuman H, Holzbaur E L F and Goldman Y E SFB 1027. 2008 Kinesin and dynein-dynactin at intersecting micro- tubules: motor density aﬀects dynein function Biophys. J. 94 3115-3125 [20] Weiss M, Elsner M, Kartberg F and Nilsson T 2004 Anomalous subdiﬀusion is a measure for cytoplasmic ∗ Electronic address: \markcomm{footercolor}{anne. crowding in living cells Biophys. J. 87 3518-3524 hafner@lusi.uni-sb.de} [21] Snider J, Lin F, Zahedi N, Rodionov V, Yu C C and Gross † Electronic address: \markcomm{footercolor}{h. S P 2004 Intracellular actin-based transport: how far you rieger@mx.uni-saarland.de} go depends on how often you switch P. Natl. Acad. Sci. [1] Schliwa M and Woehlke G 2003 Molecular motors Nature USA 101 13204-13209 422 759-765 [22] B´enichou O, Loverdo C, Moreau M and Voituriez R 2011 [2] Vale R D and Milligan R A 2000 The way things move: Intermittent search strategies Rev. Mod. Phys. 83 81-129 looking under the hood of molecular motor proteins Sci- [23] Angus K L and Griﬃths G M 2013 Cell polarisation and ence 288 88-95 the immunological synapse Curr. Opin. Cell Biol. 25 [3] Alberts B, Johnson A, Lewis J, Raﬀ M, Roberts K and 85-91 Walter P 2008 Molecular Biology of the Cell New York [24] Grakoui A, Bromley S K, Sumen C, Davis M M, Shaw Garland Science A S, Allen P M and Dustin M L 1999 The immunological ##LTLine## ##LTLine## ##LTLine##

text: Eghiaian

text: F,

text: Rigato

text: A

text: and

text: Scheuring

text: S

text: Structural,

text: mechanical,

text: and

text: dynamical

text: variability

text: of

text: the

text: actin

text: cortex

text: So

text: far

text: we

text: have

text: investigated

text: a

text: model

text: for

text: intracellular

text: in

text: living

text: cells

text: Biophys.

text: J.

text: search

text: strategies

text: in

text: two-dimensional

text: cells.

text: However,

text: Salbreux

text: G,

text: Charras

text: G

text: and

text: Paluch

text: E

text: Actin

text: cortex

text: a

text: generalization

text: of

text: our

text: approach

text: to

text: three-dimensional

text: mechanics

text: and

text: cellular

text: morphogenesis

text: Trends

text: Cell

text: Biol.

text: spherical

text: cells

text: is

text: straightforward

text: and

text: the

text: former

text: prin-

text: ciples

text: stay

text: valid.

text: If

text: we

text: assume

text: a

text: cell

text: radius

text: of

text: µm,

text: Rebhun

text: J

text: I

text: Structural

text: aspects

text: of

text: saltatory

text: particle

text: which

text: is

text: consistent

text: with

text: the

text: typical

text: size

text: of

text: a

text: CTL,

text: the

text: movement

text: J.

text: Gen.

text: Physiol.

text: opening

text: angle

text: of

text: α

text: (α

text: leads

text: to

text: an

text: B´alint

text: Sˇ,

text: Vilanova

text: I

text: V,

text: A´lvarez

text: A´

text: S

text: and

text: Lakadamyali

text: M

text: ×target

text: target

text: Correlative

text: live-cell

text: and

text: superresolution

text: microscopy

text: arc

text: length

text: of

text: µm

text: nm

text: µm).

text: For

text: instance

text: reveals

text: cargo

text: transport

text: dynamics

text: at

text: microtubule

text: intersec-

text: the

text: diameter

text: of

text: an

text: immunological

text: synapse

text: is

text: of

text: the

text: order

text: tions

text: P.

text: Natl.

text: Acad.

text: Sci.

text: USA

text: of

text: microns

text: We

text: further

text: assume

text: that

text: motors

text: move

text: Hancock

text: W

text: O

text: Bidirectional

text: cargo

text: transport:

text: mov-

text: processively

text: at

text: intersections

text: with

text: probability

text: and

text: ing

text: beyond

text: tug

text: of

text: war

text: Nat.

text: R

text: their

text: velocity

text: typically

text: is

text: about

text: µm/s.

text: A

text: transition

text: Vershinin

text: M,

text: Carter

text: B

text: C,

text: Razafsky

text: D

text: S,

text: King

text: S

text: J

text: and

text: rate

text: to

text: the

text: waiting

text: state

text: of

text: thus

text: leads

text: to

text: a

text: mesh

text: size

text: Gross

text: S

text: P

text: Multiple-motor

text: based

text: transport

text: and

text: its

text: of

text: nm,

text: which

text: is

text: biologically

text: reasonable

text: Under

text: regulation

text: by

text: tau

text: P.

text: Natl.

text: Acad.

text: Sci.

text: USA

text: Welte

text: M

text: A

text: Bidirectional

text: transport

text: along

text: micro-

text: these

text: conditions,

text: ﬁgure

text: reveals

text: an

text: optimal

text: width

text: of

text: the

text: tubules

text: Curr.

text: Biol.,

text: actin

text: cortex

text: which

text: varies

text: from

text: approximately

text: δ

text: opt

text: Gross

text: S

text: P,

text: Tuma

text: M

text: C,

text: Deacon

text: S

text: W,

text: Serpinskaya

text: A

text: S,

text: µm

text: to

text: δ

text: µm

text: in

text: and

text: δ

text: µm

text: in

text: This

text: is

text: opt

text: opt

text: Reilin

text: A

text: R

text: and

text: Gelfand

text: V

text: I

text: Interactions

text: and

text: regu-

text: in

text: good

text: qualitative

text: agreement

text: to

text: biological

text: data

text: lation

text: of

text: molecular

text: motors

text: in

text: Xenopus

text: melanophores

text: J.

text: Cell.

text: Biol.

text: In

text: summary,

text: our

text: model

text: indicates

text: that

text: the

text: spatial

text: orga-

text: Rodionov

text: V,

text: Yi

text: J,

text: Kashina

text: A,

text: Oladipo

text: A

text: and

text: Gross

text: S

text: P

text: Switching

text: between

text: microtubule-

text: and

text: actin-

text: nization

text: of

text: the

text: cytoskeleton

text: of

text: spherical

text: cells

text: with

text: a

text: cen-

text: based

text: transport

text: systems

text: in

text: melanophores

text: is

text: controlled

text: trosome

text: minimizes

text: the

text: characteristic

text: time

text: necessary

text: to

text: by

text: cAMP

text: levels

text: Curr.

text: Biol.

text: detect

text: small

text: targets

text: on

text: the

text: cell

text: membrane

text: by

text: random

text: Mallik

text: R

text: and

text: Gross

text: S

text: P

text: Molecular

text: motors:

text: strate-

text: intermittent

text: search

text: processes

text: along

text: the

text: cytoskeletal

text: ﬁla-

text: gies

text: to

text: get

text: along

text: Curr.

text: Biol.

text: ments

text: (see

text: also

text: for

text: similar

text: ﬁndings

text: in

text: a

text: model

text: Suomalainen

text: M,

text: Nakano

text: M

text: Y,

text: Boucke

text: K,

text: Keller

text: S

text: and

text: with

text: intermittent

text: diﬀusive

text: search).

text: The

text: minimization

text: is

text: Greber

text: U

text: F

text: Adenovirus-activated

text: PKA

text: and

text: achieved

text: by

text: a

text: small

text: width

text: of

text: the

text: actin

text: cortical

text: layer

text: and

text: pathways

text: boost

text: microtubule-mediated

text: nu-

text: clear

text: targeting

text: of

text: virus

text: EMBO

text: J.

text: by

text: regulation

text: of

text: the

text: motor

text: activity

text: and

text: behavior

text: at

text: net-

text: Gross

text: S

text: P,

text: Guo

text: Y,

text: Martinez

text: J

text: E

text: and

text: Welte

text: M

text: A

text: A

text: work

text: intersections.

text: Remarkably

text: a

text: thin

text: actin

text: cortex

text: is

text: also

text: determinant

text: for

text: directionality

text: of

text: organelle

text: transport

text: in

text: more

text: economic

text: than

text: distributing

text: cytoskeletal

text: ﬁlaments

text: drosophila

text: embryos

text: Curr.

text: Biol.

text: in

text: all

text: directions

text: over

text: the

text: cell

text: body.

text: Thus,

text: our

text: work

text: out-

text: Bressloﬀ

text: P

text: C

text: and

text: Newby

text: J

text: M

text: Stochastic

text: models

text: of

text: lines

text: that

text: a

text: thin

text: conﬁnement

text: of

text: the

text: actin

text: cortex,

text: besides

text: intracellular

text: transport

text: Rev.

text: Mod.

text: Phys.

text: its

text: advantages

text: concerning

text: cell

text: stability

text: or

text: motility,

text: also

text: Ali

text: M

text: Y,

text: Krementsova

text: E

text: B,

text: Kennedy

text: G

text: G,

text: Mahaﬀy

text: R,

text: serves

text: as

text: an

text: eﬃcient

text: key

text: to

text: intracellular

text: narrow

text: escape

text: Pollard

text: T

text: D,

text: Trybus

text: K

text: M

text: and

text: Warshaw

text: D

text: M

text: Myosin

text: Va

text: maneuvers

text: through

text: actin

text: intersections

text: and

text: diﬀuses

text: problems.

text: along

text: microtubules

text: P.

text: Natl.

text: Acad.

text: Sci.

text: USA

text: Acknowledgements

text: Ross

text: J

text: L,

text: Ali

text: M

text: Y

text: and

text: Warshaw

text: D

text: M

text: Cargo

text: trans-

text: port:

text: molecular

text: motors

text: navigate

text: a

text: complex

text: cytoskeleton

text: This

text: work

text: was

text: funded

text: by

text: the

text: German

text: Research

text: Foun-

text: Curr.

text: Opin.

text: Cell

text: Biol.

text: dation

text: (DFG)

text: within

text: the

text: Collaborative

text: Research

text: Center

text: Ross

text: J

text: L,

text: Shuman

text: H,

text: Holzbaur

text: E

text: L

text: F

text: and

text: Goldman

text: Y

text: E

text: SFB

text: Kinesin

text: and

text: dynein-dynactin

text: at

text: intersecting

text: micro-

text: tubules:

text: motor

text: density

text: aﬀects

text: dynein

text: function

text: Biophys.

text: J.

text: Weiss

text: M,

text: Elsner

text: M,

text: Kartberg

text: F

text: and

text: Nilsson

text: T

text: Anomalous

text: subdiﬀusion

text: is

text: a

text: measure

text: for

text: cytoplasmic

text: Electronic

text: address:

text: \markcomm{footercolor}{anne.

text: crowding

text: in

text: living

text: cells

text: Biophys.

text: J.

text: hafner@lusi.uni-sb.de}

text: Snider

text: J,

text: Lin

text: F,

text: Zahedi

text: N,

text: Rodionov

text: V,

text: Yu

text: C

text: C

text: and

text: Gross

text: Electronic

text: address:

text: \markcomm{footercolor}{h.

text: S

text: P

text: Intracellular

text: actin-based

text: transport:

text: how

text: far

text: you

text: rieger@mx.uni-saarland.de}

text: go

text: depends

text: on

text: how

text: often

text: you

text: switch

text: P.

text: Natl.

text: Acad.

text: Sci.

text: Schliwa

text: M

text: and

text: Woehlke

text: G

text: Molecular

text: motors

text: Nature

text: USA

text: B´enichou

text: O,

text: Loverdo

text: C,

text: Moreau

text: M

text: and

text: Voituriez

text: R

text: Vale

text: R

text: D

text: and

text: Milligan

text: R

text: A

text: The

text: way

text: things

text: move:

text: Intermittent

text: search

text: strategies

text: Rev.

text: Mod.

text: Phys.

text: looking

text: under

text: the

text: hood

text: of

text: molecular

text: motor

text: proteins

text: Sci-

text: Angus

text: K

text: L

text: and

text: Griﬃths

text: G

text: M

text: Cell

text: polarisation

text: and

text: ence

text: the

text: immunological

text: synapse

text: Curr.

text: Opin.

text: Cell

text: Biol.

text: Alberts

text: B,

text: Johnson

text: A,

text: Lewis

text: J,

text: Raﬀ

text: M,

text: Roberts

text: K

text: and

text: Walter

text: P

text: Molecular

text: Biology

text: of

text: the

text: Cell

text: New

text: York

text: Grakoui

text: A,

text: Bromley

text: S

text: K,

text: Sumen

text: C,

text: Davis

text: M

text: M,

text: Shaw

text: Garland

text: Science

text: A

text: S,

text: Allen

text: P

text: M

text: and

text: Dustin

text: M

text: L

text: The

text: immunological

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 synapse: a molecular machine controlling t cell activation ends Nat. Commun., 5 4778 Science 285 221-227 [35] Mullins R D,Heuser J A and Pollard T D 1998 The In- [25] Qu B, Pattu V, Junker C, Schwarz E C, Bhat S S, teraction of Arp2/3 complex with actin: nucleation, high Kummerow C, Marshall M, Matti U, Pfreundschuh M, aﬃnity pointed end capping, and formation of branch- Becherer U, Rieger H, Rettig J and Hoth M 2011 Dock- ing networks of ﬁlaments P. Natl. Acad. Sci. USA 95 ing of lytic granules at the immunological synapse in hu- 6181-6186 man CTL required Vti1b-dependent pairing with CD3 [36] Risca V I, Wang E B, Chaudhuri O, Chia J J, Geissler endosomes J. Immunol. 186 6894-6904 P L and Fletcher D A 2012 Actin ﬁlament curvature [26] Ritter A T, Angus K L and Griﬃths G M 2013 The role of biases branching direction P. Natl. Acad. Sci. USA 109 the cytoskeleton at the immunological synapse Immunol. 2913-2918 Rev. 256 107-117 [37] Schuss Z, Singer A and Holcman D 2007 The narrow [27] Alberts P and Galli T 2003 The cell outgrowth secretory escape problem for diﬀusion in cellular microdomains P. endosome (COSE): a specialized compartment involved Natl. Acad. Sci. USA 104 16098-16103 in neuronal morphogenesis Biol. Cell 95 419-424 [38] Schuss Z 2012 The narrow escape problem - a short [28] Chada S R and Hollenbeck P J 2003 Mitochondrial move- review of recent results J. Sci. Comput. 53 194-210 ment and positioning in axons: the role of growth factor [39] B´enichou O, Grebenkov D, Levitz P, Loverdo C and Voi- signaling J. Exp. Biol. 206 1985-1992 turiez R 2010 Optimal reaction time for surface-mediated [29] Andrews N W, Almeida P E and Corrotte M 2014 Dam- diﬀusion Phys. Rev. Lett. 105 150606 age control: cellular mechanisms of plasma membrane [40] Ando D, Korabel N, Huang K C and Gopinathan A 2015 repair Trends Cell Biol. 24 734-742 Cytoskeletal network morphology regulates intracellular [30] McNeil P L and Kirchhausen T 2005 An emergency transport dynamics Biophys. J. 109 1574-1582 response team for membrane repair Nat. Rev. Mol. Cell [41] Papadopulos A, Tomatis V M, Kasula R and Meunier Bio. 6 499-505 F A 2013 The cortical acto-myosin network: from dif- [31] Arcizet D, Meier B, Sackmann E, Ra¨dler J O and Hein- fusion barrier to functional gateway in the transport of rich D 2008 Temporal analysis of active and passive neurosecretory vesicles to the plasma membrane Front. transport in living cells Phys. Rev. Lett. 101 248103 Endocrinol. 4 1-11 [32] Slepchenko B M, Semenova I, Zaliapin I and Rodionov [42] Clark A G, Dierkes K and Paluch E 2013 Monitoring V 2007 Switching of membrane organelles between cy- actin cortex thickness in live cells Biophy. J. 105 570- toskeletal transport systems is determined by regulation 580 of the microtubule-based transport J. Cell Biol. 179 [43] Schwarz K, Schro¨der Y, Qu B, Hoth M and Rieger H 2016 635-641 Optimality of spatially inhomogeneous search strategies [33] Zaliapin I, Semenova I, Kashina A and Rodionov V 2005 Phys. Rev. Lett. 117 068101 Multiscale trend analysis of microtubule transport in [44] Schwarz K, Schr¨oder Y and Rieger H 2016 melanophores Biophys. J. 88 4008-4016 Numerical analysis of homogeneous and in- [34] Lo´pez M P, Huber F, Grigoriev I, Steinmetz M O, homogeneous intermittent search strategies Akhmanova A, Koenderink G H and Dogterom M 2014 http://lanl.arxiv.org/abs/1607.00892 Actin-microtubule coordination at growing microtubule

text: synapse:

text: a

text: molecular

text: machine

text: controlling

text: t

text: cell

text: activation

text: ends

text: Nat.

text: Commun.,

text: Science

text: Mullins

text: R

text: D,Heuser

text: J

text: A

text: and

text: Pollard

text: T

text: D

text: The

text: In-

text: Qu

text: B,

text: Pattu

text: V,

text: Junker

text: C,

text: Schwarz

text: E

text: C,

text: Bhat

text: S

text: S,

text: teraction

text: of

text: complex

text: with

text: actin:

text: nucleation,

text: high

text: Kummerow

text: C,

text: Marshall

text: M,

text: Matti

text: U,

text: Pfreundschuh

text: M,

text: aﬃnity

text: pointed

text: end

text: capping,

text: and

text: formation

text: of

text: branch-

text: Becherer

text: U,

text: Rieger

text: H,

text: Rettig

text: J

text: and

text: Hoth

text: M

text: Dock-

text: ing

text: networks

text: of

text: ﬁlaments

text: P.

text: Natl.

text: Acad.

text: Sci.

text: USA

text: ing

text: of

text: lytic

text: granules

text: at

text: the

text: immunological

text: synapse

text: in

text: hu-

text: man

text: CTL

text: required

text: pairing

text: with

text: Risca

text: V

text: I,

text: Wang

text: E

text: B,

text: Chaudhuri

text: O,

text: Chia

text: J

text: J,

text: Geissler

text: endosomes

text: J.

text: Immunol.

text: P

text: L

text: and

text: Fletcher

text: D

text: A

text: Actin

text: ﬁlament

text: curvature

text: Ritter

text: A

text: T,

text: Angus

text: K

text: L

text: and

text: Griﬃths

text: G

text: M

text: The

text: role

text: of

text: biases

text: branching

text: direction

text: P.

text: Natl.

text: Acad.

text: Sci.

text: USA

text: the

text: cytoskeleton

text: at

text: the

text: immunological

text: synapse

text: Immunol.

text: Rev.

text: Schuss

text: Z,

text: Singer

text: A

text: and

text: Holcman

text: D

text: The

text: narrow

text: Alberts

text: P

text: and

text: Galli

text: T

text: The

text: cell

text: outgrowth

text: secretory

text: escape

text: problem

text: for

text: diﬀusion

text: in

text: cellular

text: microdomains

text: P.

text: endosome

text: (COSE):

text: a

text: specialized

text: compartment

text: involved

text: Natl.

text: Acad.

text: Sci.

text: USA

text: in

text: neuronal

text: morphogenesis

text: Biol.

text: Cell

text: Schuss

text: Z

text: The

text: narrow

text: escape

text: problem

text: a

text: short

text: Chada

text: S

text: R

text: and

text: Hollenbeck

text: P

text: J

text: Mitochondrial

text: move-

text: review

text: of

text: recent

text: results

text: J.

text: Sci.

text: Comput.

text: ment

text: and

text: positioning

text: in

text: axons:

text: the

text: role

text: of

text: growth

text: factor

text: B´enichou

text: O,

text: Grebenkov

text: D,

text: Levitz

text: P,

text: Loverdo

text: C

text: and

text: Voi-

text: signaling

text: J.

text: Exp.

text: Biol.

text: turiez

text: R

text: Optimal

text: reaction

text: time

text: for

text: surface-mediated

text: Andrews

text: N

text: W,

text: Almeida

text: P

text: E

text: and

text: Corrotte

text: M

text: Dam-

text: diﬀusion

text: Phys.

text: Rev.

text: Lett.

text: age

text: control:

text: cellular

text: mechanisms

text: of

text: plasma

text: membrane

text: Ando

text: D,

text: Korabel

text: N,

text: Huang

text: K

text: C

text: and

text: Gopinathan

text: A

text: repair

text: Trends

text: Cell

text: Biol.

text: Cytoskeletal

text: network

text: morphology

text: regulates

text: intracellular

text: McNeil

text: P

text: L

text: and

text: Kirchhausen

text: T

text: An

text: emergency

text: transport

text: dynamics

text: Biophys.

text: J.

text: response

text: team

text: for

text: membrane

text: repair

text: Nat.

text: Rev.

text: Mol.

text: Cell

text: Papadopulos

text: A,

text: Tomatis

text: V

text: M,

text: Kasula

text: R

text: and

text: Meunier

text: Bio.

text: F

text: A

text: The

text: cortical

text: acto-myosin

text: network:

text: from

text: dif-

text: Arcizet

text: D,

text: Meier

text: B,

text: Sackmann

text: E,

text: Ra¨dler

text: J

text: O

text: and

text: Hein-

text: fusion

text: barrier

text: to

text: functional

text: gateway

text: in

text: the

text: transport

text: of

text: rich

text: D

text: Temporal

text: analysis

text: of

text: active

text: and

text: passive

text: neurosecretory

text: vesicles

text: to

text: the

text: plasma

text: membrane

text: Front.

text: transport

text: in

text: living

text: cells

text: Phys.

text: Rev.

text: Lett.

text: Endocrinol.

text: Slepchenko

text: B

text: M,

text: Semenova

text: I,

text: Zaliapin

text: I

text: and

text: Rodionov

text: Clark

text: A

text: G,

text: Dierkes

text: K

text: and

text: Paluch

text: E

text: Monitoring

text: V

text: Switching

text: of

text: membrane

text: organelles

text: between

text: cy-

text: actin

text: cortex

text: thickness

text: in

text: live

text: cells

text: Biophy.

text: J.

text: toskeletal

text: transport

text: systems

text: is

text: determined

text: by

text: regulation

text: of

text: the

text: microtubule-based

text: transport

text: J.

text: Cell

text: Biol.

text: Schwarz

text: K,

text: Schro¨der

text: Y,

text: Qu

text: B,

text: Hoth

text: M

text: and

text: Rieger

text: H

text: Optimality

text: of

text: spatially

text: inhomogeneous

text: search

text: strategies

text: Zaliapin

text: I,

text: Semenova

text: I,

text: Kashina

text: A

text: and

text: Rodionov

text: V

text: Phys.

text: Rev.

text: Lett.

text: Multiscale

text: trend

text: analysis

text: of

text: microtubule

text: transport

text: in

text: Schwarz

text: K,

text: Schr¨oder

text: Y

text: and

text: Rieger

text: H

text: melanophores

text: Biophys.

text: J.

text: Numerical

text: analysis

text: of

text: homogeneous

text: and

text: in-

text: Lo´pez

text: M

text: P,

text: Huber

text: F,

text: Grigoriev

text: I,

text: Steinmetz

text: M

text: O,

text: homogeneous

text: intermittent

text: search

text: strategies

text: Akhmanova

text: A,

text: Koenderink

text: G

text: H

text: and

text: Dogterom

text: M

text: Actin-microtubule

text: coordination

text: at

text: growing

text: microtubule