Current distribution for a two-species particle model from firstprinciples

Jan de Gier

10 September 2019, OsakaIn collaboration withZeying Chen
Iori Hiki
Tomohiro Sasamoto

arXiv:1803.06829
Phys. Rev. Lett. 120, 240601 (2018)

Jan de Gier

Current distribution for a two-species particle model from first principles

September 20191 / 31

KPZ growth

1+1D growth

Jan de Gier

Current distribution for a two-species particle model from first principles

September 20192 / 31

KPZ growth

KPZ growth
Stochastic growth normal to the surface
Kardar-Parisi-Zhang (KPZ) 1986
Basic object: (random) height function h(x, t)
KPZ equation (nonlinear stochastic PDE):KPZ equation∂t h(t, x) = 12 ∂2x h(t, x) + 12 (∂x h(t, x))2 + ξ(t, x)

Claim:Diffusion + non-linearity + space-time Gaussian white noisecorrectly describes 1+1D surface growth
Realisation in liquid crystal growth: Takeuchi lab

Jan de Gier

Current distribution for a two-species particle model from first principles

September 20193 / 31

KPZ growth

1+1D GrowthTakeuchi and Sasamoto, Tokyo 2017

Jan de Gier

Current distribution for a two-species particle model from first principles

September 20194 / 31

KPZ growth

1+1D Growth

Flat
Flat initial conditions

Jan de Gier

Current distribution for a two-species particle model from first principles

September 20195 / 31

KPZ growth

KPZ growth
Theorem (Non-Gaussian fluctuations)h ∼ vt + ct1/3 ,t→∞Transformation to Stochastic Heat Equation (SHE):h(t, x) := log z(t, x).
SHE equation∂t z(t, x) =Jan de Gier

1 2
∂ z(t, x) + ξ(t, x)z(t, x)
2 x

Current distribution for a two-species particle model from first principles

September 20196 / 31

KPZ growth

Fluctuations
The Laplace transform formula for z(t, x) can be written as a Fredholm determinant

Theorem (Laplace transform of SHE)

ZKζ (η, η 0 ) =E[e−ζz(t,0) ] = det(I − Kζ )L2 (R+ )ζ

1/2
R ζ + e−ξtAi(ξ + η)Ai(ξ + η 0 )dξ.

Theorem (Fluctuations of SHE)
lim Pt→∞h(t, x) − t<st1/3

= FGUE (s).

FGUE (s) is the Tracy-Widom distribution of the Gaussian Unitary Ensemble of random matrix
theory.
Jan de GierCurrent distribution for a two-species particle model from first principles

September 20197 / 31

Exclusion Process

Universality
The Tracy-Widom distribution also appears in the asymmetric exclusion process on Z.

Let Ny (t) be the number of particles to have crossed a given site y after time t.
p
Let Qy (t) = τNy (t) with τ = qande y (t) = Qy (t) − Qy−1 (t) .
Q
τ−1Jan de Gier

Current distribution for a two-species particle model from first principles

September 20198 / 31

Exclusion Process

Fundamental solution of ASEP
Theorem (Fundamental solution of ASEP)IIe x (t) · · · Qe x (t)] = · · ·E[Qk1Y16i<j6kzi − zj Y −λ(zj )tezi − τzjk

j=1

1 + zj /τ
1 + zjxj +1dzj

Fluctuations of particle flow across the origin follow KPZ statistics given by the Tracy-Widom
distribution:

Theorem (Fluctuations of ASEP)
lim Pt→∞

Jan de Gier
N0 (t) − vt
> −s = FGUE (s).
1/3tCurrent distribution for a two-species particle model from first principles

September 20199 / 31Exclusion Process

Summary
Ingredients:
Integrable stochastic lattice model
Observable expressed in terms of k-fold integral
Asympotics for large k → Fredholm determinant
Saddle point analysis
New results:
Rank two model
Dynamic poles in integral (from nested Bethe ansatz)
Combination of Gaussian and GUE modes

Jan de Gier

Current distribution for a two-species particle model from first principles

September 201910 / 31Multi-species model

AHR model
Introduced by Arndt-Heinzl-Rittenberg, the transition rates arep:(•, · ) → ( · , •)p0 :( · , ◦) → (◦, · )1:(•, ◦) → (◦, •)Throughout we will take p + p 0 = 1 → factorised steady state.

Jan de Gier

Current distribution for a two-species particle model from first principles

September 201911 / 31Multi-species model

Nonlinear Fluctuating Hydrodynamics
Continuity equation∂u(x, t)∂j(u(x, t))+= 0,∂t∂x
where u(x, t) = (ρ+ , ρ− ) and j(u) = (j+ , j− ) given by non-linear flows
j+ (u) = ρ+ (1 − ρ+ − ρ− ) + 2ρ+ ρ− ,j− (u) = −(1 − ρ+ − ρ− )ρ− − 2ρ+ ρ− .

Adding diffusion and noise, heuristic non-linear fluctuating hydrodynamics (NLFHD) leads toPcrossing (t) ∼ FGUE (s+ )FGauss (s− ),where s± are eigenmodes.

Aim of this talk is to rigorously prove this.

Jan de Gier

Current distribution for a two-species particle model from first principles

September 201912 / 31Multi-species modelGreen’s function
Let ~z=~x∪~y be coordinates of + and − particles.
Definition
The Green function satisfies the master equationXdM~z;~z 0 G(~z 0 , t);G(~z; t) =dt0t > 0,~zand initial condition
G(~x; ~y; 0) =nY

δx ,x0ii=1imYδy ,y0jj=1jExplicit form can be determined by Bethe ansatz using boundary conditions:Exclusion:

G(x, x; ~y; t) = G(x, x + 1; ~y; t),

Scattering: G(y; y + 1; t) = qG(y + 1; y + 1; t) + pG(y; y; t).

Jan de Gier

Current distribution for a two-species particle model from first principles

September 201913 / 31Multi-species modelGreen’s function
Initial conditions: assume x0j < y0k , i.e. at t = 0 all • particles are to the left of all ◦ particles.
Final condition: xj > yk , i.e. at time t all ◦ particles are to the left of all • particles.
ThenG(xj , yk , t) =IYnd zj


× det
nm Y
Yd wk eΛt

k=1 j=1

k=1

j=1

× det

mY

zj − 1zi − 1

j−1

wk − 1w` − 1

xzi j

m−k!

1qzj + pwk−x0j −1zj!
−yw` ky0 −1wk k,with all contours around the origin, and with eigenvalue
Λ=pnmXX(z−1−1)+q(w−1jk − 1).
j=1

Jan de Gier

k=1

Current distribution for a two-species particle model from first principles

September 201914 / 31

Current distribution

Current distribution: step initial condition
Given the following step initial condition
ntimesmtimes∞X−nXz }| { z }| {······• • ··· • •◦ ◦ ··· ◦ ◦······ ,ThenP(x1 (t) > s) =∞X···

x1 =s−1X···xn =xn−1 +1 y1 =−∞G({xj }; {yk }; t),yn =yn−1 +1

Proposition

P(x1 (t) > 0) =YIYnd zj

j=1mY

d wk eΛt

k=1

16i<j6nnY(zj − 1)j=1

Y(zi − zj )(wl − wk )16k<l6mmYnY

zn−j+sjj=1mY

wk−1
kk=1n Ym 
Y(wk − 1)
qzj + pwk

n+1−j

k=1,

k

j=1 k=1

with all contours around the origin.
Jan de Gier

Current distribution for a two-species particle model from first principles

September 201915 / 31

Current distribution

Current distributioneΛt produces an essential singularity at origin: Λ = pPnPm−1−1
j=1 (zj − 1) + q
k=1 (wk − 1).

Deform w-contours to lie around poles other than the origin
Only (simple) poles at w = 1 give nonzero contribution
After evaluating the residues in w, we getProposition
YP(x1 (t) > 0) =j=1

Jan de GiernY

zn−jjd zj Λ̃t
16i<j6nj=1enn
mYY
2πi
j=1
(zj − 1)n+1−j
qzj + pIYn(zi − zj )j=1

Current distribution for a two-species particle model from first principles

September 201916 / 31Current distributionTASEP limit
Corollary
When n = m and p = q = 1/2 we retrieve the same distribution as for the single species TASEP understep initial condition, i.e.
Pn,n,1 (t) =1n!

IYnQd xj eEt

j=1

where the contours are still around the origin and E =16i<j6n (xi − xj )Qnn
j=1 (xj − 1)

2,

Pn

−1
j=1 (xj − 1).

This is made explicit by symmetrising and the simple change of variable zj = xj /(2 − xj ).
From known analysis (Tracy-Widom) this probability converges to the GUE distribution asn, t → ∞.

Jan de Gier

Current distribution for a two-species particle model from first principles

September 201917 / 31

Current distribution

Step-Bernoulli condition

Let the distance among positive particles be independently distributed with parameter ρ 0 ,Proposition
The total exchange probability Pn,m,ρ (t) with Bernoulli initial data is given byPn,m,ρ (t) =IYn
j=1

d zjmYd wk eΛn,m t ×

k=1

ρnYYnYzn−jjmYwk−1
k
16k<l6mj=1
k=1nmn Ym ,YYY(zj − 1)n+1−j 1 − ρ 0 zj
(wk − 1)kqzj + pwk
j=1k=1j=1 k=1(zi − zj )(wl − wk )16i<j6nwith all contours around the origin.

The w-contours can be readily evaluated if n > m but not when n < m
Jan de Gier

Current distribution for a two-species particle model from first principles

September 201918 / 31

Current distribution

Exchange

Jan de Gier

Current distribution for a two-species particle model from first principles

September 201919 / 31

Asymptotics

Asymptotics
Non-linear fluctuating hydrodynamics (KPZ formalism) suggests a scaling limit of the formn = j1 t + αt1/3 + βt1/2 ,m = j2 t + γt1/3 + δt1/2 ,where j1,2 , α, β, γ, δ are known functions of ρ 0 , and n < m.
Need to analyseIIPn,m,ρ (t) = . . . factorised integrand| {z }
n×m

where n, m, t are large.
Trick: Convert to Fredholm determinant:Pn,m,ρ (t) = det(I − AB)m×m = det(I − BA)L2 (R) ,where n, m, t all occur as parameters in BA.

Jan de Gier

Current distribution for a two-species particle model from first principles

September 201920 / 31

Asymptotics

ExampleAfter symmetrising and deforming contours, P involves integrals of the form
I=1m!

I
0,1,−1Q

2
16k<`6m (w` − wk ),Qm 1mmn
k=1 wk (wk − 1)
k=1 ( 2 (1 + 1/wk ))eΛ̃m tdm w Qm

P
where Λ̃m = 21 m
k=1 (wk − 1).
This expression can be written as

I=1m!

I

dm w
Qm
0,1,−1
k=1 wkQmYgn,m (wk , 0)16k,`6m (1 − wk /w` )Qmgn,m (ak , 0)k,`=1 (1 − a` /wk )

with

k=1

gn,m (w, x) =and ak = 1,ww+1

nwx−m ewt/2 ,1 6 k 6 m.

Assuming first that |ak | < 1, the integral I can be written as a Fredholm determinant
I = det(I − K(x, y)).
Jan de Gier

Current distribution for a two-species particle model from first principles

September 201921 / 31

Asymptotics

ExampleThe kernel
K(x, y) =m−1X

φk (x)ψk (y),k=0withI k+1
1+w nwe−wt/2 ,w
1−w
wx+11
n 
I1 − w k wt/2w
ψk (x) =e,d w wx−21+ww
0,−1
φk (x) =dw

1

in which the pole at w = 1 is separated from the poles at w = −1, 0.
We rewrite this integralIφk (x) =dw

1 2πief(w)twith k ∼ m and
f(w)t = n log
Jan de Gier

1+ww

+ (m − κ1 t1/3 log



w
1−w



− ξ1 t1/3 log(w) −Current distribution for a two-species particle model from first principles

1wt,2September 201922 / 31

Asymptotics

In the scaling limit, f(w) has the double saddle point at w∗ = ρ 0 /2:−10× ∗w

1Contours for φk (x), around 1, and ψk (x); joint saddle at w∗ = ρ 0 /2.
f(w)t = f(w∗ )t − c1 (w − w∗ )t1/3 + c2 (w − w∗ )3 t + . . .
Define the scaling variable v byw − w∗ =iv
c3
t1/3

The function f(w) behaves asymptotically asf(w)t = f(w∗ )t − iv3 − iv κ + ξ + O(t−1/6 ).

Jan de Gier

Current distribution for a two-species particle model from first principles

September 201923 / 31

Asymptotics

This leads to∗φm−t1/3 κ (t1/3 ξ) ∼ef(w )t Ai κ + ξ .
The scaled kernel therefore has the long time limit
Z∞  Kt (ξ, ζ) →Ai κ + ξ Ai κ + ζ d κ := A(ξ, ζ)

as t → ∞.

0Proposition
The functions φk and ψk behave asymptotically as Airy functions. In the scaling limit the integral I
therefore satisfies
I ∼ det(1 − K(ξt1/3 , ζt1/3 ) = det(1 − Kt (ξ, ζ)),Z∞

such thatAi(κ + ξ)Ai(κ + ζ) d κ.

lim Kt (ξ, ζ) =t→∞0

Non-Gaussian distribution function

Jan de Gier

Current distribution for a two-species particle model from first principles

September 201924 / 31

Asymptotics

Nested poles
The crossing probability involves mixed integrals of the form
Iz × Iw :=QQn
20
16i<j6n (zi − zj )j=1 (1 − ρ zj )×
1nm
1
j=1 (zj − 1) ( 2 (1 + zj ))QI
2
1
16k<l6m (wl − wk )×dm w eΛ̃m t Qm
m−nm! 0,±1,−ρ 0(wk − 1)m
k=1 wkI

1n!

dn zeΛn tQn

mn−1 mY
1 + 1/ρ 0 Y Y 1 + zj.
01 + wk /ρ
1 + zj w kk=1

j=1 k=1

The integral Iw now depends on {zj }. The Fredholm kernel isK(x, y) =m−1X

φk (x)ψk (y),

k=0

andI
1

Jan de Gier

1 + w/ρ 0 Y 1 + zj w
w
wx+1 (w − 1)n−1dw

φk (x) =

j=1

w
1−w

ke−wt/2 .

Current distribution for a two-species particle model from first principles

September 201925 / 31

Asymptotics

Dominant contributionConsider the contributions arising from the combined poles at zj = 1 and wk = 1 in Iz × Iw
QIdn z1

16i<j6n (zi − zj )Qnn
j=1 (zj − 1)

QI

2dm w

×1

16k<l6m (wl − wk )Qmm
k=1 (wk − 1)

m
2 n−1
YY
j=1 k=1

1 + zj.
1 + zj wk

The mixed factors in φk expand near z = 11+z
2=+ O((z − 1)(w − 1)).
1 + zw
w+1
For the combined poles, we can set zj = 1 in φk (x)⇒ the z and w integration factorises

Jan de Gier

Current distribution for a two-species particle model from first principles

September 201926 / 31

Asymptotics

Final result
The z-integration produces a Gaussian and the w-integration a GUE distribution.
TheoremIn the appropriate scaling limit
lim Pn,m,ρ (t) = FGUE (s+ )FGauss (s− ),t→∞
1 (1 + ρ)n − (3 − ρ)m + 12 (1 − ρ)(1 − (1 − ρ)2 /4)t ,c2 t1/31 
s+ (n, m; t) =:− 2(2 − ρ)n + 2ρm + (2 − ρ)(1 − ρ)ρt ,cg t1/2
s− (n, m; t) =:Jan de Gier

Current distribution for a two-species particle model from first principles

September 201927 / 31

Asymptotics

Conclusion(First) proof of Nonlinear Fluctuating Hydrodynamics for a two-component mixture
Using integrability
Mix of Gaussian and KPZ modes
Dynamic poles in integrandJan de Gier

Current distribution for a two-species particle model from first principles

September 201928 / 31

Nagel-Schreckenberg

Universality in Nagel-Schreckenberg model
Joint work with Andreas Schadschneider, Johannes Schmidt, Gunter Schütz.
NaSch definition:1

Acceleration:2

Deceleration: If vn > dn i.e.

3Randomisation:

vn → min (vn + 1, vmax )vn → min (vn , dn )psvn → max (vn − 1, 0) .
4Vehicle movement:
x n → x n + vn

At vmax = 1 this is equivalent to (discrete time) TASEP.
Does KPZ universality survive when vmax > 1?

Jan de Gier

Current distribution for a two-species particle model from first principles

September 201929 / 31

Nagel-Schreckenberg

Structure functionS(x, t) = hu(x, t)u(0, 0)i ' t−2/3 fPS t−2/3 (x − vt)with the Prähofer-Spohn scaling function fPS .

Jan de Gier

Current distribution for a two-species particle model from first principles

September 201930 / 31

Nagel-Schreckenberg

Integrated current

Zt
Jt

=vcol
Ztu(x, 0)dx

[j(0, s) − j(ρ)]ds −00
P(J, t) ' t−1/3 FBR −J · t−1/3
with the Baik-Rains scaling function FBR (x).
For vmax = 3:

Jan de Gier

Current distribution for a two-species particle model from first principles

September 201931 / 31