searching for the tracy-widom distribution in nonequilibrium...
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Searching for the Tracy-Widom Distribution inNonequilibrium Processes(joint work with Herbert Spohn)
Christian B. Mendl
Stanford University, USA
October 19, 2016
Outline and general concepts
Multiple scales:
microscopic FPU-type chains(molecular dynamics)
mesoscopic KPZ partial differentialequation
macroscopic fluid dynamics(hyperbolic conservation laws)
Uncover statistical Tracy-Widomdistribution in one-dimensional particlesystems
FPU-type anharmonic chains
Particles with positions qj and momenta pjHamiltonian:
H =∑j
1
2mp2j + V (qj+1 − qj)
Interaction potential depends only on difference qj+1 − qj momentum conservation
Equations of motion
ddt rj = 1
m (pj+1 − pj)ddt pj = V ′(rj)− V ′(rj−1)
with the stretch rj = qj+1 − qj
FPU-type anharmonic chains
Particles with positions qj and momenta pjHamiltonian:
H =∑j
1
2mp2j + V (qj+1 − qj)
Interaction potential depends only on difference qj+1 − qj momentum conservation
Equations of motion
ddt rj = 1
m (pj+1 − pj)ddt pj = V ′(rj)− V ′(rj−1)
with the stretch rj = qj+1 − qj
FPU-type anharmonic chains: Conserved fields
Conserved fields:
~g(j) =
r jpje j
stretchmomentumenergy
with stretch r j = qj+1 − qj and energy e j = 12mp2j + V (rj)
Microscopic currents
~J (j) =
− 1mpj
−V ′(rj−1)− 1
mpjV′(rj−1)
microscopic conservation law
ddt~g(j , t) + ~J (j + 1, t)− ~J (j , t) = 0
FPU-type anharmonic chains: Conserved fields
Conserved fields:
~g(j) =
r jpje j
stretchmomentumenergy
with stretch r j = qj+1 − qj and energy e j = 12mp2j + V (rj)
Microscopic currents
~J (j) =
− 1mpj
−V ′(rj−1)− 1
mpjV′(rj−1)
microscopic conservation law
ddt~g(j , t) + ~J (j + 1, t)− ~J (j , t) = 0
Local thermodynamic equilibrium
Canonical ensemble distribution depending oninverse temperature β, pressure P and average momentum ν:
1
Zexp[−β(
12m (pj − ν)2 + V (rj) + Prj
)]dpj drj
rj , pj are independent (ensemble distribution factorizes)
Using integration by parts: P = −〈V ′(x)〉P,β
u(x,t)
~u = (r , ν, e), with total energy e = e + 12ν
2
From microscopic to macroscopic conservation laws
microscopic ddt gα(j , t) + Jα(j + 1, t)− Jα(j , t) = 0
↓macroscopic ∂tuα(x , t) + ∂x jα(~u(x , t)) = 0
~u = (r , ν, e) and α = 1, 2, 3.
u(x,t)
Macroscopic Euler currents: (local) thermal averages of themicroscopic currents:
~j(r , ν, e) = 〈 ~J 〉P,ν,β =(−ν,P(r , e− 1
2ν2), νP(r , e− 1
2ν2))
From microscopic to macroscopic conservation laws
microscopic ddt gα(j , t) + Jα(j + 1, t)− Jα(j , t) = 0
↓macroscopic ∂tuα(x , t) + ∂x jα(~u(x , t)) = 0
~u = (r , ν, e) and α = 1, 2, 3.
u(x,t)
Macroscopic Euler currents: (local) thermal averages of themicroscopic currents:
~j(r , ν, e) = 〈 ~J 〉P,ν,β =(−ν,P(r , e− 1
2ν2), νP(r , e− 1
2ν2))
From hydrodynamics to KPZ
Generic Euler equation (hyperbolic conservation law)
∂t u(x , t) + ∂x j(u(x , t)) = 0
Expand current to second order in u, add dissipation plus noise Langevin (stochastic Burgers) equation
∂tu(x , t) + ∂x(
j′(u)︸︷︷︸velocity c
u + 12 j′′(u)u2︸ ︷︷ ︸
nonlinear current
− D∂xu︸ ︷︷ ︸dissipation
+ ξ︸︷︷︸noise
)= 0,
where ξ(x , t) is space-time white noise.
Interpret as derivative of height function: u(x , t) = ∂xh(x , t) KPZ equation
From hydrodynamics to KPZ
Generic Euler equation (hyperbolic conservation law)
∂t u(x , t) + ∂x j(u(x , t)) = 0
Expand current to second order in u, add dissipation plus noise Langevin (stochastic Burgers) equation
∂tu(x , t) + ∂x(
j′(u)︸︷︷︸velocity c
u + 12 j′′(u)u2︸ ︷︷ ︸
nonlinear current
− D∂xu︸ ︷︷ ︸dissipation
+ ξ︸︷︷︸noise
)= 0,
where ξ(x , t) is space-time white noise.
Interpret as derivative of height function: u(x , t) = ∂xh(x , t) KPZ equation
Kardar-Parisi-Zhang (KPZ) and 1D surface growth
Growing interfaces described by the Kardar-Parisi-Zhang equation
Growing interfaces of liquid-crystalturbulence (Takeuchi and Sano 2010;Takeuchi et al. 2011)
Bacteria growth (APS Physics,Yunker et al. 2013)
Kardar Parisi Zhang (1986)
∂th(x , t) = 12λ(∂xh)2︸ ︷︷ ︸
tilt-dependent growth
+ D∂2xh︸ ︷︷ ︸dissipation
+ ξ︸︷︷︸noise
Tracy-Widom distribution for wedge initial geometry
-1 0 1x
1
h(x,t)
-6 -4 -2 2 4x
0.1
0.2
0.3
0.4
F2(x)
(T)ASEP particle process: integrated current Φ(x , t) correspondsto height function h(x , t), converges to GUE Tracy-Widomdistribution F2(x):
Φ(0, t) ' 14 t −
(2−4t
)1/3ξTW
(Johansson 2000 for TASEP, Tracy and Widom 2009 for ASEP)
Wedge initial geometry translated to anharmonic chains
Derivative u = ∂xh(x , t) has a jump for “wedge” initial conditions:
-1 0 1x
1
h(x,0)
-1 1
x
-1
1
∂xh(x,0)
Remember: ~u = (r , ν, e) are the local average field variables “domain wall” initial conditions, i.e., solve Riemann problem
~u(x , 0) = ~u− for x ≤ 0, ~u(x , 0) = ~u+ for x > 0
Integrated current and TW for anharmonic chains
Integrated current for anharmonic chains (corresponds to h(x , t)):
Φ(x , t) =
∫ t
0j(x , t ′) dt ′ −
∫ x
0u(x ′, 0)dx ′
(path independent since (−u, j) is curl free)
u- u
+
-2 -1 0 1 2x
1
t
In analogy to (T)ASEP, for integration within rarefaction wave:expecting
Φ(vt, t) ' a0t + (Γt)1/3ξTW
Linearization of the current for several fields
Explicitly consider three-component field ~u (stretch, momentum,energy):
A(~u) =∂j(~u)
∂~u=
0 − 1m 0
∂rP − νm∂eP ∂eP
νm∂rP
1mP − ( νm )2 ∂eP
νm∂eP
with pressure P(r , e)
A(~u) has eigenvalues 0 and ±c(~u), with c(~u) the adiabatic speedof sound:
c(~u)2 = 1m (−∂rP + P ∂eP)
Generalization to several fields
Three-component noisy Burgers equation:
∂t~u + ∂x(A~u + 1
2〈~u, ~H~u〉 − ∂x D~u + B~ξ(x , t))
= 0
Hessians: Hαγγ′ = ∂uγ∂uγ′ jα, jα = 〈Jα〉
Initial correlations: 〈uα(x , 0); uα′(x ′, 0)〉 = Cαα′ δ(x − x ′)
Diagonalization (transformation to normal modes):
~φ = R~u, RAR−1 = diag(−c , 0, c), RCRT = 1
∂tφα + ∂x(cαφα + 1
2〈~φ,Gα~φ〉 − ∂xDφα + B~ξ(x , t)
)= 0
α = 1α = -1
α = 0
-1000 -500 500 1000x
0.002
0.004
0.006
0.008
Sαα# (x,t)
Generalization to several fields
Three-component noisy Burgers equation:
∂t~u + ∂x(A~u + 1
2〈~u, ~H~u〉 − ∂x D~u + B~ξ(x , t))
= 0
Hessians: Hαγγ′ = ∂uγ∂uγ′ jα, jα = 〈Jα〉
Initial correlations: 〈uα(x , 0); uα′(x ′, 0)〉 = Cαα′ δ(x − x ′)
Diagonalization (transformation to normal modes):
~φ = R~u, RAR−1 = diag(−c , 0, c), RCRT = 1
∂tφα + ∂x(cαφα + 1
2〈~φ,Gα~φ〉 − ∂xDφα + B~ξ(x , t)
)= 0
α = 1α = -1
α = 0
-1000 -500 500 1000x
0.002
0.004
0.006
0.008
Sαα# (x,t)
Riemann problem: Rarefaction curves
Rarefaction curves: solve the following Cauchy problem in statespace (see e.g. Bressan 2013)
∂τ ~u = ψα(~u),
α = 0,±1, where ψα are the right eigenvectors of A = ∂j(~u)∂~u .
Specifically for eigenvalue σc , σ = ±1:
∂τ
rve
=
− 1mσc
1mσP + ν
mc
u- u
+
-2 -1 0 1 2x
1
t
Riemann problem solution for periodic boundaries
Analytical solution of the periodic Riemann problem (anharmonicchain of hard-point particles with alternating masses):
R1S1 R0S-1
u0 u1
u2 u3
u1
-L
2-L
40
L
4
L
2
x
1
t
Riemann problem: Molecular dynamics
Average profiles of the stretch rj(t) and momentum pj(t) att = 1024
orange dots: molecular dynamicsblack curve: analytical solution of the Riemann problem
-2000 -1000 0 1000 2000j
1.0
1.1
1.2
1.3
⟨r j ⟩
-2000 -1000 1000 2000j
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
⟨pj ⟩
Transformation to normal modes
Projected current components (with asymptotic value subtracted):
Φ]σ(t) =
⟨ψσ|~Φ(vt, t)− t(~j(~uv)− v~uv)
⟩with ψσ the left eigenvectors of A(~uv)
Φ]1(t) results from perturbations propagating with velocity v along
the magenta observation ray expecting
Φ]1(t) ' (Γ1t)1/3ξTW
Components σ = −1, 0 pick up samples from essentiallyindependent space-time regions central limit type behavior
Φ]σ(t) ' (Γσt)1/2ξG, σ = 0,−1,
with ξG a normalized Gaussian random variable
Transformation to normal modes
Projected current components (with asymptotic value subtracted):
Φ]σ(t) =
⟨ψσ|~Φ(vt, t)− t(~j(~uv)− v~uv)
⟩with ψσ the left eigenvectors of A(~uv)
Φ]1(t) results from perturbations propagating with velocity v along
the magenta observation ray expecting
Φ]1(t) ' (Γ1t)1/3ξTW
Components σ = −1, 0 pick up samples from essentiallyindependent space-time regions central limit type behavior
Φ]σ(t) ' (Γσt)1/2ξG, σ = 0,−1,
with ξG a normalized Gaussian random variable
Results based on molecular dynamics simulations
Top row: statistical distribution of (Γ1t)−1/3 Φ]1(t)
Bottom row: projection (Γ−1t)−1/2 Φ]−1(t)
-6 -4 -2 2 4Φ1
#-0.35
0.1
0.2
0.3
0.4
probability
-6 -4 -2 0 2 4Φ1
#-0.35
10-6
10-4
10-2
1probability
-4 -2 2 4Φ-1
#+0.04
0.1
0.2
0.3
0.4
probability
-6 -4 -2 0 2 4 6Φ-1
#+0.04
10-510-410-310-210-1
probability
Summary and conclusions
Nonlinear fluctuating hydrodynamicsidentified with KPZ equation
Tracy-Widom GUE distributionemerges for “wedge” initial condition,translates to Riemann problem foranharmonic chains
Observing Tracy-Widom GUE forprojected current componentintegrated within rarefaction wave
u- u
+
-2 -1 0 1 2x
1
t
-6 -4 -2 2 4Φ1
#-0.35
0.1
0.2
0.3
0.4
probability
References I
Bressan, A. (2013). “Modelling and Optimisation of Flows on Networks,Cetraro, Italy 2009”. In: vol. Lecture Notes in Mathematics 2062.Springer. Chap. Hyperbolic conservation laws: An illustrated tutorial,pp. 157–245.
Johansson, K. (2000). “Shape fluctuations and random matrices”. In:Commun. Math. Phys. 209, pp. 437–476.
Mendl, C. B. and Spohn, H. (2013). “Dynamic correlators of FPU chainsand nonlinear fluctuating hydrodynamics”. In: Phys. Rev. Lett. 111,p. 230601.
– (2014). “Equilibrium time-correlation functions for one-dimensionalhard-point systems”. In: Phys. Rev. E 90, p. 012147.
– (2015). “Current fluctuations for anharmonic chains in thermalequilibrium”. In: J. Stat. Mech. 2015, P03007.
– (2016a). “Searching for the Tracy-Widom distribution innonequilibrium processes”. In: Phys. Rev. E 93, 060101(R).
– (2016b). “Shocks, rarefaction waves, and current fluctuations foranharmonic chains”. In: J. Stat. Phys.
References II
Spohn, H. (2014). “Nonlinear fluctuating hydrodynamics for anharmonicchains”. In: J. Stat. Phys. 154, pp. 1191–1227.
Takeuchi, K. A. and Sano, M. (2010). “Universal Fluctuations ofGrowing Interfaces: Evidence in Turbulent Liquid Crystals”. In: Phys.Rev. Lett. 104, p. 230601.
Takeuchi, K. A. et al. (2011). “Growing interfaces uncover universalfluctuations behind scale invariance”. In: Sci. Rep. 1, p. 34.
Tracy, C. A. and Widom, H. (2009). “Asymptotics in ASEP with stepinitial condition”. In: Commun. Math. Phys. 290, pp. 129–154.
Yunker, P. J. et al. (2013). “Effects of particle shape on growth dynamicsat edges of evaporating drops of colloidal suspensions”. In: Phys. Rev.Lett. 110 (3), p. 035501.