with P.L. Krapivsky and Arkady Vilenkin Prof. Giulio Racah - - PowerPoint PPT Presentation

Survival of a target in a gas of diffusing particles with exclusion

Baruch Meerson

Racah Institute of Physics, Hebrew University of Jerusalem

Advances in Nonequilibrium Statistical Mechanics, GGI, Firenze, July 4 2014 with P.L. Krapivsky and Arkady Vilenkin

• Prof. Giulio Racah

1909-1965 He was born in Firenze, and died here

Plan

Macroscopic Fluctuation Theory of diffusive lattice gases MFT in non-stationary settings: examples

Survival of a target against “searchers”

• a. Stationary fluctuations: d>2, long times
• b. Non-stationary fluctuations: d=1, and any d for intermediate times

Extensions and summary

Diffusive lattice gases

SSEP: simple

symmetric exclusion process

RWs, ZRP: a=a(ni)

random walkers; zero-range process

Large-scale behavior: fluctuating hydrodynamics

 ,

) , ( ) ( ) ( t D

t

x ξ           

x: Gaussian noise, delta-correlated in x and t Diffusive lattice gases are fully characterized, at large scales, by the diffusivity D() and mobility () Spohn 1991, Kipnis and Landim 1999

D() and () are related to the equilibrium free energy density F():

) ( ) ( 2 ) (

2 2

     D d F d 

 

) , ( ) ( ) ( t D

t

x ξ           

When noise is ignored: diffusion equation

 

        ) ( D

t

Macroscopic Fluctuation Theory (MFT)

2

) )( ( 2 1 ) ( )] , ( ), , ( [ p q p q q D d t p t q H           H H, x x x

MFT can be derived from fluctuating hydrodynamics via saddle-point expansion of a proper path integral (Tailleur, Kurchan, Lecomte 2007). This leads to a minimization problem that can be cast into a classical Hamiltonian field theory for the particle density q(x,t) and conjugate “momentum” density p(x,t):

2 2

) )( ( ' 2 1 ) ( )] ) ( ) ( [ p q p q D p p q q q D q

t t

              

Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim (2001, …) Large parameter: number of particles in a relevant region of space. Generalizes the weak-noise WKB theory of Freidlin and Wentzel to fields Similar in spirit: Elgart and Kamenev (2004), M and Sasorov (2010) – WKB approximation to master equation for random walk on lattice and on-site reactions. Large parameter: number of particles on a single site

, / , / q H p p H q

t t

        

Mean-field (noiseless) limit: p(x,t)=0: downhill trajectories Fluctuations: p(x,t)≠0: uphill trajectories, the optimal density history The probability density of a large deviation is given by the mechanical action along a proper uphill trajectory:

2 2

) )( ( ' 2 1 ) ( )] ) ( ) ( [ p q p q D p p q q q D q

t t

              

 

   

      

T T t

p q dt d t q t p dt d S

2

) )( ( 2 1 ) , ( ) , ( ln  x x x x H P

 

q q D q

t

     ) (

Boundary conditions, in x and t, are determined by specific problem. If the initial condition is random, one should also find the optimal initial density profile and add to S the Boltzmann-Gibbs free energy “cost” of creating it

MFT emerged in the context of non-equilibrium steady states of lattice gases

ρ+ ρ- L

 

           L) (x 0) (x const / ) ( dx d D

 

1 )] / ( [ exp ~ )] ( [   L L x F L x   P

Density fluctuations Expected density profile solves the steady-state mean-field problem

)] / ( [ L x F 

large deviation functional

Reviews: Derrida 2007, Jona-Lasinio 2010, Bertini et al 2014

MFT emerged in the context of non-equilibrium steady states of lattice gases

ρ+ ρ- L

L A J ) , (

 

  

1 )], , , ( exp[ ~ ) (  

 

L J S L J   P

Fluctuations of current Average current

) , , (

  

 J S

large deviation function

Reviews: Derrida 2007, Jona-Lasinio 2010, Bertini et al. 2014

What is the most probable density profile for given J ?

MFT emerged in the context of non-equilibrium steady states of lattice gases

ρ+ ρ- L

Reviews: Derrida 2007, Jona-Lasinio 2010, Bertini et al. 2014

• Non-locality: long range correlations
• Uphill trajectory is different from time-reversed downhill trajectory
• Non-smooth parameter dependence of large deviation

function/functional: “phase transitions”

Example 1: Formation of void of size L at time T in an initially uniform gas

Krapivsky, M and Sasorov 2012

n

1 , 1 )], , ( exp[ ~ ) , (

2 /

   L T n T L S T T L

d d

P

) , ( n T L Sd

large deviation function; Most probable density history d: dimension of space

Non-stationary settings are also interesting

Non-stationary settings are also interesting

Example 2: Fluctuations of mass/energy transfer in finite time

Derrida and Gerschenfeld 2009a,2009b, Sethuraman and Varadhan 2011, Krapivsky and M 2012, M and Sasorov 2013, 2014, Vilenkin, M and Sasorov 2014

 

1 )], , , ( exp[ ~ ) (  

 

T T M S T M

T T

  P

? ) , , ( 

  

 T M S

T

1 ,   

 

T T MT   

for Random Walkers (RWs), SSEP and KMP

What is the most probable history of the density field conditional on MT ? Large deviation function Even is nontrivial

   

 dx x T x MT )] , ( ) , ( [  





 

Example 3 (this talk): Target survival problem

Non-stationary settings

R n0

What is the probability that no particle hit the target until t=T? What is the most probable density history of the gas conditional on the non-hitting? For a given lattice gas, the answers depend on three parameters:

, , n d DT R l 

Diffusion-controlled reactions Smoluchowski 1917

The T→∞ asymptotic of the target survival probability is known for ideal gas (RWs), see references in

Bray, Majumdar and Schehr, Adv. Phys. 62, 225 (2013)

Most probable density histories have not been found even for ideal gas. For non-ideal gases such as SSEP there are no previous results, except for some bounds.

MFT formulation is similar to that for the mass transfer: + spherical symmetry Boundary condition: The process is conditional on N absorbed particles by time T:

2 2

) )( ( ' 2 1 ) ( )] ) ( ) ( [ p q p q D p p q q q D q

t t

              

This integral constraint calls for a Lagrangian multiplier  and leads to additional boundary condition (in time) coming from the minimization of action:

) ( ) , ( R r T t r p    

N T r q n r dr d

R d d

  



 

)] , ( [ ) 2 / ( 2

1 2 /



) , (   t R r q

The parameter  is ultimately fixed by N=0 M and Redner 2014

Once q(r,t) and p(r,t) found:

) , ( n t R r q   

Deterministic, or quenched, initial condition Random, or annealed initial condition introduces two changes:

• the initial condition becomes p-dependent:

(Derrida and Gerschenfeld 2009)

• when evaluating the probability, one should add to S the Boltzmann-Gibbs free energy

“cost” of creating the optimal initial density profile q(r,0) described by Eq. (1)

 

    

  T r R d d

p q r dr dt d

2 1 2 /

) )( ( ) 2 / ( S(N) ln   P

) 1 ( ) ( ) ( ) ( 2 ) , (

) , ( 1 1 1

R r q q D dq r p

r q n

  



 

MFT equations are invariant under rescaling

The radius of absorber becomes

 

           

  1 2 1 2 / 2 / 2 /

) )( ( ) 2 / ( , , ) ( , ) ( S ln p q r dr dt d s n DT N l s DT

r l d d d d

  P

Dynamic scaling of the absorption probability

x x   DT t T t / , /

DT R l / 

We are interested in the limit N->0

d=1: S is independent of R, so s doesn’t depend on l leading to survival probability

) ( ) ( ln

1 2 / 1

n s DT   P

for all diffusive lattice gases The T1/2 scaling signals that the 1d-problem is non-stationary. An important consequence is that s1(n0) depends on whether the initial condition is deterministic or random.

 

dr dp v v q v r dr d r q D v q dr dq q D

d d

/ , ) ( ' 2 1 ) ( times all at flux zero ) ( ) (

2 1 1

     

 

 

Long-time asymptotics for d>2: stationary fluctuations

This leads to a single nonlinear ODE for q(r):

                       

 

dr d r dr d r q dr dq D D q

d d r r 1 1 2 2 2

1 where 0, 2 ' '  

SSEP

The nonlinear ODE becomes

0. ) 1 ( 2 1 2

2 2

           dr dq q q q q

r

q)

• 2Dq(1

(q) const, ) (     D q D

A simple change of variables brings this equation to

u(r) sin ) (

2

 r q

0.

2

  u

r

The solution, in the variable q, is

2 , , arcsin 1 sin ) (

2 2 2

                 

 

d DT R l n r l r q

d d

SSEP, d>2

                                    

      2 2 1 2 2 2 2

arcsin 1 2 sin arcsin ) 2 ( 2 ) 1 ( 2 1 ) ( , arcsin 1 sin ) ( n r l r n l d dr dq q q dr dp r v DT R l n r l r q

d d d d d d

 

n

• n

depend t doesn' times; all at singular 1 ) ( zero is target the flux to that the so quadratic 1 arcsin ) 2 ( ) (

2 2 2

r l l l r v l r n d l l r q               

Asymptotics of q and v near the target, r-R<<R:

SSEP, d>2

Taking n0<<1 we get the results for ideal gas”

                                    

      2 2 1 2 2 2 2

arcsin 1 2 sin arcsin ) 2 ( 2 ) 1 ( 2 1 ) ( , arcsin 1 sin ) ( n r l r n l d dr dq q q dr dp r v DT R l n r l r q

d d d d d d 1 2 2 2 2

1 2 ) ( , 1 ) (

   

                            

d d d

l r r d dr dp r v DT R l r l n r q

Survival probability for SSEP

differs from the result for ideal gas only by the density dependence. The LDF increases much faster with the density, but remains finite.

The stationary solution does not satisfy the boundary conditions in time. As a result, boundary layers at t=0 and t=1 develop

Numerical solution: iterations of q forward in time, p backward in time

Chernykh and Stepanov 2001

Stationary solution for d=3: dots t=0: dashed line t=0.25, 0.5 and 0.75: three green lines (coincide) t=1: black line d=3 l=5  10-3 n0=0.5 stheor= 3.876…10-2 snum= 3.9210-2

The stationary solution also implies that the survival probability is independent, at d>2 and l<<1, on whether the initial condition is quenched

• r annealed

For ideal gas (RWs) this prediction is verified by microscopic calculations

M,Vilenkin and Krapivsky 2014

d=2: critical dimension for long-time asymptotic

L~1. In the original variables L~(DT)1/2 Logarithmic accuracy

d=1: Non-stationary fluctuations, SSEP

We have been unable to solve the complete non-stationary problem analytically

• 1. We solved it in the ideal gas limit n0<<1
• 2. We calculated finite-density corrections perturbatively
• 3. We solved the problem numerically for different gas densities

and determined s1(n0).

2 2

) ( ) 2 ( p p p p q q q

t t

            

Ideal gas limit: non-interacting Random Walkers (RWs)

q q q D 2 ) ( , 1 ) (   

Hopf-Cole canonical transformation 



   



P q dx P q dx e P qe Q

q p

ln ) , ( ,

New Hamiltonian

P Q d t P t Q H       H , H x x x )] , ( ), , ( [

P P Q Q

t t 2 2

      

New Hamilton equations are linear and uncoupled

Solution for ideal gas, quenched initial condition

                     







) 1 ( 4 erf ) 1 ( e ) , ( erf e e ) 1 ( 4 erf ) , (

t)

• 4(1
• t
• t
• 2

2 ) (x/2 2 )

• (x/2

t x t x p t x v d t x t n t x q

x

   

 

t x x t x v    1 , 1 ) , (

t=0, 1/3, 2/3, 1

=x/2

A singularity of v at x=0 is present at all times; universal asymptotic, solves steady state eqn. for p:

2

) ' ( ' ' p p   

Action for ideal gas, quenched initial condition:

T n P n d n sRW

1 1 1

ln ... 06883 . 2 , erf 2          





 

This is different from

T n P 2 ln   

• btained for annealed initial condition

The two results (quenched and annealed) can be also obtained from the microscopic model:

Example of quenched initial condition: particles are arranged periodically in space. Annealed initial condition: random distribution. The microscopic theory also gives pre-exponential factors M,Vilenkin and Krapivsky 2014

Solution for ideal gas, annealed initial condition

                          ) 1 ( 4 erf ) 1 ( e ) , ( ) 1 ( 4 erf 4 erf ) , (

t)

• 4(1
• 2

t x t x p t x v t x t x n t x q

x



t x x t x v    1 , 1 ) , (

t=0, 1/4, 1/2, 3/4, 1 A singularity of v at x=0 is present at all times; universal asymptotic, solves steady state eqn. for p:

2

) ' ( ' ' p p   

symmetric around t=1/2, no overshot same as for quenched

Finite density correction

2

) )( 1 ( p q q p q

x x x

      H

Split the SSEP Hamiltonian D=1 in two parts: the ideal gas Hamiltonian,

2

) ( p q p q h

x x x

    

and small correction

2 2 1

) ( p q h

x

  

coming from exclusion interaction. The small correction to action can be computed

• perturbatively. For the quenched initial condition:

where the integration is over unperturbed (that is, ideal gas) trajectory. The final result is

s T P d n n s             





ln ... 08337 . 1 ..., 06883 . 2 erf 2 ...,

2 1 2 2 1

 

Finite density correction

2

) )( 1 ( p q q p q

x x x

      H

Split the SSEP Hamiltonian D=1 in two parts: the ideal gas Hamiltonian,

2

) ( p q p q h

x x x

    

and small correction

2 2 1

) ( p q h

x

  

coming from exclusion interaction. The small correction to action can be computed

• perturbatively. For the annealed initial condition one also needs to calculate the small

correction to the Boltzmann-Gibbs free energy cost. The final result is

 

) ( ln . ... ) 1 2 ( 2 ) (

2

n s T P n n n s

an an

      

That is, for d=1 one obtains different n0-dependences of the survival probability for the SSEP in the quenched and annealed case

The n0

2 correction in the annealed case agrees with Santos and Schütz (2001). They solved a different

problem: of particle injection into a semi-infinite line. Thei problem, however, is directly related to the target survival problem. Thanks to Gunter Schütz for this comment!

Arbitrary densities: numerical solution, quenched initial condition n0=0.8

t=0, 0.25, 0.5 and 1 (0.75 for v) 1/x Numerically found action vs. density

1

ln s T P  

s1 apparently diverges as

1 as ) 1 (

2 / 1 1

  



n n s

l>>1: Intermediate asymptotic of the target survival probability in any dimension

For the SSEP this becomes

   

     

    1 2 1 2 / 1 2 1 2 /

) )( ( ) 2 / ( ) )( ( ) 2 / ( p q dr dt d l p q r dr dt d s

r l d d r l d d

    dimension any in , ) 2 / ( ) ( 2 ln , ) 2 / ( ) ( 2

1 1 2 / 1 1 2 /

d T R n s S d n s l s

d d d d

     

 

  P

Quenched and annealed are different for any d in this limit!

Extensions to other lattice gases

2 ' '

2 2

                dr dq D D q

r

 

• Example. ZRP with departure rate

2 ' ' q (q) , ) ( 2 1 ) (

2 2 2

          q D D q q D n n

r i i

   a

1 2 2 2

1 2 ) ( , 1 ) (

   

                           

d d d

l r r d r v DT R l r l n r q

different n0 dependence

Conjecture (cf. with additivity principle of Derrida): if solution obeying q(l)=0 and q(∞)=n0 exists, it yields P(T)

Extensions to other lattice gases

2 ' '

2 2

                dr dq D D q

r

  2

• 2

condition necessary a yields which , 2

• 2

2 term nonlinear by the balanced is ) ( ' ' ..., ) ( const ) ( for Look as 2 2 2 ' ' as ~ ) ( , ~ ) ( Let                   a  a    a   

  a

r q l r l l r q q q D D q q q q q D

When this condition is met, the action is bounded

Summary

MFT makes it possible to evaluate (in some cases, quite easily) the target survival probability for a class of interacting host gases where no previous results existed Possible applications for diffusion-controlled reactions in crowded environments. One more example of efficiency and versatility of the MFT One more example of ever-lasting effect of initial condition in 1d MFT equations are usually hard to solve. More examples should be worked

• ut to gain experience and intuition

Large deviations in non-stationary problems provide a fascinating insight into non-equilibrium stochastic systems

Thank you