Diffusion of Interacting Particles in One dimension Deepak Kumar School of Physical Sciences Jawaharlal Nehru University New Delhi IITM, Chennai Nov. 9,

Diffusion of Interacting Particles in One dimension

Deepak KumarSchool of Physical Sciences

Jawaharlal Nehru UniversityNew Delhi

IITM, ChennaiNov. 9, 2008

Outline• Introduction and History• Single Particle Diffusion: Role of Boundary conditions• Two-Particle Problem• Bethe’s Ansatz: N-Particle Solution• Tagged Particle Diffusion• Correlations• Applications

• Reference: Phys. Rev. E 78, 021133 (2008)

Introduction• The concept of ‘Single File Diffusion’ was

introduced in a biological context to describe flow of ions through channels in a cell membrane.

• These channels are crowded and narrow so that the ions diffuse effectively in one dimension and cannot go past each other.

• The lattice version of the problem was first considered by T. E. Harris (J. Appl. Probability 2, 323 (1965)

History• Harris showed that the hard-core interaction

introduces a qualitative new feature in the diffusion of particles in one dimension.

• Mean square displacement

• This result received a lot of attention, and has been derived using a number of physical arguments.

• Notably, Levitt used the exact methods of one-dimensional classical gas to obtain this result. Phys. Rev. A 8, 3050 (1973)

2/12 tx

Earlier Work• Numerical studies of the problem also showed

the sub-diffusive behavior of type under the condition of constant density of particles.

• (P. M. Richard, Phys. Rev. B 16, 1393, 1977; H. van Beijeren et al., Phys. Rev. B 28, 5711, 1983)

Now there are some exact results. Rödenbeck et al., (Phys. Rev. E 57, 4382, 1998) obtained the one-particle distribution function

for a nonzero density by averaging over initial positions. They obtained the above behavior.

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Earlier Work• Ch. Aslangul (Europhys. Lett. 44, 284, 1998)

gave the exact solution for N particles on a line with one initial condition: all particles are at one point at t=0.

• Here we give an exact solution for arbitrary initial conditions. We calculate one particle moments and two-particle correlation functions as expansion in powers of .2/1t

Single Particle Solution

]4

)(exp[

4

1),(

)(

)()0,(;)(),(

);sin(or)cos(or)(

)(),(

),(;0),(),(

),(),(

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2

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ikx

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Single Particle Diffusion

)(}]4

)(exp{}

4

)([exp{

4

1),(

)sin()(;)sin()(

0),0( :conditionBoundary

0at xAbsorber

)(]}4

)(exp[]

4

)({exp[

4

1

)cos()();cos()(),(

0/),(condition;Boundary

0at Wall

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0

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tP

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xtxP

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tDk

tDk

Two Particle Solution

)(21

21

22

21

)()(212121

21021

2121

2121

21

21

2211

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21

),(

Xwith

])(exp[

])[,(),,(

;0)- (:ConditionBoundary

Meetingon Reflect Particles :nInteractio Core-Hard

)()()0,,(

),,()(),,(

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txxQ

Two Particle Solution

agnet.antiferroman of

sexcitation ofcontext in thesolution particle two thenggeneralizi

by problem quantum particle-N theessentialy solved Bethe

.at conditionsboundary quantum theusingby obtained

isfactor phase The .degenerate are here added solutions twoThe

][),(),,(

isxfor bosons identical of problem quantum for theSolution

)()]4

)()(exp(

)4

)()([exp(

4

1),,(

21

),(2/)(212121

21

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221

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Two Vicious Walkers

)(}]4

)(

4

)(exp{

}4

)(

4

)([exp{

4

1),,Q(x

0|),,Q(x :conditionBoundary

meeting.on other each annhilate Walkers

12

212

221

222

211

21

21 21

xxDt

Xx

Dt

Xx

Dt

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tx xx

N-Particle Solution

N-Particle Distribution Function

Tagged Particle Diffusion

Large Time Expansion

Mean Displacement

Mean Displacement

Mean Square Displacement

Correlations rkkN xxrkkC ),(

Correlations

Correlations: Central Particle to Others

Correlations: End Particle to Others

An Open Problem

The N particle solution obtained by us and Aslangul shows that the one-particle moments behave as

but the coefficients vanish as N tends to infinity. It is not clear what emerges in the infinite N limit. Properly one should take a finite line and the go over to nonzero density limit.

However, in the present calculations some further conditions like constant density or averaging over initial conditions are imposed, to obtain the sub-diffusive behavior.

2/);( mN

mi tmiDx

);( miDN

Experiments• Diffusion of colloidal particles has been studied in one-

dimensional channels constructed by photolithography (Wei et al., Science 287, 625,2000; Lin et al., Phys. Rev. Lett. 94, 216001, 2005) and by optical tweezers (Lutz et al., Phys. Rev. Lett. 93, 026001, 2004).

• Diffusion of water molecules through carbon nanotubes (Mukharjee et al., Nanosci. Nanotechnol. 7, 1, 2007)

• The experiments track the trajectories of single particles and show a transition from normal behaviour at short times to sub-diffusive behavior at large times.

Applications: Single File DiffusionBiological Applications1. Flow of ions and water through molecular-sized channels in

membranes.2. Sliding proteins along DNA3. Collective behaviour of biological motors

Physical and Chemical Applications4. Transport of adsorbate molecules through pores in zeolites5. Carrier migration in polymers and superionic conductors6. Particle flows in microfluidic devices7. Migration of adsorbed molecules on surfaces8. Highway traffic flows

• Thank You

HistoryThis problem was first investigated on a linear lattice by T. E.

Harris. (J. Appl. Prob. 2, 323, 1965). He obtained a qualitatively nontrivial and important result, i.e.

subdiffusive behaviour of a tagged particle.

He derived the result for an infinite number of particles on an infinite lattice with finite density.

Many workers rederived this result in many ways and checked it numerically for systems with uniform density. Some experiments have investigated the diffusion of colloidal particle through 1D channels created by photolithography or optical tweezers. Another experiment has studied the water diffusion through carbon nanotubes. There is good support for the subdiffusive behaviour.

2/12 txi

Random Thoughts

Life as a random walk

Embrace Randomness

Thank you

Diffusion of Interacting Particles in One Dimension

Outline1. Random Walk and Diffusion2. Boundary Conditions: Method of Images3. Two Interacting Particles on a Line4. N Interacting Particles: Bethe’s Ansatz5. Tagged Particle Diffusion6. Correlations in Non-equilibrium Assembly7. Physical Applications Reference: Phys. Rev. E 78, 021133 (2008)

Random Walk and DiffusionA particle jumps in each step a distance ‘a’ to the right or to the left

on a line with equal probability. Displacement X after N steps

2

2

2

22

2

222

),(),(

2),(),(),(

)],(),([2

1)1,(

),( t,or time steps Nafter Xat being ofy Probabilit

/

;/

0

x

txPD

t

txP

X

PatXP

t

PtXPtXP

NaXPNaXPNXP

NXP

aD

NtDttaNaX

X

xXi

i

Single Particle Solution

)(]}4

)(exp[]

4

)({exp[

4

1

)cos()();cos()(),(

0/),(condition;Boundary ;0at Wall

]4

)(exp[

4

1),(

)(

)()0,(;)(),(

);sin(or)cos(or)(

)(),(

0),(),(

;),(),(

20

20

0

0

20

0

2

2

2

2

0

2

xDt

xx

Dt

xx

Dt

kxkAkxekdkAtxP

xtxPx

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xx

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ekA

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xetxP

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txJ

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txP

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txP

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ikx

tDkikx

ikx

t

Two Particle Solution

21021

222

211

21

)(212121

22

21

)(212121

2121

21

21

;0)- (:ConditionBoundary

Meetingon Reflect Particles :nInteractio Core-Hard

]4

)(

4

)(exp[

4

1),,(

),();()()0,,(

)](exp[),(),,(

Particles ginteractin-Non

),,()(),,(

21

2211

2211

xxyy

Q

x

Q

x

Q

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Xx

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Xx

DttxxQ

ekkAXxXxxxQ

kkDekkAdkdktxxQ

txxQxx

Dt

txxQ

xx

XkXki

xkxki

Two Particle Solution

21021

222

211

21

)(212121

22

21

)(212121

2121

21

21

;0)- (:ConditionBoundary

Meetingon Reflect Particles :nInteractio Core-Hard

]4

)(

4

)(exp[

4

1),,(

),();()()0,,(

)](exp[),(),,(

Particles ginteractin-Non

),,()(),,(

21

2211

2211

xxyy

Q

x

Q

x

Q

Dt

Xx

Dt

Xx

DttxxQ

ekkAXxXxxxQ

kkDekkAdkdktxxQ

txxQxx

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txxQ

xx

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Two Particle Solution

21021

222

211

21

)(212121

22

21

)(212121

2121

21

21

;0)- (:ConditionBoundary

Meetingon Reflect Particles :nInteractio Core-Hard

]4

)(

4

)(exp[

4

1),,(

),();()()0,,(

)](exp[),(),,(

Particles ginteractin-Non

),,()(),,(

21

2211

2211

xxyy

Q

x

Q

x

Q

Dt

Xx

Dt

Xx

DttxxQ

ekkAXxXxxxQ

kkDekkAdkdktxxQ

txxQxx

Dt

txxQ

xx

XkXki

xkxki

Two Particle Solution

agnet.antiferroman of

sexcitation ofcontext in thesolution particle two thenggeneralizi

by problem quantum particle-N theessentialy solved Bethe

.at conditionsboundary quantum theusingby obtained

isfactor phase The .degenerate are here added solutions twoThe

][),(),,(

isxfor bosons identical of problem quantum for theSolution

)()]4

)()(exp(

)4

)()([exp(

4

1),,(

, andcondition initialearlier with the

][),(),,(

21

),(2/)(212121

21

12

212

221

222

211

21

21

)(212121

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21

1221221122

21

xx

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x

xxDt

XxXx

Dt

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XX

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