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Diffusion of Interacting Particles in One dimension Deepak Kumar School of Physical Sciences Jawaharlal Nehru University New Delhi IITM, Chennai Nov. 9, 2008
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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.
2/1t
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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0
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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
22
0
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0
0
2
2
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tP
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xx
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xtxP
x
tDk
tDk
Two Particle Solution
)(21
21
22
21
)()(212121
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),(
Xwith
])(exp[
])[,(),,(
;0)- (:ConditionBoundary
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)()()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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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
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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
x
txJ
t
txP
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txP
tDk
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
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ekkAXxXxxxQ
kkDekkAdkdktxxQ
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txxQ
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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
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xxyy
Q
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kkDekkAdkdktxxQ
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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
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Q
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ekkAXxXxxxQ
kkDekkAdkdktxxQ
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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
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