MASTER EQUATION OF MANY-PARTICLE SYSTEMS IN A

FUNCTIONAL FORM

Wipsar Sunu Brams Dwandaru

Matthias Schmidt

CORNWALL, 6-8 MARCH 2009

A special many-particle system: totally asymmetric exclusion process (TASEP).

Motivation: why study the TASEP?

The master equation of the TASEP.

conclusion

outlook

What will be discussed in the talk?

totally asymmetric exclusion process

X=1

2 3 … N

chosen site

time t time t + dt

chosen site

time t + 2dt

The TASEP in one dimension (1D) is an out of equilibrium driven system in which (hard core) particles occupy a 1D lattice. A particle may jump to its right nearest neighbor site as long as the neighbor site is not occupied by any other particle.

Dynamical rule: shows how particles move in the 1D lattice sites.

Boundary condition: open boundaries.

motivation: everyday life

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motor protein

motivation: everyday life

protein synthesis http://oregonstate.edu/instruction/bb331/lecture12/Fig5-20.html

motivation: everyday life

Yogyakarta, Indonesia

Jakarta, Indonesia

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Prof. David Mukamel,Weizmann Institute, Israel

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Dr. Debasish Chowdhury,Physics Dept., IIT, India

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Prof. Royce K.P. Zia,Virgina Tech., US

Prof. Beate Schmittmann,Virgina Tech., US

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Prof. Dr. Joachim Krug,Universitat zu Koln, Germany

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Prof. Dr. rer. nat. Gunter M. Schutz,Universitat Bonn, Germany

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The master equation is a first order DE describing the time evolution of the probability of a system to occupy each one of a discrete set of states.

The gain-loss form of the master equation:

(1) where wmn is the transition rate from state n to m. Pn(t) is the probability to

be in state n at time t. n,m = 1, 2, 3, …, N. N is the total number of microscopic states.

The matrix form of the master equation: (2)

where

master equation of the TASEP

acknowledgement

Prof. Matthias SchmidtProf. R. EvansMorgan, Jon, Gavin, Tom, and PaulOverseas Research Student (ORS)All of you for listening

relationship between TASEP and the lattice fluid

mixture1. Identify TASEP particles and their movements as species in the lattice fluid mixture, hence the relationship.

2. Do calculations in the static lattice fluid mixture via DFT.

3. Apply the correspondence to obtain the desired TASEP properties.

[Dwandaru W S B and Schmidt M 2007 J. Phys. A: Math. Theor. 40 13209-13215]

1. Identify TASEP particles and their movements as species in the lattice fluid mixture, hence the relationship.

X

Y1 2 … N

1

…

N

ρ2(x,y)

ρ3(x,y)

ρ1(x,y) ρ(x,y)

jr(x,y)

ju(x,y)

particle 112

particle 2

particle 3

3

kr(x,y)

ku(x,y)

A correspondence between the fluids mixture and the TASEP in 2D:

.

i = 1, 2

( ) ( ),,,L yxjyx ii→ρ

( ) ( ) ( )tyxyx yx ),(S ,, τρρ =→

( ) ( )[ ] ( ),,,, SL yxke i

yxVyxVi →

−β

The linearized density profiles, i.e. ( ) 0L →rl

rρ

2. Calculations in the static lattice fluid mixture, yields:

( ) ( )[ ],,1, SSS yxeyx V ρρ β −=

( ) [ ] ( ) ( )[ ],,11,, SSLS1L

1yxyxeyx

VV+−=

−ρρρ

β

( ) [ ] ( ) ( )[ ].1,1,, SSLS2L

2+−=

−yxyxeyx

VVρρρ

β

3. Apply the correspondence to get into the TASEP.

( ) ( ) ( ) ( )[ ],,11,,, 11 yxyxyxkyxj +−= ρρ

( ) ( ) ( ) ( )[ ].1,1,,, 22 +−= yxyxyxkyxj ρρ

a steady state result: density distribution of the

TASEP in 2D

x

y

16

1116

S1

S6

S11

S16

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

density

sites (x) [alpha2, beta2]

sites (y) [alpha1,beta1]

2D TASEP with Open Boundaries (alpha1 = 0.9; beta1 = 0.1; alpha2 = 0.1; beta2 = 0.9; k = 0.5)

0.9-1

0.8-0.9

0.7-0.8

0.6-0.7

0.5-0.6

0.4-0.5

0.3-0.4

0.2-0.3

0.1-0.2

0-0.1

1 = 0.9

2 = 0.1

2 = 0.9

1 = 0.1

2 = 0.1

1 = 0.4

2 = 0.4 2D Junction TASEP with Open Boundaries (Alpha1 = 0.1; Beta1 = 0.4;

Alpha2 = 0.4; Beta2 = 0.1)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100

sites

density

HD_lane(x); k = 0.5; alpha2 = 0.4; beta2= 0.1

LD_lane(y); k = 0.5; alpha1 = 0.1; beta1= 0.4; 10^7 time steps

LD_lane(y); k = 0.5; alpha1 = 0.1; alpha= 0.4; 10^8 time steps

2 = 0.1

1 = 0.1

2 = 0.4

1 = 0.0