MASTER EQUATION OF MANY-PARTICLE
SYSTEMS IN A FUNCTIONAL FORM
Wipsar Sunu Brams Dwandaru
Matthias Schmidt
CORNWALL, 6-8 MARCH 2009
What will be discussed in the talk?
 A special many-particle system: totally
asymmetric exclusion process (TASEP).
 Motivation: why study the TASEP?
 The master equation of the TASEP.
 conclusion
 outlook
totally asymmetric exclusion process
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.
time t + 2dt
time t
chosen
time t + dt
chosen
site
site

X=1 2
3 …
Dynamical rule: shows how particles move in the 1D lattice sites.
Boundary condition: open boundaries.

N
motivation: everyday life
motor protein
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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
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Prof. Beate Schmittmann,
Virgina Tech., US
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Prof. Dr. Joachim Krug,
Universitat zu Koln, Germany
Prof. Dr. rer. nat. Gunter M. Schutz,
Universitat Bonn, Germany
master equation of the TASEP
 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
acknowledgement
 Prof. Matthias Schmidt
 Prof. R. Evans
 Morgan, Jon, Gavin, Tom, and Paul
 Overseas Research Student (ORS)
 All of you for listening
relationship between TASEP
and the lattice fluid mixture
1. 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.
particle 2
N
2
1
particle 1
…
3
Y
1
1
2
…
N
X
particle 3
ρ1(x,y)
ρ(x,y)
ρ2(x,y)
jr(x,y)
ρ3(x,y)
ju(x,y)
kr(x,y)
ku(x,y)
A correspondence between the fluids mixture and the TASEP
in 2D:
S x, y    x, y    ( x , y ) t 
 L  x, y   ji  x, y ,
i
i = 1, 2
.
e

 VLi  x , y VS  x , y 

 ki x, y ,
2. Calculations in the static lattice fluid mixture, yields:
The linearized density profiles, i.e.
S x, y   e
 L  x, y   e
VS
1  S x, y ,




S x, y 1  S x, y  1.
 VL1 VS
1
 L  x, y   e
2

 Ll r   0
 VL2 VS
S x, y 1  S x  1, y ,
3. Apply the correspondence to get into the TASEP.
j1 x, y   k1 x, y  x, y 1   x  1, y ,
j2 x, y   k2 x, y  x, y 1   x, y  1.
a steady state result: density
distribution of the TASEP in 2D
2 = 0.9
1 = 0.9
y
x
2 = 0.1
2 = 0.1
1 = 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)
1
HD_lane(x); k = 0.5; alpha2 = 0.4; beta2
= 0.1
0.9
1 = 0.1
0.8
LD_lane(y); k = 0.5; alpha1 = 0.1; beta1
= 0.4; 10^7 time steps
0.7
LD_lane(y); k = 0.5; alpha1 = 0.1; alpha
= 0.4; 10^8 time steps
density
0.6
0.5
0.4
2 = 0.1
0.3
2 = 0.4
0.2
0.1
0
0
20
40
60
sites
1 = 0.0
80
100