MASTER EQUATION OF
MANY-PARTICLE SYSTEMS IN A
FUNCTIONAL FORM
Wipsar Sunu Brams
Dwandaru
Matthias
Schmidt
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
X= 1
2 3 … N
chose n 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.
motivation: everyday
life
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motor
motivation: everyday
life
protein
motivation: everyday
life
Yogyakarta,
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
acknowledgeme
nt
Prof. Matthias Schmidt
Prof. R. Evans
Morgan, Jon, Gavin, Tom, and Paul
Overseas Research Student (ORS)
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.
1. Identify TASEP particles and their movements as species
in the lattice fluid mixture, hence the relationship.
X Y
1 2 … N
1
…
N
ρ2(x,y)
ρ3(x,y)
ρ1(x,y) ρ(x,y)
jr(x,y)
ju(x,y) particle 1
1 2
particle 2
particle 3 3
kr(x,y)
A correspondence between the fluids mixture and the TASEP in 2D:
.
i = 1, 2
,
,
,
L
ix
y
→
j
i
x
y
ρ
x
y
x
y
(x,y)
t
S
,
ρ
,
τ
ρ
→
=
L,
S,
,
,
y
x
The linearized density profiles, i.e. L
r
→
0
l
r
ρ
2. Calculations in the static lattice fluid mixture, yields:
,
1
S
,
,
S
x
y
e
Sx
y
V
ρ
ρ
=
β−
,
S
,
1
S
1
,
,
L
S 1 L
1
x
y
e
x
y
x
y
V V
+
−
=
−ρ
ρ
ρ
β
,
S
,
1
S
,
1
.
L
S 2 L
2
=
−
+
−
y
x
y
x
e
y
x
V Vρ
ρ
ρ
β3. Apply the correspondence to get into the TASEP.
,
1,
,
1
1
,
,
1
x
y
k
x
y
x
y
x
y
j
=
ρ
−
ρ
+
,
2,
,
1
,
1
.
2
x
y
=
k
x
y
x
y
−
x
y
+
a steady state result:
density distribution of the
TASEP in 2D
x
y
1 6 11 16 S1 S6 S11 S16 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 densitysites (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
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
