master equation of many-particle systems in a functional form wipsar sunu brams dwandaru matthias...
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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