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=123…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 motor protein

motivation: everyday life protein synthesis

motivation: everyday life Yogyakarta, Indonesia Jakarta, Indonesia

Prof. David Mukamel, Weizmann Institute, Israel Dr. Debasish Chowdhury, Physics Dept., IIT, India Prof. Royce K.P. Zia, Virgina Tech., US Prof. Beate Schmittmann, Virgina Tech., US Prof. Dr. Joachim Krug, Universitat zu Koln, Germany Prof. Dr. rer. nat. Gunter M. Schutz, Universitat Bonn, Germany

 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 w mn is the transition rate from state n to m. P n (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 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 ]

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)ρ2(x,y) ρ3(x,y)ρ3(x,y) ρ1(x,y)ρ1(x,y)ρ(x,y)ρ(x,y) jr(x,y)jr(x,y) ju(x,y)ju(x,y) particle particle 2 particle 3 3 kr(x,y)kr(x,y) ku(x,y)ku(x,y)

A correspondence between the fluids mixture and the TASEP in 2D:. i = 1, 2

The linearized density profiles, i.e. 2. Calculations in the static lattice fluid mixture, yields: 3. Apply the correspondence to get into the TASEP.

a steady state result: density distribution of the TASEP in 2D  1 = 0.9  2 = 0.1  2 = 0.9

 1 = 0.1  2 = 0.1  1 = 0.4  2 = 0.4  2 = 0.1  1 = 0.1  2 = 0.4  1 = 0.0