Mean-field approximations for TASEP
A talk in the Kolloquium Mathematische Physik series by
Thomas Kriecherbauer
| Abstract: | The Totally Asymmetric Simple Exclusion Process (TASEP) is a fundamental model in
statistical physics that was rst introduced in biology in the context of ribosome dynamics
along mRNA chains. Generally speaking, TASEP describes a uni-directional random
movement of particles along a chain. Particles may jump to the neighboring site. These
jumps are triggered by independent Poisson point processes where the dynamics is limited
by the constraint that each lattice site may be occupied by at most one particle at any
time.
We consider nite lattices with open boundaries. These are formulated as Markov processes
on a state space of a size that grows exponentially with lattice length $N$. This
makes the computation of the evolution of the state space probability distribution very
expensive even for moderate system sizes such as $N = 30$. The well-known Ribosome Flow
Model (RFM), a nonlinear system of size N, yields for many practical purposes excellent
approximations if one is interested in the occupation probabilities of each site. However,
there are also cases where this approximation fails.
In this talk we discuss a family of mean-field approximations indexed by m, $1 < m < N$,
that interpolates between the RFM ($m = 1$) and the full master equation for the Markov
process (m = N) that is of dimension $2N$. The system size of the model of order m is
bounded above by $2m+1N$. Numerical experiments show that even for small values of m
these models yield far better approximations than RFM whenever RFM fails. Moreover,
the numerics show that all these systems are contractive and therefore globally stable on
the relevant phase spaces. So far, stability proofs exist only in the cases $m = 1$ and $m = N$.
For $1 < m < N$ we provide an analytical framework and rst small steps towards a stability
result. Within the CRC this talk is associated to the project(s): C6 |