MA 368A: Exclusion processes
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Instructor:
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Arvind Ayyer
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Office:
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X-15 (new wing)
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Phone number:
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(2293) 3215
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Email:
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(First name) at math dot iisc dot ernet dot in
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Class Timings:
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Tuesdays and Thursdays, 2:00–3:30pm.
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Classroom:
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LH 3, Mathematics Department (ground floor)
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Office hours:
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By appointment
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Textbook:
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Exactly solved models in statistical mechanics
by Rodney J. Baxter
ISBN-13 - 978-1-4832-65940
An exactly soluble non-equilibrium system: The asymmetric simple exclusion process
by Bernard Derrida
Physics Reports 301 (1998), 65—83.
Exclusion processes with drift
by Arvind Ayyer
LPS 2017 lecture notes
Freely downloadable from
here
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Course Prerequisites
MA 261 or equivalent.
Course Description
Review of discrete and continuous time Markov chains, review of equilibrium
and nonequilibrium statistical mechanics, Ising model in one dimension,
Glauber dynamics, Bethe ansatz, Yang-Baxter equation, asymmetric simple
exclusion processes with periodic and open boundary conditions,
multispecies exclusion processes, zero range processes, Schur and Macdonald processes
Exams
All exams will be closed book, closed notes, and
no calculators or electronic devices are allowed.
No communication among the students will be tolerated.
There will be no make up exams.
The date for the midterm and final will be announced later.
Grading
Here are the weights for the homework and exams.
All marks will be posted online
on Moodle.
- 5% – Class participation
- 15% – Homeworks
- 30% – Midterm
- 50% – Final
Tentative Class Plan
Week 1: Review of discrete time Markov chains
Definitions, Chapman-Kolmogorov equation, classification of states,
irreducibility, aperiodicity, stationary distribution, mixing time, reversibility
Week 2: Review of continuous time Markov chains
Exponential random variables, transition probability functions,
Kolmogorov backward and forward equations, interacting particle systems
on graphs, definitions of ASEP on a ring
Week 3: ASEP on a ring and with open boundary conditions
Stationary distribution of the ASEP on the ring, current and density,
Bethe ansatz for the ASEP on the ring with 1, 2 and 3 particles
Symmetries of the open ASEP, the ASEP algebra, stationary
distribution, representations of the ASEP algebra, limiting case of SSEP,
Homework set 1
Week 4: TASEP with open boundary conditions
TASEP algebra, stationary distribution, Catalan and ballot numbers,
formulas for the partition function and current
Week 5: TASEP with open boundary conditions
Asymptotics for the partition function and current, phase diagram
for the current, exact solutions for small systems in Mathematica.
Density asymptotics, phase diagram, shock line.
Week 6: Mid-term week
Week 7: Two-species ASEP on a ring
Basic properties, stationary distribution,
matrix ansatz, colouring, current.
Homework set 2
Week 8: Semipermeable TASEP
Irreducibility, matrix ansatz, partition function
and its generating function, density and current.
Week 9: Semipermeable TASEP
Asymptotics of the partition function and density,
phase diagram for the density, fat shock on the shock line,
intuitive explanation in all phases, exchangeability.
Week 10: Multispecies ASEP on a ring
Basic properties, colouring, sectors and their poset.
Week 11: Multispecies ASEP on a ring
Matrix ansatz using tensor products for TASEP and ASEP,
hat relations
Week 12: Multiline queues
Ferrari-Martin multiline process, bully path projection,
two-point nearest neighbour correlations.
Week 13: Finals week