# Fundamental models of interacting particles

Fundamental models of interacting particles, such as those occurring in mathematical physics and queuing theory, are widely studied to understand non-equilibrium behavior in physical systems consisting of large numbers of particles, to study large classes of transport phenomena, scheduling mechanisms and interface growth.

We aim to derive precise predictions from microscopic first principles for properties that encode emerging and collective behaviour. Such properties include important stationary and dynamical properties of model systems such as exclusion processes and queuing systems. Tools to be used and further developed lie in the realm of stochastic calculus, queuing theory, integrable probability, and include matrix product states.

Important relations have been established between the stationary distribution of multi-species asymmetric exclusion processes (mASEP) and the theory of many-variable polynomials.  For example, we have established that the notoriously difficult stationary distribution of mASEP with open particle reservoirs at the boundaries is described by Koornwinder polynomials corresponding to root system $(C_n,\check{C}_n)$.

Our appoach has led to important new advances in the theory of multi-variable polynomials. Using techniques from stochastic processes, such as matrix product formulas, we have obtained completely new and explicit formulas for Macdonald polynomials. An exciting and well received result using this methodology is the explicit formulation of a new class of polynomials based on bosonic solutions to the quantum Knizhnik-Zamolodchikov equation, unifying Macdonald with Borodin-Petrov polynomials. This result is expected to lead to a new class of integrable stochastic processes.

A recent result of Chen, de Gier and Wheeler, in preparation, is the development of a general theory for constructing stochastic dualities based on the connection with integrability. These dualities relate systems with many degrees of freedom to those with just a few, and hence provide a very significant reduction in computational complexity for natural observables.

An ambitious part of the project is to extend our precise understanding of stochastic processes to two spatial dimensions. To achieve this we need to develop new mathematical techniques to effectively implement quantum toroidal algebras and their representations as building blocks.

## Related Publications

#### 2018

Zhang, L., Finn C., Garoni T. M., & de Gier J. (2018).  Behaviour of traffic on a link with traffic light boundaries. Physica A: Statistical Mechanics and its Applications. 503, 116 - 138. doi: 10.1016/j.physa.2018.02.201

#### 2017

Zhang, L., Stuart C., Rajapaksha S., White G., & Garoni T. M. (2017).  Study of Cross-Correlations in Traffic Networks with Applications to Perimeter Control. Transportation Research Record: Journal of the Transportation Research Board. 2623, 108 - 116. doi: 10.3141/2623-12
Bedini, A., Zhang L., & Garoni T. M. (2017).  A case study of a continuous flow intersection and its impact on public transport. 2017 IEEE 20th International Conference on Intelligent Transportation Systems (ITSC)2017 IEEE 20th International Conference on Intelligent Transportation Systems (ITSC). 959 - 964. doi: 10.1109/ITSC.2017.8317892

#### 2016

Kutadinata, R., Moase W., Manzie C., Zhang L., & Garoni T. M. (2016).  Enhancing the performance of existing urban traffic light control through extremum-seeking. Transportation Research Part C: Emerging Technologies. 62, 1 - 20. doi: 10.1016/j.trc.2015.10.016