Goldsmiths’ 2014 Mathematics

Tuesday 22nd July

Dr Rosemary Harris

From Applied Maths to Transport Modelling

(via non-equilibrium statistical mechanics)


Transport processes


Traffic, ants and molecular motors

Stochastic Markovian dynamics

In probability theory and statistics, a Markov process or Markoff process, named after the Russian mathematician Andrey Markov, is a stochastic process that satisfies the Markov property. A Markov process can be thought of as ‘memoryless’: loosely speaking, a process satisfies the Markov property if one can make predictions for the future of the process based solely on its present state just as well as one could knowing the process’s full history. i.e., conditional on the present state of the system, its future and past are independent.

It can be used to model a random system that changes states according to a transition rule that only depends on the current state.


Markov process example


Andrey (Andrei) Andreyevich Markov (14 June 1856 N.S. – 20 July 1922) was a Russian mathematician. He is best known for his work on stochastic processes. A primary subject of his research later became known as Markov chains and Markov processes.

Interacting particles

Discrete space, configurations labelled by σ (t) (“where the particles are at a given time”)

Dynamics (“how the particles move”)

Are they in boxes?

– Memoryless – Markov

– Inherently random – Stochastic

– Continuous time

Transition rates (probabilities per unit time) kσ′,σ

• Probability distribution (probability is of great interest)

P (σ, t) changes in time according to Master Equation:


Aside: Can also be written in matrix formulation


Equilibrium versus non-equilibrium

Master equation again


Conservation of probability


Long-time/stationary distribution


Equilibrium, detailed balance



– Broken detailed balance

– Stationary state has non-zero currents

• (So far) non-equilibrium statistical mechanics (especially in physics) not well understood…

…insight from “toy” models, e.g., asymmetric simple exclusion process


What is the probability the balls move from the pots

Asymmetric Simple Exclusion Process

Model defined in continuous time:


• Site occupancies


Results look roughly the same for all p q

• Let’s make life easier and look at the Totally Asymmetric case (TASEP) with q = 0

• What happens at the boundaries…?

Periodic boundary conditions


Total number of particles is constant

• Let’s look at average quantities, denoted by angular brackets, e.g. ˂X˃

• Average density (same on all sites)


How does current depend on density?

• Average current



which is a quadratic

Fundamental diagram

How does current depend on density?



Open boundary conditions


Model has phase transitions


Modelling of transport

Models are a simplification of reality


Onion picture, build up layers: – Simplest possible toy model – Progressively add more details – Use computer!

• ASEP (Asymmetric Simple Exclusion Process) used as starting point for various transport processes…

In probability theory, the asymmetric simple exclusion process (ASEP) is an interacting particle system introduced in 1970 by Frank Spitzer in Interaction of Markov Processes. Many articles have been published on it in the physics and mathematics literature since then, and it has become a “default stochastic model for transport phenomena”


Frank Ludvig Spitzer (July 24, 1926 – February 1, 1992) was an Austrian-born American mathematician who made fundamental contributions to probability theory, including the theory of random walks, fluctuation theory, percolation theory, the Wiener sausage, and especially the theory of interacting particle systems. Rare among mathematicians, he chose to focus broadly on “phenomena”, rather than any one of the many specific theorems that might help to articulate a given phenomenon. His book Principles of Random Walk, first published in 1964, remains a well-cited classic.


Vehicular traffic

ASEP is toy model for single-lane traffic


Phase diagram already shows some features of real traffic


Adding details

Variation in road-surface adds disorder


Different speeds, rules for acceleration/deceleration (e.g., Nagel-Schreckenberg)

The Nagel–Schreckenberg model is a theoretical model for the simulation of freeway traffic. The model was developed in the early 1990s by the German physicists Kai Nagel and Michael Schreckenberg. It is essentially a simple cellular automaton model for road traffic flow that can reproduce traffic jams, i.e., show a slowdown in average car speed when the road is crowded (high density of cars). The model shows how traffic jams can be thought of as an emergent or collective phenomenon due to interactions between cars on the road, when the density of cars is high and so cars are close to each other on average.


The above image shows a plot of the average velocity, <v>, as a function of the density of cars per cell, r, in the Nagel–Schreckenberg model. The black curve is for p = 0, i.e., for the deterministic limit, while the red curve is for p = 0.3.

• Discrete time easier for computer modelling – cellular automata

• Different geometries – More lanes – Road networks

Experiments on real traffic

Not usually periodic boundary conditions…

Fundamental diagram

Note “metastable states” (metastability describes the extended time spent by an isolated system in a long lived configuration other than the system’s state of least energy or stability)

Can construct simple models with similar effects


Real life: measurement

Real life: prediction


Real life: validation


Ant-trail mode

Modify model to include chemical signals (pheromones)


Ant fundamental diagrams

Uni-directional versus bi-directional movement


Schadschneider et al. ’03

Model predicts formation of “platoons”.

Intracellular transport

Kinesin on axonal microtubule.


from Klumpp & Lipowsky

A kinesin is a protein belonging to a class of motor proteins found in eukaryotic cells. Kinesins move along microtubule filaments, and are powered by the hydrolysis of ATP (thus kinesins are ATPases). The active movement of kinesins supports several cellular functions including mitosis, meiosis and transport of cellular cargo, such as in axonal transport. Most kinesins walk towards the plus end of a microtubule, which, in most cells, entails transporting cargo from the centre of the cell towards the periphery. This form of transport is known as anterograde transport. In contrast, dyneins are motor proteins that move toward the microtubules’ minus end.

Key features: – Preferred direction – Discrete steps – Exclusion – Attachment and detachment

At coarse-grained level can model by stochastic exclusion process…

Breakdown of intracellular transport

Higher than usual concentration of tau protein in Alzheimer’s patients

• Experiments on tau, e.g., [Trinczek et al.]:

– Doesn’t affect speed of motors on microtubules.

– But reduces their absorption probability

• Theoretical calculations for simple model, simulation for more complicated model

[Grzeschik, Harris & Santen ’08]:

• System robust to low concentration of tau but at higher densities

– Mean current strongly reduced

– Fluctuations increased…

Current fluctuations

Often (relatively) easy to calculate mean current

But finite-time observations can yield average current larger or smaller than real mean

Characterizing these fluctuations is important in applications

Also of theoretical importance – Current large deviations analogous to free energy in equilibrium


Exhibit particular symmetry (fluctuation theorem) – Insight into structure of non-equilibrium statistical mechanics


From Applied Maths…

• Probability

• Differential equations

• Matrices

…to Transport Modelling

• Can model important real-life situations – Transport failure in Alzheimer’s disease – Pedestrian dynamics at the Hajj – Gas network via non-equilibrium statistical mechanics

• Non-equilibrium physics typically very different to equilibrium physics

• ASEP is a simple test model allowing exact calculations

Acknowledgments / References [traffic state in NRW] [simulations/videos, articles on Hajj]

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