Project III (MATH3382) 2017-18

Many, many, many, many, ..., really very many collisions.

Bernard Piette

Description

When two particles interact with each other, say via a string or the gravitation force, we can easily write down Newton's equation for the system and solve the equations if we know the initial position and speed of the objects. When three particles are involved, one can easily write Newton's equations, but solving them is nearly always impossible analytically.

So what can we do if we have millions, billions or even more particles like the molecules in a milks bottle? This looks hopeless. Not only will writing the equations impossibly tedious but even if we dared to try, there is no way we can possibly know the initial positions and speeds of every molecule at a given time.

This looks hopeless, but as a matter of fact having a very large number of molecules make the problem a lot simpler. This is because when we study a gas for example, we are not interested in the detail knowledge of all the molecules. We are instead interested in global properties of the gas, like the density, displacement speed, pressure, temperature. We thus need to look at the gas as a large number of particles which interact very rapidly and a huge number of time, allowing us to consider the system statistically and focus only on the global properties of interest.

The aim of the project is to study how one can study dynamical system made out of a huge number of particles and how this can be used to describe systems like gas, chemical reactions, biological systems or properties of spin systems called phase transitions.

The project can involve some python programming to perform some simulations, but this is not a requirement.

Prerequisites

  • Programming and Dynamics (MATH1041)

Resources

email: Bernard Piette

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