A Level HE3 course discussing the basis and applications of statistical methods and information theory to microscopic systems.

Dealing with Uncertainty

i.                To explain that as scientists we almost always only have incomplete knowledge of a system, and that this forces use to use the theory of probability; examples are: we usually do not know the microscopic state a system such as a gas or semiconductor is in, and we are often unable to measure a physical variable, such as radioisotope lifetime, very accurately;

ii.              To demonstrate that we need quantitative tools, such as the Central Limit Theorem of statistics and Bayes’ theorem, to enable us, not only get the maximum understanding from the data we do have, but to quantify the uncertainty that remains;

Applications of Statistical Physics

i.                To introduce the statistical physics of phase transitions, by showing how the interactions between spins in the Ising model cause the paramagnet-to-ferromagnet phase transition. Also, the fundamental relation between fluctuations and thermodynamic functions, such as heat capacities, will be demonstrated;

ii.              To introduce simple time-dependent statistical processes such as diffusion in a gas, and to show how statistical physics relates microscopic phenomena such as the random motion of molecules to thermodynamic phenomena such as an increase in entropy;

iii.             To show how fluctuations limit the accuracy with which experiments can measure some properties, with an application to experiments on the limits to human night vision;

iv.            To introduce the basics of nanomotors. These are machines only approx. 10nm across, and so are small enough to diffuse. Nevertheless they expend energy and in doing so perform mechanical work and move in a directed fashion;

On successful completion of the module, students will be familiar with Bayes’ theorem and its uses. They will also appreciate that probabilities and free energies are intimately related. They will understand the Ising model and the elements of how its phase transition is a result of interactions between the constituent spins of the magnet. They will also be able to calculate the diffusion constant of a gas and show how large scale diffusion is related to the motion of the gas molecules. They will understand what places the limit on how many photons are required before we can detect light, and how this understanding can be generalized to other attempts to measure weak signals. They will know how a machine perhaps 10nm across can do work but only by burning a fuel.

This module will build upon previous study of statistical physics by introducing general and powerful methods of dealing with systems where there we lack complete knowledge and so must use probabilistic, statistical, methods. It will also apply these statistical methods in a number of more advanced applications.

20 hours of lectures will introduce:

i.         Bayes’ theorem for finding the most likely value from an experiment;

ii         A derivation of the Gaussian distribution.

Iii        Correlation and causation.

Applications of statistical physics, including:

i.                The Ising model of the paramagnet-to-ferromagnet phase transition;

ii.              The relationship between fluctuations and response functions, e.g., heat capacities and magnetisabilities;

iii.             Diffusion in a gas;

iv.            The experiments of Hecht and co-workers on human night vision and their interpretation;

v.              Nanomotors and transport by molecules.

vi.            If time permits, a further application, such as molecular switches will be considered.

24 contact hours in lecture delivery format.

Recommended Texts:

i.                K. Huang, Introduction to Statistical Physics, Taylor and Francis.

ii.              P. Nelson, Biological Physics: Energy, Information, Life, Freeman.

iii.             S.J. Blundell and K.M. Blundell, Concepts in Thermal Physics, Oxford .