Advanced Statistical Physics (U01426)

? Credit Points : 10 ? SCQF Level : 11 ? Acronym : PHY-5-AdvStatPh

In this course we are concerned with the statistical physics of interacting particles in systems which range from the imperfect gas to the full many-body problem posed by phase transitions. We begin by formulating statistical physics in terms of quantum ensembles and the density matrix representation. Then we set up a general attack on the problem using perturbation theory, and establish the limitations of this approach. After that, the main aim of the course is to extend perturbation theory by resorting to methods such as mean-field theory, linear response theory, variational methods, scaling theory, use of control parameters and the renormalization group (RG) approach. The course concludes with the real-space RG.

Entry Requirements

? Pre-requisites : At least 80 credit points accrued in courses of SCQF Level 9 or 10 drawn from Schedule Q. Prior/concurrent attendance at Statistical Physics (PHY-4-StatPh) is desirable.

Variants

? This course has variants for part year visiting students, as follows

Advanced Statistical Physics (VS1) (Not being delivered)

Subject Areas

Home subject area

Undergraduate (School of Physics), (School of Physics, Schedule Q)

Delivery Information

? Normal year taken : 5th year

? Delivery Period : Not being delivered

? Contact Teaching Time : 2 hour(s) per week for 11 weeks

All of the following classes

Type	Day	Start	End	Area
Lecture	Monday	11:10	12:00	KB
Lecture	Thursday	11:10	12:00	KB

Summary of Intended Learning Outcomes

Upon successful completion of this course it is intended that a student will be able to:
1)Express the expectation value of an observable in terms of the density matrix and obtain the density operator for the canonical ensemble
2)Discuss the phenomenology of first- and second-order phase transitions with particular reference to the liquid-gas and para-ferromagnetic transitions
3)State the defining relationships for the six principal critical exponents
4)Explain linear response theory and use it to obtain both
correlations and connected correlations
5)Explain the implementation of perturbation theory using a control parameter and carry out high-temperature expansion for the Ising model and low-density expansion for an imperfect gas
6)Discuss the formulation of theoretical models, particularly for magnetism
7)Obtain exact solutions for the Ising model in d=1, both linear chain and ring, and for bond percolation in d=2
8)Show how mean-field theory can be used with a variational
principal to obtain the two-point correlations
9)Show that mean-field theory of the Ising model is equivalent to an assumption that each spin interacts equally with every other spin
10)Discuss the validity of mean-field theory in terms of upper critical dimension and give an heuristic argument to suggest dc=4
11)State the static scaling hypothesis and use it to derive
relationships among critical exponents
12)Explain Kadanoff's theory of block spins and show how this may be combined with simple dynamical systems theory to shed light on critical phenomena
13)State the RG transformation and discuss the nature of its fixed points for a symmetry-breaking phase transformation
14)Apply the RG transformation to a one-dimensional magnet or two-dimensional percolation
15)Obtain a linearized form of the RG transformation and derive relationships between critical exponents
16)Apply the linearized RGT to two-dimensional percolation and the two-dimensional magnet