## Statistical Physics II

**Requirements**:

Course(s) on Statistical Physics, Classical Mechanics,

Quantum Mechanics; elements of Classical Thermodynamics

**Description**:

The most spectacular consequence of interactions among many molecules, or subsystems in general, is the appearance of new states of matter whose collective behavior bears little resemblance to that of a few subsystems. How do the subsystems then selforganize from one macroscopic state to a completely different one?

A purpose of this course will be to show the answer given by statistical mechanics of interacting systems. We start from classical systems (gases, simple liquids, soft matter and crystals) to show that selforganization in these systems can be understood using language of Local density functional (LDF) theory supplemented by symmetry arguments.We will also concentrate on critical points, where new symmetry emerges, known as scale invariance.

Description of systems near critical points requires new approach - the Renormalization Group- which we will illustrate for the Ising model. Then we introduce general concept of order parameters, which will bring us to a very powerful symmetry description of phases and phase transitions, known as Landau theory. Landau's approach is highly universal and applies both to classical and quantum systems, although it fails near critical point. Its connection to LDF is apparent.

Drawbacks of the Landau' s description can be cured by including fluctuations of the order parameter. This generalized theory coined the name Landau-Ginzburg-Wilson theory or Statistical Field theory for short. General aspects of this theory will be discussed and illustrated with a few examples.

Fluctuations or external forces can bring a given system's state out of equilibrium. Last lectures will be devoted to a description of such perturbed systems, mostly using concept of random Gaussian-Markoff processes. They not only serve as a useful illustration of irreversible system's behaviour, but also represent a good approximation to a class of real processes. We shall discuss general properties of such processes.

This course will cover the following topics:

1. Introduction to phase transitions: general mechanisms, examples (magnets, nematics, smectics A, superconductors, TGB phases);

a) discrete symmetry breaking and domain walls

b) continuous symmetry breaking and Goldstone modes

c) models of condensed matter with local gauge symmetries: Higgs mechanism.

d) Statistical, microscopic description of condensed matter; calculation of averages; distribution functions;

2. Classical systems: model interaction potentials; quantum corrections in classical regime.

3. Cluster (Virial expansion) for gases and simple liquids; equation of state; connection with Van-der-Waals equation

4. Local density functional (LDF) theory : general formulation; case of noninteracting systems; low density expansion and mean field approximation; properties of one-particle distribution function and concept of order parameters;

LDF description of spontaneous symmetry breaking in classical systems. Examples:

a) Ising model and Blume-Emery-Griffiths models

b) Meyer-Saupe mean field model of isotropic-nematic phase transition

c) Onsager model and excluded volume effects.

5. Near critical point:

a) scaling hypothesis, critical exponents and their properties;

b) Kadanoff's blocks, scale invariance and renormalization group (RG)

c) RG: implementation for an Ising magnet and percolation; cumulant expansion and critical exponents (Ising)

7. Beyond microscopic models of phase transitions: general concept of primary order parameters and Landau theory; connection with DFT, universality of mechanisms of spontaneous symmetry breaking in Landau's approach; classification of singularities; multicritical phenomena. Example: magnets and nematics

8. Spatially varying order parameters and Landau-Ginzburg theory. Example: superconductors in an external magnetic field and global U(1) symmetry breaking; role of local gauge symmetry;

9. Defects and Landau-Ginzburg theory (example: hedgehogs and disclinations in nematics, domain walls in magnets).

10. Landau-Ginzburg hamiltonians and fluctuations.

11. Path integrals formulation of Statistical Mechanics:

a) Gaussian theory;

b) basic properties of Green (correlation) functions; connection with susceptibilities;

c) distinction between discrete and continuous (Global) symmetry breaking,

d) Goldstone theorem; lower critical dimension and Wilson's renormalization Group.

12. Elements of nonequilibrium statistical mechanics: Gaussian-Markoff processes, Master equation and Metropolis algorythm,

13. Onsager's relations, entropy production, fluctuation dissipation theorem, Kramers-Kronig relations

**Bibliography**:

1. P.M.Chaikin and T. C. Lubensky Principles of condensed matter physics

2. M. Plischke and B. Bergersen Equilibrium Statistical Physics, 3rd Edition

3. Nigel Goldenfeld Lectures on phase transitions and the RG

**Supplementary literature**:

5. P.G. de Gennes and J. Prost The Physics of Liquid Crystals

6. H. E. Stanley Introduction to Phase Transitions and Critical Phenomena

7. S. R. de Groot and P. Mazur Non-equilibrium Thermodynamics