Study-Unit Description

Study-Unit Description


CODE PHY2245

 
TITLE Statistical Mechanics

 
UM LEVEL 02 - Years 2, 3 in Modular Undergraduate Course

 
MQF LEVEL 5

 
ECTS CREDITS 6

 
DEPARTMENT Physics

 
DESCRIPTION Classical thermodynamics encapsulates the study of systems due to heat or temperature. Despite its extensive application notably through the use of macroscopic quantities, thanks to recent advancements and understanding of the microscopic world, classical thermodynamics fails to account phenomena where such microscopic effects become dominant, a feature commonly encountered at very low temperatures.

Accounting for these microscopic effects as means to describe the macroscopic behaviour of said thermodynamic systems falls under statistical mechanics. Throughout this study-unit, classical thermodynamical results are derived from a microscopic viewpoint while properly accounting for cases when strong quantum effects are present. In particular, the branch of equilibrium thermodynamics is explored.

Study-unit Aims:

This study-unit aims to provide the student with a practical understanding of the principles of statistical mechanics. The study-unit aims to provide:

1. An introduction to the basis concepts encountered in statistical mechanics including:
- the fundamental statistical postulate of equilibrium statistical mechanics;
- the difference between macrostates and microstates;
- introduction to probability, combinatorics and types of statistics (Bose-Einstein, Fermi-Dirac and Maxwell-Boltzmann).

2. A detailed overview of the Microcanonical Ensemble which includes:
- the relationship between microstates and entropy;
- thermodynamics of isolated systems;
- derivation of the relevant thermodynamical relations;
- applications including two level systems and the Einstein crystal.

3. Application and use of the Canonical Ensemble including:
- application to closed systems;
- definition and use of the canonical partition function to obtain the thermodynamical behaviour of closed systems;
- factorisability of the canonical partition function;
- introduction to density of states;
- application to gases (1-, 2- and 3D).

4. A detailed overview of ideal gases which focuses on:
- Gibbs Paradox;
- investigation of diatomic ideal gases from the viewpoint of quantum mechanics and the canonical ensemble.

5. Introduction to the Grand Canonical Ensemble in aims of:
- describing the thermodynamics of open systems;
- derive and use the grand canonical partition function;
- brief introduction to chemical reactions in chemical equilibrium.

6. Understanding Ideal Quantum Gases using the Grand Canonical Ensemble.

(a) Fermi Gases
- derivation of the Fermi-Dirac distribution;
- definition of Fermi level and Fermi energy/momentum;
- distinction between the classical and strong quantum regimes of a Fermi gas;
- application of the Sommerfield expansion to closely investigate the strong quantum regime;
- application of Fermi gases including the free electrons in a metal.

(b) Bose Gases
- derivation of the Bose-Einstein distribution;
- distinction between the classical and strong quantum regimes of a Bose gas;
- detailed description of the Bose-Einstein condensation phenomenon;
- using Bose gases to derive Planck’s Law for blackbody radiation.

7. Introduction to Paramagnetism and Paramagnetic Systems, including:
- application of the previous statistical ensembles to obtain the thermodynamic behaviour of such magnetic systems;
- identify and distinguish between paramagnetic, ferromagnetic and antiferromagnetic systems;
- description of Curie’s Law, Curie temperature and Neel temperature;
- introduction to the simplified 1D Ising model (free and periodic boundary conditions) and the transfer method;
- qualitative discussion of mean field theory in particular that of Weiss' mean field approximation to describe higher dimensional Ising systems including its limitations, predictions and relation to the Curie-Weiss law.

Learning Outcomes:

1. Knowledge & Understanding
By the end of the study-unit the student will be able to:

- Explain thermodynamics as logical consequences of the postulates of equilibrium statistical mechanics;
- Describe entropy in relation to the microstates of a system;
- Understand and describe each statistical ensemble and identify when the thermodynamics derived from each ensemble can match;
- Understand the importance of density of states and its use to determine thermodynamic quantities;
- Explain what is Gibbs paradox and how it is resolved;
- Discuss how characteristic temperatures help to identify when the degrees of freedom of a system become dominant particularly in the context of diatomic ideal gases;
- Distinguish between ideal Fermi and Bose quantum gases against ideal gases;
- Explain the difference between Fermi level and Fermi energy;
- Describe the phenomenon Bose-Einstein condensation;
- Discuss how a Bose gas can be applied to photons to derive Planck's Law for blackbody radiation;
- Distinguish between ferromagnetic, antiferromagnetic and paramagnetic materials and how these properties relate to Curie-Weiss Law and to Curie and Neel temperatures;
- State the main features of the Ising model and discuss its limitation to obtain analytical solutions in higher dimensional structures;
- Discuss how the mean field theory in particular the Weiss mean field approximation can provide further insight beyond 1D magnetic structures to predict the existence of phase transitions while also highlighting the main limitations of the approach.

2. Skills
By the end of the study-unit the student will be able to:

- Identify and quantify the number of macrostates and microstates of various physical systems using mathematical techniques;
- Associate probability with thermodynamic quantities depending on the nature of the system (isolated, closed or open);
- Derive and use the respective thermodynamical relations for each statistical ensemble;
- Identify and apply the suitable statistical ensemble given a system;
- Compute and use the density of states depending on the nature of the system;
- Perform suitable approximations including obtaining the classical and strong quantum limiting behaviours of a system;
- Ability to define suitable characteristic temperatures and using them to describe the features of a system;
- Calculate the Fermi energy and Bose condensation temperature;
- Show whether a system can undergo Bose condensation;
- Perform calculations to identify the nature of a magnetic system and whether it undergoes a phase transition at some critical temperature;
- Use and apply the transfer method particularly in paramagnetic systems;
- Perform suitable mean field approximations to investigate paramagnetic systems.

Main Text/s and any supplementary readings:

Recommended Main Textbooks:

• Thermodynamics and an introduction to Thermostatistics, H. B. Callen, (John Wiley & Sons, Singapore, 2nd Edition, 1985)
• Statistical Mechanics, K. Huang (John Wiley & Sons, 2nd Edition, 1987)
• Statistical and Thermal Physics: With Computer Applications, H. Gould and J. Tobochnik (Princeton University Press, 2010)

Supplementary Textbooks:

• Thermodynamics, Kinetic Theory, and Statistical Thermodynamics, F. W. Sears and G. H. Salinger (Addison-Wesley, 3rd Edition, 1975)
• Statistical Mechanics, R. K. Pathria (Elsevier, 2nd Edition, 1996)

 
ADDITIONAL NOTES Pre-Requisite qualifications: Good understanding of classical Thermodynamics and basic understanding of quantum mechanics

Pre-Requisite Study-units: Thermodynamics and Kinetic Theory

 
STUDY-UNIT TYPE Lecture and Tutorial

 
METHOD OF ASSESSMENT
Assessment Component/s Assessment Due Sept. Asst Session Weighting
Assignment SEM2 Yes 20%
Progress Test SEM2 Yes 30%
Examination (2 Hours) SEM2 Yes 50%

 
LECTURER/S Gabriel Farrugia

 

 
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