Statistical mechanics with applications to materials science (2018-2019)
Intro
The goal of the course is providing an introduction to the basic concepts and applications of equilibrium statistical mechanics.
IMPORTANT: students interested in attending the lectures should send an email to giuseppe.florio@poliba.it
IMPORTANT: Interested students can receive updates and info through the Telegram channel: https://t.me/joinchat/AAAAAFVHpw17kKAHwwfnTg
IMPORTANT: the lectures of April 30. and may 6 will NOT take place and will be postponed. New dates will be arranged.
The course will start on April 29, h.15-h.18 (Building B01, 2nd floor, Seminar room) and will follow the following calendar (some modifications can be due to unexpected duties of the teacher):
April 29 h15-h18
April 30, h15-h18 (postponed)
May 6, h15-h18 (postponed)
May 7, h15-h18
May 13, h15-h18
May 27, h15-h18
May 28, h15-h18
June 10, h15-h18
June 11, h15-h18
June 17, h15-h18
Syllabus
INTRODUCTION
Observables; probability distribution; free energy and entropy; partition function.
FLUCTUATIONS AND LINEAR RESPONSE
Brownian motion and diffusion; correlation functions; fluctuation-dissipation theorem
PHASE TRANSITIONS
Critical points; symmetry breaking and order parameter; Landau theory and mean field.
MODELS
Ising and Heisenberg model; random ferromagnets; polymers; liquid crystals
BIBLIOGRAPHY
D. Chandler, Introduction to Modern Statistical Mechanics (Oxford University Press, New York, 1987)
J. P. Sethna: Statistical Mechanics: Entropy, Order Parameters, and Complexity, Oxford University Press (2006)
P. M. Chaikin and T.C.Lubensky, Principles of Condensed Matter Physics (Cambridge U.P., Cambridge, 2000).
J. H. Weiner, Statistical Mechanics of Elasticity (Dover, 2002)
T. Kawakatsu, Statistical Physics of Polymers: An Introduction (Springer, 2004)
Detailed program and lecture diary
April 29, h15-h18.30
1) INTRODUCTION
Thermodynamical systems: states and equations of state; transformations. First law of thermodynamics, internal energy. Second law of thermodynamics; thermal engines and efficiency; entropy. Third law of thermodynamics. Foundations of classical statistical mechanics. Observables and averages; probability distribution. Postulate of equal-a-priori probability. Boltzmann's formula for entropy, connection with thermodynamics.
May 7, h15-h18.30
Microcanonical ensemble. Systems at constant temperature. Canonical ensemble. Boltzmann distribution. Helmoltz free energy and partition function. Mechanically isolated systems at fixed temperature. Pressure. Partition function of a gas, equation of state. System at constant pressure and fluctuations of volume. Gibbs free energy. Fluctuation of particles. Chemical potential. Grand canonical ensemble. Grand partition function. Simple model of absorption. Constraint on an observable; Landau free energy.
May 13, h15-h18.30
2) FLUCTUATIONS AND LINEAR RESPONSE
Brownian motion and diffusion; correlation functions; Langevin theory
May 27, h15-h18.30
Properties of time correlation functions. Fourier transform and spectral analysis of fluctuations.
May 28, h15-h18.30
Linear response. Fluctuation-dissipation theorem. Damped harmonic oscillator.
June 10, h15-h18.30
3) PHASE TRANSITIONS
Classification of phase transitions. Order parameter and critical exponents. Van der Waals' equation: Maxwell's construction, phase coexistence, critical point of the liquid-gas transition.
Landau's theory of phase transition; mean field theory; Application to magnetic systems.
4) MODELS
Classical Ising model.
June 11, h15-h18.30
Explicit solution on the Ising model in one dimension and transfer matrix. Correlation functions. Absence of phase transition for the classical Ising model in one dimension. Ising model in two dimensions, spontaneous magnetization. Mean field solution of the Ising model. Long range interactions and the Curie-Weiss model.
June 17, h15-h18.30
Ising model in a random field, phase diagram. Heisenberg model. Landau theory for I order phase transitions: nematic liquid crystals.
June 18, h9-h11
Models for polymers: freely jointed chain and wormlike chain.