Statistical Mechanics,
Meccanica statistica
Martin Hasenbusch
Fri Oct 20 12:58:18 CEST 2006
outline of the lecture
Density matrix, von Neumann equation, ensembles, ideal Fermi and Bose gas,
the distribution functions, equation of state, phonons in a solid,
Debye theory of solids, Bose-Einstein condensation, Liquid He-4,
metal electrons;
Some examples: white dwarfs, quantum Hall effect, Landau diamagnetism,
De Haas-Van Alpine effect, Pauli theory of Paramagnetism;
Path integral formulation of quantum statistical mechanics;
Ising model, quantum spin models;
Phase transitions, symmetry breaking, Monte Carlo simulation;
disordered systems, spin glasses;
Polymers, interfaces and membranes;
Meissner effect, Introduction to the BCS theory of superconductivity;
Linear response and transport theory
scripts in the internet
Script on statistical mechanics in English and German, F. Wegner
Statistical Mechanics I: Statistical Mechanics of Particles, Mehran Kardar
Statistical Mechanics II: Statistical Mechanics of Fields, Mehran Kardar
Lecture notes, Mark Tuckerman
Lecture Notes in Statistical Mechanics, I Vilfan
Lecture notes in German, G. Schoen and M. Eschrig.
Books e.g.
R.K. Pathria, Statistical Mechanics, Second edition
(Butterworth-Heinemann 1996).
M. Plischke and B. Bergersen, Equilibrium Statistical Physics, Third Edition;
(World Scientific, Singapore, 2006).
Useful links:
constants of nature
Diary:
Mon Sep 25 2006: The density matrix, I follow
the Script of F.Wegner page 4 and 5; see also script of
M.Kardar lect20 p.132-135.
Tue Sep 26 2006: The density matrix continued: The reduced density matrix,
The von Neumann equation;
Script of F.Wegner page 5 and 6;
see also M.Kardar lect20 p.135.
The microcanonical, canonical, grand canonical ensemble (brief reminder):
M.Kardar lect20 p.136, see also F.Wegner pages 17, 21, 30;
Systems of N noninteracting identical particles in Quantum mechanics;
Construction of the states by proper symmetrization of tensor products
of single particle states. Pauli principle.
See M.Kardar p 138,139,140
Wed Sep 27 2006:
Fock space on wikipedia
Average occupation number in the grand canonical ensemble
F.Wegner pages 39,40, M.Kardar p. 144,145
Energy levels of a single free particle in a cuboid;
F.Wegner page 17; M.Kardar lect20 p.137.
Polylogarithm, wikipedia
Polylogarithm, Mathworld
Mon Oct 01 2006: Discussion of the occupation number; in limit of
low chemical potential we recover the Maxwell Boltzmann case.
Compute the grandcanonical potential for The ideal quantum gas.
See Wegner pages 39,40, Kardar 144,145.
Tue Oct 02 2006: Particle density, internal energy and Pressure
of the Bose and Fermi gas, discussion of the low density limit. See
Wegner pages 42 ; Kardar 147,148
Bose-Einstein condensation;
article on wikipedia
Wed Oct 03 2006:
Blackbody radiation; Wegner page 43; 44;
Phonons in a
Solid ; Wegner page 45.
Mon Oct 09 2006: Finish Einstein spectrum; start with Debye approximation
Debye model 2nd script of Kardar page 2,...,9.
Tue Oct 10 2006: Finish Debye approximation; Wegner page 46.
Wed Oct 11 2006: Electrons in a metal; Sommerfeld approximation; Wegner page 53.
Mon Oct 16 2006: White dwarf
White dwarf, Wikipedia
White dwarf, NASA
Sphere Gravitational Force
Sphere Gravitational Potential Energy
Nobel price lecture of Prof. Chandrasekhar
Tue Oct 17 2006: Pauli paramagnetism; see e.g. Pathria 2nd edition page 202
or
script of S. Teitel lecture 20
Wed Oct 18 2006: Landau diamagnetism; see e.g. Pathria 207 or
script of S. Teitel lecture 20 and 21
More on Euler's integration formula
Mon Oct 23 2006: finish Landau diamagnetism
Tue Oct 24 2006: Definition of the Ising model; Solution of the Ising model
in one dimension with the transfer matrix method. (See e.g. script of tuckerman
lec. 26)
History of the Ising model: S.G. Brush, History of the Lenz-Ising model,
Rev. Mod. Phys. 39 (1967) 883. \\
Wed Oct 25 2006: Solution of the Ising model in the meanfield approximation
Mon Oct 30 2006: Summary of the meanfield results, Ehrenfest classification
of
phase transitions.
Tue Oct 31 2006: Correlation function in the 1 dimensional Ising model,
Transfermatrix for systems with dimension larger than one, summary of results
from the exact solution for the two dimensional Ising model at vanishing
external field. Free energy and correlation length:
L. Onsager, Phys.Rev.65 (1944) 117, Bruria Kaufman, Phys.Rev. 76 (1949) 1232.
Magnetisation: C.N. Yang, Phys.Rev.85 (1952) 808. (See also Kardar,
Statistical Mechanics II, Lecture 19)
A Book for those who want to go deeper into the subject:
B.M. McCoy and T.T. Wu, The Two-dimensional Ising model (1973)
Harvard University Press.
Duality transformation of the two dimensional Ising model. (Goes back to
H.A. Kramers and G.H. Wannier, Phys. Rev. 60 (1941) 252, here I follow R.Savit,
Rev. Mod. Phys. 52 (1980) 453; See also Kardar Statistical Mechanics II,
lecture 16)
Mon Nov 06 2006: Define some lattice models: O(N)-invariant vector models:
XY-model, Heisenberg model; (For a review see
Andrea Pelissetto, Ettore Vicari, Critical Phenomena and
Renormalization-Group Theory,
cond-mat/0012164,
Phys.Rept. 368 (2002) 549-727)
solid on solid (SOS) models, Potts model
For duality of 2 dim. XY and SOS models see
R.Savit, Rev. Mod. Phys. 52 (1980) 453
Tue Nov 07 2006: Meanfield solution of the q-state Potts model,
High temperature (HT) series expansion (for series expansion
see e.g. Kardar Statistical Mechanics II, lecture 15.
In particular VI.B; HT series for the 2 and 3 dimensional
Ising model (and related models) can be found e.g.
in P. Butera and M. Comi,
A library of extended high-temperature expansions of basic observables
for the spin S Ising models on two- and three-dimensional lattices
hep-lat/0204007,
J.Statist.Phys. 109 (2002) 311-315)
Wed Nov 08 2006: Introduction to Monte Carlo simulations (see
my script pages 16-21)
Example program 2D Ising model simulated with the Metropolis
algorithm. It should be compiled like e.g.
gcc -O ising_metro.c -lm -o ising_metro.exe
Example program 2D Ising model, generates all configurations
can be used for small lattices only; 5x5 or smaller .
It should be compiled like e.g.
gcc -O ising.c -lm -o ising.exe
Mon Nov 13 2006: Analysis of Monte Carlo data: Variance, statistical
error, Central limit theorem, Autocorrelation function, Autocorrelation time,
Binning.
Tue Nov 14 2006: Landau Theory of critical Phenomena, see script of
Wegner p. 86, 87, ... , see e.g. Kardar Statistical Mechanics II, lecture 2.
Tue Nov 15 2006: Landau theory for tricritical points , correlation function,
Wegner p. 88, ... , see e.g. Kardar Statistical Mechanics II, lecture 2.
Mon Nov 20 2006: Landau-Ginzburg criterion, Blume-Emery-Griffiths model
Tue Nov 21 2006: Real Space Renormalization Group (RG) Transformations;
exact calculation for the one dimensional Ising model. Discussion of the
general case; RG-exponents and scaling fields: relevant, marginal and
irrelevant. See my script on MC-simulations pages 63 to 69 or
Kardar Statistical Mechanics II, lecture 7.
Tue Nov 22 2006:
Finish Real Space Renormalization Group (RG) Transformations;
scaling relations ; i.e. relate alpha,gamma,nu,beta,delta with the
RG-exponents y_t, y_h ; Concept of Universality.
Fortran program to
compute a truncated RG-transformation for the 2 D Ising model.
Compile with e.g. g77 -O rg.f -o rg.exe ; The truncated coupling vector
has three components. Only even terms are implemented in the program
(i.e. no external field).
Mon Nov 27 2006: Path integral formulation, see script of M. Tuckerman;
advanced stuff:
D. M. Ceperley, Rev. Mod. Phys. 67 (1995) 279,
Path integrals in the theory of condensed helium
Trotter formula
Tue Nov 28 2006: Path integral formulation continued, Many
identical particles, Heisenberg spin model.
Wed Nov 29 2006: Solution of the Gaussian model on a lattice I, See also
Kardar Statistical Mechanics II, Lecture 5 and 8.
Mon Dec 04 2006: No Lecture ; perche non ho tempo
Tue Dec 05 2006: Solution of the Gaussian model on a lattice II
Wed Dec 06 2006: Last lecture, summary