The PDF covers the
following topics related to Mathematical Physics : Introduction to
statistical mechanics, Canonical Ensembles for the Lattice Gas,
Configurations and ensembles, The equivalence principle, Generalizing
Ensemble Analysis to Harder Cases, Concavity and the Legendre transform,
Basic concavity results, Concave properties of the Legendre transform, Basic
setup for statistical mechanics, Gibbs equilibrium measure, Introduction to
the Ising model, Entropy, energy, and free energy, Large deviation theory,
Free energy, Basic Properties, Convexity of the pressure and its
implications, Large deviation principle for van Hove sequences, 1-D Ising
model, Transfer matrix method, Markov chains, 7 2-D Ising model, Ihara graph
zeta function, Gibbs states in the infinite volume limit, Conditional
expectation, Symmetry and symmetry breaking, Phase transitions, Random field
models, Proof of symmetry-breaking of continuous symmetries, The spin-wave
perspective, Infrared bound, Reflection positivity.
The purpose of this note is to present standard and widely used mathematical methods in Physics, including
functions of a complex variable, differential equations, linear algebra and special functions associated with eigenvalue problems of ordinary and
partial differential operators.
note covers the following topics: Series of Functions, Binomial Theorem,
Series Expansion of Functions, Vectors, Complex Functions, Derivatives,
Intergrals, and the Delta Function, Determinants, Matrices, Vector Analysis,
Vector Differentiation and Integration, Integral Theorems and Potential
Theory, Curvilinear Coordinates, Tensor Analysis, Jacobians and Differential
Forms, Vectors in Function Spaces, Gram-Schmidt Orthogonalization and
Operators, Transformations, Invariants, and Matrix Eignevalue Problems,
Hermitian and Normal Matrix Eigenvalue Paroblems, Ordinary Differential
Equations, Second-Order Linear ODEs, Green's Functions.
The main focus of this note is on theoretical
developments rather than elaborating on concrete physical systems, which the
students are supposed to encounter in regular physics courses. Topics covered
includes: Newtonian Mechanics, Lagrangian Mechanics, Hamiltonian Mechanics,
Hilbert Spaces, Operators on Hilbert spaces and Quantum mechanics.
Durhuus and Jan Philip Solovej
This note covers the following topics: Prologue, Free Fall and Harmonic Oscillators, ODEs and SHM, Linear Algebra,
Harmonics - Fourier Series, Function Spaces, Complex Representations, Transform
Techniques, Vector Analysis and EM Waves, Oscillations in Higher Dimensions.