This is the second episode of the Toward-QMC series. I discuss how interactions enter the thermodynamics of quantum systems from an operational standpoint. The Hubbard-Stratonovich transformation allows us to take a step forward by decomposing the interaction factor as an integral over external fields. Check it out! More details on the algebra next time.Video. Notes.
Monthly Archives: January 2023
Toward QMC: Trotter-Suzuki factorization
Here’s a first installment of an introductory series dedicated to quantum Monte Carlo at finite temperature. First step: a brief technical note on the Trotter-Suzuki factorization. Video. Notes.
Roots on the complex plane
I derive in detail the specific form of 2D Newton-Raphson for a simple root-finding problem on the complex plane. Try it out and create your own fractals!Video. Notes.
Roots in 2D
Generalization of the Newton-Raphson method to 2D, i.e. two equations with two unknowns. Basic algebraic setup, schematics of the algorithm, potential pitfalls, and generalizations. Video. Notes.
Roots using the fixed-point method
A very brief introduction to the fixed-point method to calculate roots. I discuss the convergence criterion and its relation to the Newton-Raphson method. Video. Notes.
Roots via Newton-Raphson
This is a very brief explanation of the Newton-Raphson method, its geometry and derivation, and the schematics of the algorithm.Video. Notes.
Green’s function for diffusion in 1D. Part 2.
I finalize the discussion started in Part 1 with an example showing the behavior for a specific source, which is a simple Gaussian in space and in time. I comment on how to make sure the desired initial conditions are satisfied by adding a solution to the homogeneous equation. Video. Notes.
Green’s function for diffusion in 1D. Part 1.
I solve for the Green’s function of the diffusion operator on the whole real line. To that end, I use Fourier transforms in space and time and contour integration (more on that later!).Video. Notes.