# Idea of Path integral Monte Carlo

- Computation of Thermodynamic (equilibrium) quantities of a quantum many-particle system
- Transformation of the Quantum Partition function into Feynman's path integral picture
- Decomposition (Factorization) of the unknown N-particle density matrix into a product of many

(known) high-temperature density matrices
- (Anti-)symmetrization of the result for bosons (fermions)
- Computation of the multidimensional integrals using Monte Carlo methods

## Exchange of Fermi particles

Illustration of the Fermionic exchanges: Shown are 5 non-interacting electrons in a
two-dimensional plane in real space. Each line (color) corresponds to the "trajectory"
(or probability density) of one electron.

**Left figure**: neglecting spin. Thick dots are initial and final point of the trajectory
(they coincide)

**right figure**: with spin included. Notice that different lines ("polymers") are now
connected - this is fermionic exchange in the path integral picture.

Formulas and simulation by Alexei Filinov, 2000