Knowlpedia

Definition

Normalized torus average

Integration over a lattice cell divided by its volume gives a translation-invariant probability average.

For an integrable function on the Rn/BZn\mathbb R^n/B\mathbb Z^n, its normalized torus average is

⟨f⟩=1∣det⁡B∣∫B[0,1)nf([x]) dx=∫[0,1)nf([By]) dy.\langle f\rangle=\frac1{|\det B|}\int_{B[0,1)^n}f([x])\,dx =\int_{[0,1)^n}f([By])\,dy.

In particular ⟨1⟩=1\langle1\rangle=1. The same integral results from any measurable fundamental domain.

Translation invariance

Translating the domain and partitioning it into lattice translates of pieces of the original domain shows ⟨f(⋅+a)⟩=⟨f⟩\langle f(\cdot+a)\rangle=\langle f\rangle. Thus this is the probability normalization of on the torus. On the unit torus a smooth function's average is its . Averaging independent auxiliary variables does not mean averaging physical coordinates after an auxiliary map has been evaluated.