Girvin, Yang Modern Condensed Matter Physics 2019 SSP · Problem 3.16

Lattices and Symmetries

Physics — Condensed Matter · cumulant-expansion

Problem

Consider a general probability distribution function of an arbitrary number of variables, and an observable

A

that depends on these variables in an arbitrary way. We may always write

\langle e^A \rangle = e^C, \tag{3.69}

where

C

is given by the so-called cumulant expansion

C \equiv \sum_{n=1}^{\infty} \frac{1}{n!} c_n. \tag{3.70}

Without loss of generality, we may subtract a constant from

A

to impose the constraint that

\langle A \rangle = 0

. Under this constraint show that the lowest few cumulants obey

\begin{aligned} c_1 &= 0, \tag{3.71} \\ c_2 &= \langle A^2 \rangle, \\ c_3 &= \langle A^3 \rangle, \\ c_4 &= \langle A^4 \rangle - 3\langle A^2 \rangle^2. \end{aligned}

If the distribution of

A

is close to Gaussian then we expect the cumulant series to converge rapidly.