102 Microcanonical ensemble
x
f(x)
s
Fig. 3.3Gaussian distribution given in eqn. (3.8.11).
erally, ifxis a Gaussian random variable with zero mean, its probability distribution
is
f(x) =
(
1
2 πσ^2
) 1 / 2
e−x
(^2) / 2 σ 2
, (3.8.11)
whereσis the width of the Gaussian (see Fig. 3.3). Here,f(x)dxis the probability
that a given value of the variable,x, will lie in an interval betweenxandx+ dx. Note
thatf(x) satisfies the requirements of a probability distribution function:
f(x)≥ 0
∫∞
−∞
dx f(x) = 1. (3.8.12)
The cumulative probability that a randomly chosen value ofxlies in the interval
x∈(−∞,X) for some upper limitXis
P(X) =
∫X
−∞
dx f(x) =
(
1
2 πσ^2
) 1 / 2 ∫X
−∞
dxe−x
(^2) / 2 σ 2
. (3.8.13)
SinceP(X) is a number between 0 and 1, the problem of samplingf(x) consists,
therefore, in choosing a probabilityξ∈[0,1] and solving the equationP(X) =ξ,
the probability thatx∈(−∞,X] forX. The resulting value ofX is known as a