1549380323-Statistical Mechanics Theory and Molecular Simulation

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