Modern Thermodynamics
- Chapter 1
45
P(N,m) =
N!
m!(N m)!
pm
(1−
p)N−m
(A1.2.6)
Poisson Distribution: In many random processes the random variable is a number n. For example, the
number of gas molecules in a small volume within a gas will vary randomly around an average value.
Similarly, so the number of molecules undergoing chemical reaction in a given volume per unit time. The
probability of n in such processes is given by the Poisson distribution:
P(n) = e−α
α n
n!
(A1.2.7)
The Poisson distribution has a one parameter, α; it is equal to the average value of n, i.e., n = α.
Gaussian Distribution: When a random variable, x, is a sum of many variables, its probability
distribution is generally a Gaussian distribution:
P(x)dx =
1
2πσ 2






1/ 2
e

( x−x0
)2

2 dx
(A1.2.8)
The Gaussian distribution has two parameters, x0 and σ. The average value of x is equal to x0 and the
standard deviation equals σ .
Some Useful Integrals
(a)
e−ax
2
dx
0


=
1
2
π
a






1/ 2
(b)
xe
−ax
2
dx
0


=
1
a
(c)
x2e−ax
2
dx
0


=
1
4a
π1/2
a





(d)
x3e−ax2
dx
0


=
1
2a2
More generally:
(e)
x2ne−ax2
dx
0


=
1.3.5....(2n −1)
2n+1an
π
a






1/ 2
(f)
x2n+1e−ax2
dx
0


=
n!
2
1
an+1



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