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

2σ

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

π1/2

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