Introduction to Quantum Physics and Information Processing

216 Introduction to Quantum Physics and Information Processing

compatible with a given macrostate is Ω then the Boltzmann entropy of the

system is defined to be

S = k

B

ln Ω,

where k

B

is Boltzmann’s proportionality constant.

1

We think of the mi-

crostates as tiny imaginary cells dividing the total gas volume, for the gas

particles to occupy. If p

i

is the probability that the i

th

cell is occupied, the

Boltzmann entropy of the gas can be worked out to be

S = −k

b

X

i

p

i

ln p

i

. (11.3)

Example 11.1.2. Consider a set X of possible events {a, b, c, d} with the

following probabilities:

p(a) = 0.5; p(b) = 0.3; p(c) = 0.1; p(d) = 0.1

Then the entropy of this distribution