The Statistical Interpretation of Entropy

The aim of this lecture is to show that entropy can be interpreted in termsof the degree of randomness as originally shown by Boltzmann.

Boltzmann’s definition of entropy is that where is theprobability that a given state exists.

lnBS k

For example, we consider a system composed of 3 particles with energylevels where the energy of level 0 is zero, level 1 is u, level 2 is 2u and level 3 is 3u. Let the total energy of the system, U = 3u.

1 2 3, , ,o

The total energy of 3u can be present with various configurations or

microstate complexions.

Distinguishable complexions for U = 3u.

All 3 of these complexions or microstates correspond to a single “observable”macrostate.

e1

e2

eo

e3

a =1

a; all three particles in level 1; probability of occurrence 1/10

b = 3

b; one particle in level 3, 2 particles in level 0; 3/10

u

2u

3u

0

c = 6

c; one particle in level 2, 1 particle in level 1, one particle in level 0; 6/10

In general the number of arrangements or complexions within a singledistribution is given by

1 2

!

! ! !... !o i

n

n n n n

where n particles are distributed among energy levels such that no are in level

eo,

n1 are in level eo, etc. 3!

13!0!0!

3!3

2!1!0!3!

61!1!1!

a

b

c

distribution a;

distribution b;

distribution c;

The most probable distribution is determined by the set of numbers ni

that maximizes .

Since for real systems the numbers can be large (consider the number in1 mole of gas), Stirling’s approximation will be useful,

ln ! lnx x x x

The observable macrostate is determined by constraints.

1 1

1

...

...

i r

o o r r i ii o

i r

o o r ii o

U n n n n

n n n n n n

constant energy in the system

constant number of particles in the system

Any interchange of particles among the energy levels are constrained bythe conditions:

0

0

i ii

ii

U n

n n

Also using the definition and Stirling’s approximation;

1

1

!

! !... !

ln ln ! ln ! ln ! ... ln !

ln ln ln

o i

o i

i

i i ii o

n

n n n

n n n n

n n n n n n

A

B

The constraints on the particle numbers impose a condition on ,

ln ln 0i in n

What we need is to find the most likely microstate or complexion and that will be given by the maximum value of . This occurs when equations A, B and C aresimultaneously satisfied.

C

Technique of Lagrange multipliers which is a method for finding the extrema of afunction of several variables subject to one or more constraints.

We will multiply equation A by a quantity b, which has the units of reciprocalenergy.

0i ii

n D

Equation B is multiplied by a dimensionless constant a,

0ii

n E

Equations C, D and E are added to give,

ln 0.i r

i i ii o

n n

i.e.,

1 1 1ln ln ... ln 0.o o o r r rn n n n n n

This can only occur is each of the bracketed quantities are identically zero,

ln 0i in

rearranging for the ni,

iin e e

and summing over all r energy levels,

i

i r i r

ii o i o

n n e e

The quantity 1 ...i o r

i r

i o

e e e e P

is very important and occurs very often in the study of statistical mechanics. Itis called the partition function, P. Then,

/

i

i r i r

ii o i o

n n e e

e n P

This allows us to write the expression for ni in convenient form,

i

i

nen

P

So, the distribution of particles maximizing is one in which the occupancy or population of the energy levels decreases exponentially with increasing energy.

We can identify, the undetermined multiplier b using the following argument

connecting with entropy, S.

Consider two similar systems a and b in thermal contact with

entropies Sa and Sb and associated thermodynamic probabilities a and b. Since entropy (upper case) is an extensive variable, the total

entropy of the composite system is

a bS S S

The thermodynamic probability of the composite system involves a productof the individual probabilities,

a b

Since our aim is to connect with entropy, S,, we seek

S f

Then we must have

a b a bf f f f

The only function satisfying this is the logarithm, so that we must have

ln ,S k

where k is a constant.

Now we can identify the quantity b. We start with the condition,

ln ln 0i in n C

and make the substitution in C for ln 0,i in ln in from

ln i in

ln lni i i in n n

Expanding

ln i i in n

rearranging

ln i i in n = 0

ln U

and solving for b,

ln

U

But we can see that,

ln ln 1 1ln

V

d d Sk

U dU k dU k U

The constant volume condition results from the fixed number of energy states.

The from the combined 1st and 2nd Law

1;

V V

dU TdS pdV

U ST

S U T

and finally 1; Bk k

kT

Configurational and Thermal Entropy

Mixing of red and blue spheres

for unmixed state 1,

for mixing of red and blue spheres; !

! !b r

confb r

n n

n n

1

Then

!ln

! !b r

conf Bb r

n nS k

n n

The total entropy will be given by

ln ln

ln

total thermal conf

total B th B conf

total B th conf

S S S

S k k

S k

The number of spatial configurations available to 2 closed systems placed in thermal contact is unity. For heat flow down a temperature gradient weonly have th changing.

Similarly for mixing of particles A and B the only contribution to the entropychange will be S conf if the redistribution of particles does not cause anyshift in energy levels, i.e. . This would be the case of ideal mixingsince the total energy of the mixed system would be identical to the sum of the energies of the individual systems. This occurs in nature only rarely.

1 2th th