1Fundamentals of statistical mechanics

II Statistical Physics

1.6 Extensive and intensive properties

So far, we have defined a lot of different quantities —

p, V, µ, N, T, S

etc. In

general, we can separate these into two different types. Quantities such as

V, N

scale with the size of the volume, while µ and p do not scale with the size.

Definition

(Extensive quantity)

.

An extensive quantity is one that scales pro-

portionally to the size of the system.

Definition

(Intensive quantity)

.

An intensive quantity is one that is independent

of the size of the system.

Example. N, V, E, S are all extensive quantities.

Now note that the entropy is a function of

E, V, N

. So if we scale a system

by λ, we find that

S(λE, λV, λN) = λS(E, V, N).

Example. Recall that we defined

1

T

=

∂S

∂E

V,N

.

So if we scale the system by

λ

, then both

S

and

E

scale by

λ

, and so

T

does

not change. Similarly,

p = T

∂S

∂V

T,N

, µ = −T

∂S

∂N

T,V

Example. The free energy is defined by

F = E − T S.

Since

E

and

S

are both extensive, and

T

is intensive, we find that

F

is extensive.

So

F (T, λV, λN) = λF (T, V, N).

Similarly, the grand canonical potential is

Φ = F − µN.

Since F and N are extensive and µ are intensive, we know Φ is extensive:

Φ(T, λV, µ) = λΦ(T, V, µ).

This tells us something useful. We see that Φ must be proportional to

V

.

Indeed, taking the above equation with respect to λ, we find

V

∂Φ

∂V

T,µ

(T, λV, µ) = Φ(T, V, µ).

Now setting λ = 1, we find

Φ(T, V, µ) = V

∂Φ

∂V

T,µ

= −pV.

Here p is an intensive quantity, it cannot depend on V . So we have

Φ(T, V, µ) = −p(T, µ)V.

This is quite a strong result.