1Revision of equilibrium statistical physics

III Theoretical Physics of Soft Condensed Matter

1.1 Thermodynamics

A central concept in statistical physics is entropy.

Definition (Entropy). The entropy of a system is

S = −k

B

X

i

p

i

log p

i

,

where

k

B

is Boltzmann’s constant,

i

is a microstate — a complete specification

of the microscopics (e.g. the list of all particle coordinates and velocities) — and

p

i

is the probability of being in a certain microstate.

The axiom of Gibbs is that a system in thermal equilibrium maximizes

S

subject to applicable constraints.

Example.

In an isolated system, the number of particles

N

, the energy

E

and

the volume

V

are all fixed. Our microstates then range over all microstates

that have this prescribed number of particles, energy and volume only. After

restricting to such states, the only constraint is

X

i

p

i

= 1.

Gibbs says we should maximize

S

. Writing

λ

for the Lagrange multiplier

maintaining this constraint, we require

∂

∂p

i

S − λ

X

i

p

i

!

= 0.

So we find that

−k

B

log p

i

+ 1 − λ = 0

for all i. Thus, we see that all p

i

are equal.

The above example does not give rise to the Boltzmann distribution, since

our system is completely isolated. In the Boltzmann distribution, instead of

fixing E, we fix the average value of E instead.

Example.

Consider a system of fixed

N, V

in contact with a heat bath. So

E

is no longer fixed, and fluctuates around some average

hEi

=

¯

E

. So we can

apply Gibbs’ principle again, where we now sum over all states of all

E

, with

the restrictions

X

p

i

E

i

=

¯

E,

X

p

i

= 1.

So our equation is

∂

∂p

i

S − λ

I

X

p

i

− λ

E

X

p

i

E

i

= 0.

Differentiating this with respect to p

i

, we get

−k

B

(log p

i

+ 1) − λ

I

− λ

E

E

i

= 0.

So it follows that

p

i

=

1

Z

e

−βE

i

,

where Z =

P

i

e

−βE

i

and β = λ

E

/k

B

. This is the Boltzmann distribution.

What is this mysterious

β

? Recall that the Lagrange multiplier

λ

E

measures

how S reacts to a change in

¯

E. In other words,

∂S

∂E

= λ

E

= k

B

β.

Moreover, by definition of temperature, we have

∂S

∂E

V,N,...

=

1

T

.

So it follows that

β =

1

k

B

T

.

Recall that the first law of thermodynamics says

dE = T dS − P dV + µ dN + ··· .

This is a natural object to deal with when we have fixed

S, V, N

, etc. However,

often, it is temperature that is fixed, and it is more natural to consider the free

energy:

Definition

(Helmholtz free energy)

.

The Helmholtz free energy of a system at

fixed temperature, volume and particle number is defined by

F (T, V, N) = U − T S =

¯

E − T S = −k

B

T log Z.

This satisfies

dF = −S dT −P dV + µ dN + ··· ,

and is minimized at equilibrium for fixed T, V, N.