PHS702 Statistical Thermodynamics, Lecture 3

Canonical Ensemble

The canonical ensemble - N, V, T fixed.

For an ensemble,

          volume = AV
          # of particles = AV
          energy = AV

where A is the total number of particles in a system. Energy of each system - not fixed, but

There are

degenerate quantum states, and E_i is repeated.

State label = 1, 2, 3, ... l, ...

Energy = E₁, E₂, ... E_l, ...

Occupation #= a₁, a₂, ... a_l, ...

Occ. # is a number of systems of the ensemble in particular state l
A set of occupation # = distribution
Occ. # satisfies,

                                                  1)
                                                2)

The occ. #, a₁, a₂, ..., are all eqully probable (from a priori probability postulate). Therefore, different occ. # have the same weight for ensemble average.

# of ways a particular distribution of a_j can be realized by

# of ways that A distinguishable object can be arranged into groups, such that

a₁ - in the first group
a₂ - in the second group
and so on

The distribution is given by (as you have seen already)

which is just a multinomial coefficient.

Many distribution satisfies Eq. 1) and 2)

fraction of systems (members of the ensemble) in the j^th energy level
Then, the overall probability of a system in j^th quantum state is obtained by averaging a_j/A over all allowed distribution, which is

Remember that

The average mechanical property is obtained by

3)

where M_j is the j^th quantum state.

Since W(a) is a multinomial coeff. and is highly localized at maximum position (almost a d function!), if a_j are sufficiently large.

We can, then, make the spread of W(a) arbitrarily small by taking a_j to be large.

Then, a set of a_j's other than can be set small! Therefore, we have

in the limit that a_j goes to infinity.

It means that

4)

So, we now have to find distribution a^* that maximizes W(a) with constraints that:

We are going to use Lagrange's Method of Undertermined Multiplier on W(a). The resulting equation is

          
	  where j = 1, 2, 3, ...

Since

Then,

By using Stirling's approximation,

And, we arrive at

or alternatively,

5)

Since this is the maximum distribution sharply peaked at its maximum if we choose an arbitrarily large numbers, this is now equal to most probable distribution.

Determination of a and b
Rewriting Eq. 5) by using a' = 1 + a, and summing both sides over all j, we obtain

Eq. 4) then becomes

Now, we can calculate any mechanical property by using Eq. 3). Therefore, the average energy is given by

The quantity in the denominator persists in the equations in the canonical ensemble. This is the canonical ensemble partition function.

6)

Pressure as mechanical property, since

Therefore,

Evaluation of b, two ways.
1). Through Mechanical Property

Differentiate average E with respect to V w/ constant N and b.

7)

Also differentiate average p with respect to b w/ constant N and V

8)

From Eqs. 7) and 8), you should arrive at (Derive this!)

From thermodynamics, we know that

And, if we change the differential to

using the fact that

Therefore, the term b is

Proof of k being a universal constant
Consider a system
N_A, V_A, {E_Aj}, a_j
N_B, V_B, {E_Bj}, b_j
The systems A and B are in thermal contact. The number of states of the AB system with molecules in {a_j} and {b_j} is

The ensemble must satisfy

Applying most probable distribution of the AB ensemble with systems A in i^th quantum state and systems B in j^th quantum state leads to

Thus, the arbitrary systems in thermal contact have the same b. Further, since b = 1/kT, the constant k is a universal constant.

2). Another way of determining b

Through non-mechanical property
--through entropy

To be done: 1/T is integrating factor of dq_rev

Consider a fxn,

Then,

The total derivative of f is then,

which can be written in the form,

9)

Now, make physical change on the ensemble of systems such that

V + dV
T

T + dT

a_jdE_j = work done on systems

(a_j systems with energy E_j)

Total work done on all systems =

Changing population of systems

Average reversible work =

From these conditions, the term in the parenthesis on the right-hand side of Eq. 9) is interpreted as average reversible heat supplied. Therefore, we have

10)

It means that bdq_rev is the derivative of a state function, and b is an integrating factor.

The left-hand side of Eq. 10) is then dS/k, and by integrating the equation to obtain

What does all these mean?

First, in microscopic (molecular) view, the thermodynamic work done on the system is given by changing energy of the system slightly while keeping the population to be constant.

Second, thermodynamic heat exchange done on the system is changing the population of the system, but keeping the energies of the system constant.

Thermodynamic Quantities from Canonical Ensemble

We have seen again, increasing the number of available system gives arise to spontaneous process in closed system.