PHS702 Statistical Thermodynamics, Lecture 2

Equilibrium Systems

1) Microcanonical Ensemble. Isolated systems of fixed energy U containing N particles of known type or types in a fixed volume V.
2) Canonical Ensemble. Systems in volume V containing number set N of particles in thermal contact with a heat reservoir characterized by a temperature T.
3)Grand Canonical Ensemble. Systems in volume V in thermal contact with a heat reservoir characterized by a temperature T and open to exchange of matter with ther reservoir which is also considered as a source of particles characterized by a set of chemical potentials, m.

Microcanonical Ensemble

Postulate 1. Ergodic Postulate. The time average of any thermodynamic variable f in an actual systme is equal to the ensemble average of f in the limit as the number of replicas n in the ensemble goes to infinity, provided that the members of the ensemble copy precisely the thermodynamic state and environment of the actual system, that is, adhere to the specified macroscopic conditions.

In a macroscopic world

Observed Thermodynamic properties - Time averaged!

dealing with astronomical numbers!

Therefore, the energy spacing between two quantum states are: 1/N which is extremely small!! (a system with N particles)

U + DU reppresents the experimentally observable internal energy.
The DU is in a sense the measure of experimental error, and is always larger than the energy spacing of the macroscopic system.

Postulate 2. Equal A priori Probability. The many-body systems of the ensemble are distributed with equal a priori probabilities over all the possible quantum states of the many-body system consistent with the few macroscopic conditions.

If W-fold degeneracy of quantum states, probability of finding a certain particle in the state l is given by

Consider the following:

Initially, the box on the left-hand side is filled with gas, and empty on the right. Then, the removable wall is taken out to establish an equilibrium.

This leads to more microstates due to availablity of more translational energy states of the gas. Therefore W can only increase when the constraint is removed! This is an Irrevesible change!

The only way to maintain W at any stage of the process, is the reversible change. Then for fixed N, V, and U, W has a lot of similarity with entropy, doesn't it? Tentatively, we assign W with entropy by the following (Boltzmann equation),

Remember that from your Thermodynamics class that the entropy is related to change in heat in a reversible process.

For reversible change - S is constant.

For irreversible change - S is increasing.

Later, we will prove that S is the thermodynamic enetropy function.

Molecular Distribution

If we assume there is no force between molecules, then the total energy of the system is the sum of individual molecules,

and

Then, the number of ways Ndistinguishable particles to be distributed, is given by

such that total of number of distribution is M, then the total distribution is

.example
The following illustrates the relationship between the constraint of energy and and lnW. For example, consider a reaction at an equilibrium

For a system with total of 86e with molecules A, B, and C, two arbitrarily chosen states are shown below:

If enzyme is introduced to establish an equilibrium such that the following occurred.

This is no longer a reversible process because of our constraint of having U to be constant.

The probability of finding particles with the distribution a is

Approximation: W_m is a maximum distribution. Then,

for all a. The largest possible term of W can be thought of W_m multiplied by M, the total number of terms. Therefore, the entropy of the system is bounded by:

and, so

But, M is of order N, and W_m is of order e^N. Therefore, the far-right-hand side term is negligible for large ensemble. Therefore, we obatain:

This means that one can approximate the most probable distribution at its maximum distribution!

S as the thermodynamic entropy

Consider a two-part system.

Total Energy U fixed

U = U₁ + U₂
Two parts - no thermal contact Different volume
Different numbers of particles

If U₁ --> U, and a detectable energy spacing () of experiment is made to be the same as the average energy spacing of a molecule. And, the number of distribution can be kept as an integer by,

Then, the probability of particular energy distribution is

The most probable distribution is at a stationary condition with respect to a is

We can write above equation in the following way if

where it used a chain rule. Then,

Therefore, from the definition (Boltzmann equation)

Note that the condition for this relation being V₁, N₁, V₂, N₂ are constant. This may be viewed as a condition for equilibrium in two-component system. Even though there is no heat exchange between the two parts, but the most probable distribution occurs at the same values of the derivatives! This means that in general the derivative is defined as a temperature.

If we make our two-component system to allow energy exchange, that is dU₁ = - dU₂, then by using the entropy being an extensive variable, S = S₁ + S₂, and its differential should be larger than zero, then

and

Using the definition of temperature shown above, the inequality is

This is quite interesting! If dU₁ is positive T₁ is smaller than T₂, and the opposite is true for dU₁ is negative. Therefore, our definition of temperature is correct!

With similar line of thought, one can arrive at pressure and chemical potential to be,

With a constant N the total differential of S is

which can be rearrange to arrive at the familiar thermodynamic equation

This concludes the section of microcanonical ensemble. Next, we will look at canonical ensemble.