2.1: Microcanonical and Canonical Ensembles

Introduction

Statistical thermodynamics (statistical mechanics) is our connection between the microscopic behavior and properties of matter and macroscopic thermodynamic behavior. Statistical thermodynamics involves calculating weighted averages of the microscopic behavior of the constituents of the system of interest to arrive at the macroscopic properties of the system. To do these calculations we must decide which parameters we will allow to vary and which will be held constant. We will use statistical concepts to average over the quantities that can vary. In the language of statistical mechanics this means choosing which ensemble to work with. There are many, but we will consider only two.

Definition of an ensemble

An ensemble is an imaginary replication of the system of interest so that every possible arrangement of the quantities that can vary exist in one of the replications. As an example consider a system consisting of three quantum mechanical particles (A, B and C) each with a ground state and two equally spaced excited state in an isolated system (fixed volume and insulated from the surroundings). If the system contains enough energy for three quanta of excitation we can identify all the different arrangements of energy that use all of it. They are shown below using * to indicate a quantum of excitation:

There is only one real system, which would be found in one of these seven arrangements, but each one of the possible arrangements is part of the imaginary ensemble representing the system. Another name for an arrangement is a microstate.

When building an ensemble, we will make the a priori (before hand) assumption that any microstate that satisfies our constraints is equally likely. This means that if we take many measurements of the distribution of the quanta we will get each of the microstates 1/7 of the time.

Microcanonical ensemble and a statistical definition of entropy

The example ensemble above is a very small microcanonical ensemble, which are isolated systems with the total energy of the system (E), volume (V) and number of particles (N) held constant. In the above example we said nothing about the volume, but because the particles are isolated from the rest of the universe they must be contained is some isolated volume.

Notice that six of the arrangements have one doubly excited molecule, one singly excited molecule and one unexcited (ground state) molecule. If the molecules are indistinguishable any one of these microstates is equivalent (the same configuration or macrostate).  This means one doubly excited, one singly excited and one unexcited molecule is the most likely configuration for us to measure (6/7 of the time). Enumeration is the hard way to find the most likely distribution. This is equivalent to the statistical problem of distributing objects among bins, which has a well known solution. From statistics if n_i objects are in bin i and there are N total objects and k bins (meaning i = 0, 1, 2, ... k-1) the weight (W) is the number of ways of distributing those particles in groups of n_i into the k bins:

\[W = \frac{N!}{n_1!n_2!,…}.\label{EQ:statweight}\]

Think of the energy levels (0, 1 or 2 quanta) as the bins. In the first case all the particles are in the 1 quanta of excitation bin: n₁ = 3 and n₀ = n₂ = 0. This yields:

\[W_1 = \frac{3!}{3!0!0!} = \frac{3\cdot 2\cdot 1}{3\cdot 2\cdot 1\cdot 1\cdot 1} = 1. \nonumber\]

In the second case 1 particle is in the 0 state, one in the 1 state and one in the 2 state:

\[W_2 = \frac{3!}{1!1!1!} = \frac{3\cdot 2\cdot 1}{1\cdot 1\cdot 1} = 6. \nonumber\]

Because we have only two excited states there are no more bins and thus no other configurations. The total number of microstates is just the sums of all the weights (W₁ + W₂ = 7).

Entropy is a measure of disorder. A higher weighting for a configuration corresponds to more ways of getting the configuration and therefore more disorder. Thus, we would expect a larger W to indicate higher entropy. The other property that entropy must have is that it is additive. If we have two systems with entropies S_A and S_B the total entropy S_tot = S_A + S_B. But the total number of configurations we can make with two systems with W_A and W_B is W_tot = W_AW_B. Thus we need a definition in terms of weights that is additive, but also consistent with the fact that the weight of a combined system should be the product of the weights of the individual systems.

 The solution is to utilize the properties of logs. In particular ln(ab) = ln(a) + ln(b). We will define the statistical entropy of a particular configuration as S_i = k_BlnW_i, where k_B = Boltzmann constant. This means that S_A + S_B = k_Bln(W_A) + k_Bln(W_B) = k_Bln(W_AW_B), satisfying both the statistical and thermodynamic requirements for an entropy.

Thus for a microcanonical ensemble the entropy is:

\[S = k_BlnW = k_B ln \left(\frac{N!}{\prod_i{n_i !}}\right) =  k_B \left(ln N!- \sum_i{ln(n_i!)}\right).\label{microcan_S}\]

Real systems have huge numbers of particles making the factorials impractical. Luckily, there is a very good approximation \(\ln x! \approx x\ln x – x \) called Stirling's approximation, which has very small fractional errors for large x. Applying this we can rewrite equation \(\ref{microcan_S}\) as:

\[S = k_B \left(NlnN-N- \sum_i({n_i ln(n_i) -n_i})\right).\label{microcan_S_approx}\]

Since \(\sum_i{n_i} = N\) this simplifies further to:

\[S = k_B \left(NlnN-\sum_i{n_i ln(n_i)}\right) =  k_B \left(\sum_i{n_i lnN} -\sum_i{n_i ln(n_i)}\right)= k_B \left(\sum_i{n_i lnN -n_i ln(n_i)}\right) 
\\=k_B \left(\sum_i{n_i ln\left(\frac{N}{n_i}\right)}\right)  = -k_B \left(\sum_i{n_i ln\left(\frac{n_i}{N}\right)}\right) = -Nk_B \left(\sum_i{\frac{n_i}{N} ln\left(\frac{n_i}{N}\right)}\right).\label{microcan_S_prob_form}\]

Notice that the fractions \(\frac{n_i}{N}\) is the fraction of the total molecules that will be found in the energy level i. Although nice and compact this is still not a very practical way of calculating the entropy if N and thus i are large. The traditional approach is to apply the method of Lagrange multipliers to constrain the energy to our fixed value and maximize the number of microstates. Instead we will develop more generally useful expressions by working with the canonical ensemble.

Canonical ensemble¹

This ensemble is used for constant temperature systems. The volume (V), number particles (N) and temperature (T) of the system are constant. This is a chemically useful way to describe a system. In most cases chemical systems consist of a container in equilibrium with the surroundings, which have a particular temperature. At any given time the system can exchange energy with the surroundings as long as the temperature of the system remains in equilibrium with the surroundings. This is less constrained than the microcanonical ensemble where the system has a fixed total energy. By using the canonical ensemble we will be able to use the thermodynamic definition of temperature to get a very general expression for the entropy of the system. From there we will be able to go on to derive general statistical expressions for other thermodynamic quantities.

Our discussion begins by applying the idea of the weight of our configurations in a more general way. The trick is to express this weight in terms of the surroundings, which are so much larger than the system that we can use an approximate Taylor series expansion to express the entropy without significant error.

For this system in contact with the surroundings, which is part of the Universe, the total energy is fixed at E_T = E_sur + E_sys. We will refer to a particular energy configuration of the system as E_i. Thus for a particular E_i E_sur = E_T - E_i.

Start by considering the canonical equivalent of the microcanonical ratio \(\frac{n_i}{N}\), which we will call \(P(i)\):

\[P(i) = \frac{\text{number of ways of getting the system with energy }E_i}{\text{all ways of arranging}E_T} = \frac{W_{sur}(E_{sur})}{W_T (E_T)}\label{Pi1}\]

To convert this to a more useful general expression we need to express the entropy of the surroundings in terms of the total energy, E_T, and the energy of the system, E_i. Then we will use that to express P(i) in terms of E_i.