[MUSIC] Now let me turn on a light bulb. This now gives us light and heat. But what happened microscopically? Microscopically, there was the motion of electrons, photons and even phonons. The electrons were responsible for the electricity that powered the light bulb. The photons were responsible for the light that came out of the light bulb. And phonons are responsible for the heat that comes out. Now the single uniting feature of these particles is that they're non-interacting systems. By the end of this lecture, you should be able to identify distinguishable and indistinguishable particles, develop the statistical thermodynamics for a two level system, a ground state and an excited state system. And develop cononical partition function for an ideal gas, electrons, phonons, and photons. In this module, we'll study a few model systems under the frame work of statistical thermodynamics. We will begin by looking at a few non-interactive systems. First, we'll introduce the extremely key concept of distinguishability and indistinguishability. Now let's consider a system of two distinguishable particles meaning that they can be uniquely identified according to some criteria, say a fixed spatial position of particles on a surface or a location in space. Now, these two particles will have energy levels given by epsilon A with the index I running from 1 to A and epsilon B with the index I running from 1 to B. Now in the canonical ensemble, the partition function is simply given by Now this can be simplified as a product of two individual parts of a partition function, qA and qB, which are basically the single particle partition function for the two particles. Now generally, for the system with N non-interacting distinguishable particles, each with a single particle partition function Q, the overall partition function Q is simply giving us small q raised to the power N. Now, what about indistinguishable particles? Now, many systems are composed of particles that are indistinguishable from each other. For example, particles in the gas phase cannot be distinguished from each other. Now, for the system of N non-interacting indistinguishable particles, the partition function summation contains instances where particles exist in different states, but cannot be distinguished. Now let's consider one extreme case where all particles exist in different states. In this case the overcounting factor is N factorial, which is simply the number of ways of reassigning N particle labels. Now on the opposite extreme, if all of the particles are in the same state, the indistinguishability and the degeneracy factor are simply the same. So there's absolutely no overcounting factor. Now it turns out that at finite temperature, the particles are more likely to exist in a diversity of states that are over countered by this factor N factorial. Now in order to correctly account for the indistinguishability at finite temperature, the partition function is given by the single particle partition function raised to the power N but divided by N factorial to accomplishing distinguishability. Now let's study our first example to illustrate this. Let's consider a collection of non-interacting, distinguishable particles that can exist in two states, the ground state, and an excited state. Now, the excited state has an energy epsilon, now we will analyse the system using two ensembles derived earlier to show the equivalence between the two descriptions. Let's first take the perspective of a microcanonical ensemble. Now for for the system on N particles we consider a stage with a fixed energy capital E, given as epsilon times N. Where now N is the number of particles in the excited state. Now, the D generosity in the number of states with this fixed energy E is simply given by the combinatorial factor of choosing the number of particles in the excited state small N. From the total number of particles capital N. Now this is the familiar factor, capital N choose small N. Now, what is the entropy of the system? The entropy, as we divide earlier, is simply the logarithm of the number of possible states. So, this is simply given by the following expression Now, this expression is extremely tricky to evaluate. Now to approximate the entropy for large number of particles N in the thermodynamic limit we can apply something known as the Stirling's Approximation. The idea behind the Stirling's approximation is the following. Let's first write the logarithm of N factorial as a summation of log I where I now runs from one to N. Now the summation can then be converted to an integral for sufficiently large N to give the following equation. Now integrating this we get the Stirling's approximation which says that the logarithm of N factorial in the large N limit Is simply given by N log N minus N. Now, let's use the Stirling's approximation to simplify the expression that we derive for the entropy. Note that in this expression, X which is the fraction of the number of particles in the excited state now how do we evaluate the temperature. Let's go back to the definition of the temperature that we had mentioned. Temperature is simply given as the change in the internal energy as you change the entropy holding the volume and the number of particles constant. Now using this definition, and a little bit of chain rule calculus we can evaluate the temperature in a very simple way. We now have everything necessary to evaluate the Helmholtz free energy per particle. This is simply given as the internal energy minus the temperature times the entropy. Now remember in this expression, alpha is defined as the exponential of minus epsilon naught devided by K T. The Helmholtz free energy simply becomes -kB T times log q, where we now define q to be one plus exponential of minus epsilon over kbt. Now let's revisit the same problem from the perspective of a Canonical Ensemble. The Canonical Ensemble has a fixed temperature T and the number of particles N and so the energy states are allowed to fluctuate according to the Bozeman probability distribution. We can easily find the canonical solution from what we had found from the micro canonical ensemble case. We have to account for the B degeneracy factor which is simply given by the combinatorial factor. Now each of these terms has to be multiplied with a factor associated with the probability for that energy state, given by exponential of minus n epsilon not, divided by kbt. Now this can be simplified using the standard binomial theorem. To be 1 plus exponential of minus epsilon not over K-B-T raised to the power of N. Now the Helmholtz free energy once again turns out to be the exact same expression that we had derived from the microcanonical ensemble case. Now we can use the partition function Q as a generating function and use that to derive the average energy. And the average energy simply comes out to be Now let's look at some limiting cases of this now as beta, which is given as 1 over KBT. So as beta tends to 0, that is, when temperature tends to infinity, energy simply is epsilon over 2. That is we have an entropy-driven mixing of states at large temperature. Now what happens when beta turns to infinity or in other words temperature turns to zero, while the energy simply becomes zero. And this is simply because all the molecules occupy the ground state at zero temperature. Now what about the entropy? The entropy is simple given by. Now let's look at some limiting cases for the entropy as beta tends to zero that is when temperature tends to infinity, the entropy per particle simple becomes K log 2 and we have this entropy driven mixing of states at large temperature. What happens as beta tends to infinity. That is, temperature turns to zero, the entropy simply becomes zero, because all the particles are in the downstate.
