[MUSIC] Indeed, the Entropy Maximization Principle is fundamentally a Convex Optimization Problem. The Convex Optimization Problem is guaranteed because the way in which we've defined the entropy, we do guarantee a global maximum and this actually found by the Lagrange Multipliers method. Now, the Principle of Maximum Entropy states that subject to precisely stated prior data, such as a proposition that expresses testable information. The probability distribution which best represents the carnal state of knowledge is the one with maximum entropy. Now, another way of stating this, take precisely stated prior data or testable information about a probability distribution function. Now, consider a set of trial probability distributions that would encode the prior data that you've collected of those, the one with maximum information entropy is the proper distribution according to this principle. The principle was first expounded by Edwin James in two papers in 1957, where he emphasized a natural correspondence between Statistical Thermodynamics and Information Theory. In particular, James offered a new and very general rationale of why the Gibbesian approach of Statistical Thermodynamics works. Now, he argued that the entropy of Statistical Thermodynamics is principally one and the same with the information entropy concept in Information Theory. Consequently, Statistical Thermodynamics should be seen just as a particular application of a general tool of logical inference and information theory. Now, this is a wonderful question. Let's discuss this with an example. Now, take all the air in a room and start it out with a special configuration. The special configuration is where we only occupy a small corner of the room. In a short amount of time, the molecules will spread out over the room and occupy the full volume of the room. Now can the opposite happen? No. Because it violates the Second Law of Thermodynamics, right? The entropy of a confined gas is less than the entropy of the gas that occupies the entire room. But, if we time reverse every molecule of the final state, the air will rush back to the corner of the room. Now, the problem is, that if we make a tiny error in the motion of just a single molecule the error grows exponentially with what is known as the exponent. And instead of going back into the corner of the room they actually go and fill the entire room. Now, this is not to say that freak accidents don't happen. Now, if we wait long enough the air in the room will accidentally congregate in the corner. The correct statement is not that unlikely things never happen, but only that they very rarely happen. The time that you would have to wait for the unusual air event to take place is exponential in the number of molecules Now that's a great question. The many physical phenomena of interest that involve quasi-thermodynamic processes that are slightly out of equilibrium. Let's take some examples. Now, heat transport by the internal motions in a material, driven by a temperature imbalance. Now electric currents, carried by the motion of charges, in a conductor, driven by a voltage imbalance. Spontaneous chemical reactions, driven by a decrease in free energy. Friction, dissipation, quantum decoherence, and so on. Now all of these processes occur over time with characteristic rates. And these rates are of crucial importance in engineering. Now, the field of what is called Non-equilibrium Statistical Thermodynamics concerns itself with understanding these non-equilibrium processes at the microscopic level. Now, Statistical Thermodynamics, the way we have learned it can only be used to calculate the final result after all these external imbalances have been removed and the ensemble settles back in equilibrium. In principle, Non-equilibrium Statistical Thermodynamics could be exact and ensembles, for instance, for an isolated system could be evolved over time according to the doministic equations such as the Louisville's Theorem, or the Quantum Mechanical Version, the Fornierian Equation. Now, in order to make headway in to modeling this irreversible processes its necessary to add additional ingredients besides probability and reversible mechanics. Now, Non-equilibrium Statistical Thermodynamics is therefore still an active area of theoretical research as the range of validity of these additional assumptions continue to be explored. Now, this is a tricky question. As we have learned, there are Thermodynamic Potentials that govern the behavior of specific ensembles. And these Thermodynamic Potentials are state variables. Now an important theorem holds for the state variables that the second order partial derivatives with respect to these potentials, do not depend on the order in which you perform the derivative. Let's take an example. Let's take internal energy U. Now, we can take a second order derivative of the internal energy U with respect to the entropy and the volume. Now we can take this first with respect to volume, then with respect to entropy, or we can take it with entropy and then with respect to volume. Now, why is this important? It turns out that with this, you can connect the partial derivative of temperature with respect to volume to the derivative of the pressure with respect to entropy. Now, these relations are what are known as maximal relations. Now, the maximal relations are very useful for relating difficult to define Thermodynamic Quantities to ones that are more easily determined. In particular, changes in entropy, as you pointed out are difficult to find. So, it's easy to relate this to changes in pressure, volume or temperature. Wow, a tricky last question. Now, any response to the question of what is the meaning of Chemical Potential is necessarily subjective. Now what satisfies one person is probably going to be wholly inadequate to satisfy another. Now, there are three characterizations of the Chemical Potential that captures the diverse aspects of its manifold nature. Now, as a function of position the Chemical Potential measures the tendency of particles to defuse. Now, when a reaction may occur. An of some Thermodynamic Function determines equilibrium. Now, the Chemical Potential measures the contribution of this change per particle and for an individual species, the Thermodynamics function's rate of change. Now, the Chemical Potential also provides a characteristic energy, that is, the change in energy when one particle is added to the system Holding of course entropy and the volume constant. Now, I have to add that these three assertions need to be qualified by the contextual conditions under which they have been framed. Now, the first statement captures an essence especially when the temperature is uniform. Now, if this is not the case and the temperature varies spatially, diffusion is somewhat more complex. Now, the second statement that we described is valid if the temperature is uniform and fixed. If instead the total energy is fixed, and the temperature may vary from place to place then it turns out that the Chemical Potential divided by the temperature measures this contribution. Now, when one looks for conditions that describe chemical equilibrium, one may focus on each locality separately and then, the division by temperature is inconsequential. Now, the system's external parameters are the macroscopic environmental parameters. Such as the external magnetic field or the container volume that appear in the energy operator, or the energy eigenvalues. Now, all external parameters are to be held constant when the derivative in statement three that we described is formed. The subscript V, that is, volume, illustrates merely the most common situation. Note that the pressure does not appear in the Eigenvalues. So, in the present usage, pressure is not an external parameter. Thanks a lot for all the questions that you've given us in the office hours. And I hope you enjoyed the answers.