In this week, we will talk about thermodynamics, statistical mechanics, and their modern applications. It's very rich literature. But here we will keep things short and talk about one key concept of that, which is entropy. What is entropy? Entropy in thermodynamics is defined as follows. Consider an isolated system, divide it into subsystems. If we keep track of the heat transfer between different subsystems within this isolated system, and there is Delta Q, the heat transfer of the subsystem i divided by T. Summing this quantity over every subsystem, the whole isolated system, then we get a quantity Delta S as the change of enthalpy. In other words, the change of entropy is summing over the heat transfer between subsystems, weight it by the inverse temperature. This is the change of entropy and that is related to the two key laws of thermodynamics, which is the second law of thermodynamics, dE equals to TdS minus pdV. This is nothing mysterious because recall this TdS is nothing but the heat transfer of very small elements of subsystem at least. Then this energy change of the subsystem, it can generalize to the whole system. The energy change will be the heat transfer minus the work that is done within the subsystem. This is the first and all of thermodynamics, which is energy conservation. Nothing about energy conservation, so it is relatively easy to understand. There is a second law of thermodynamics as well which is the change of entropy within this whole isolated system will be non-decreasing, is either greater than zero or equal to zero. This is entropy. This is very mysterious, at least it was mysterious to me when I first learned this concept. Why? Because first of all, we were told and actually we can prove within thermodynamics that the entropy indicates the state of the system. Entropy is defined as some procedure heat transfer. But actually, it indicates the state of the system, some properties of the system, similar to volume, similar to energy, that state of the system. However, volume is very intuitive, how large it is. Energy, very intuitive. But entropy, what is entropy? What is the corresponding state? What on Earth is this state of the system that entropy describes? What is the property? The second mysterious thing about entropy is the second law of thermodynamics itself. Why it is mysterious? Because it defines for us a preferred time direction. It defines for us an arrow of time. Almost every other law of physics has no preference in the time direction. What do I mean? What I mean is if I have a law describing a particle moving in one direction, then if I revert the time direction from moving forward to moving backward, I revert the arrow of time, then equally, that law can describe the particle moving from the other direction to one direction. This is time-reversal symmetry, which is generalizing particle physics to CPT symmetry. Nevertheless, there is no preferred arrow of time for almost every other law in nature. However, for entropy, there is a clear preference that entropy is increasing in the future directions that defines past and the future for us. What is the special thing of the entropy to define for us past and the future? Number 3, there is a thought experiment, there is a paradox known as the Maxwell's demon What is the paradox? It is saying that suppose there is a demon and this demon can do one thing which is to guard this gate, and if there is a particle moving from this part of the box to this part of the box, the demon will examine the speed of this particle. If the speed is a large enough, then this demon lifts the gate, so the largest speed particle can move to the other part of the system, from left to right. However, if from the left, the speed of the particle is not that large, the demon is not going to allow this particle to move to the right part of the system. After some time, what happens? After some time you will notice that this part of the system on average has smaller speed for the molecular motions and this part of the system has larger speeds. This is because the demon by purpose only allow the larger speed particles to move toward this side. What that means? Effectively, that means heat has transferred from the low-temperature part of the system to the high-temperature part of the system almost without any other effects. Only this demon is raising and lowering the gate. Opening, closing the gate and demon can be made as small enough if you can imagine. In this case, it appears to violate the second law of thermodynamics, that heat is transferring from a low temperature to high temperature without anything else and you can also write it as Delta S is smaller than zero in this case and what is wrong? We have discussed quite a few questions and these questions cannot be answered in thermodynamics itself. We have to broaden our views to connect the thermodynamics to statistical mechanics, and also to information. Then we know they are actually very natural in the modern view of thermodynamics, statistical mechanics, and information theory. This is what we'd like to do in this week.