Simple Nature - Light and Matter

B When we run the Carnot engine in figures d-g, there are four partsof the universe that undergo changes in their physical states: the hotreservoir, the cold reservoir, the working gas, and the outside world towhich the shaft is connected in order to do physical work. Over one fullcycle, discuss which of these parts gain entropy, which ones lose entropy,and which ones keep the same entropy. During which of the four strokesdo these changes occur?5.4 Entropy As a Microscopic Quantitya / A gas expands freely, doublingits volume.b / An unusual fluctuation inthe distribution of the atomsbetween the two sides of thebox. There has been no externalmanipulation as in figure a/1.5.4.1 A microscopic view of entropyTo understand why the second law of thermodynamics is alwaystrue, we need to see what entropy really means at the microscopiclevel. An example that is easy to visualize is the free expansion ofa monoatomic gas. Figure a/1 shows a box in which all the atomsof the gas are confined on one side. We very quickly remove thebarrier between the two sides, a/2, and some time later, the systemhas reached an equilibrium, a/3. Each snapshot shows both the positionsand the momenta of the atoms, which is enough informationto allow us in theory to extrapolate the behavior of the system intothe future, or the past. However, with a realistic number of atoms,rather than just six, this would be beyond the computational powerof any computer. 2But suppose we show figure a/2 to a friend without any furtherinformation, and ask her what she can say about the system’s behaviorin the future. She doesn’t know how the system was prepared.Perhaps, she thinks, it was just a strange coincidence that all theatoms happened to be in the right half of the box at this particularmoment. In any case, she knows that this unusual situation won’tlast for long. She can predict that after the passage of any significantamount of time, a surprise inspection is likely to show roughlyhalf the atoms on each side. The same is true if you ask her to saywhat happened in the past. She doesn’t know about the barrier,so as far as she’s concerned, extrapolation into the past is exactlythe same kind of problem as extrapolation into the future. We justhave to imagine reversing all the momentum vectors, and then allour reasoning works equally well for backwards extrapolation. Shewould conclude, then, that the gas in the box underwent an unusualfluctuation, b, and she knows that the fluctuation is very unlikelyto exist very far into the future, or to have existed very far into thepast.What does this have to do with entropy? Well, state a/3 hasa greater entropy than state a/2. It would be easy to extract mechanicalwork from a/2, for instance by letting the gas expand while2 Even with smaller numbers of atoms, there is a problem with this kind ofbrute-force computation, because the tiniest measurement errors in the initialstate would end up having large effects later on.316 Chapter 5 Thermodynamics