Thermodynamics: More Entropy
From Warmup On a kind of spiritual note, this could possibly explain how God works some miracles. Supposing He could precisely determine which microstate occurs, He could heat, cool, transform, and manipulate matter in countless other ways while staying within the laws of the physical universe. I noticed at the canon commons a sign saying that kiwis are nutrient rich: they have more nutrients than calories. I thought you would find that funny. I really want to know their units. Well, I guess calories was measured in calories, but what about nutrients? I agree. I don t think the comparison makes sense. Isn t it kind of like saying 10 kg > 1 m?
From Warmup How specifically can entropy be calculated. In general it is hard. In this class we will calculate entropy two ways: 1. Explicitly count the number of microstates (using combinatorics) 2. Calculate the change in entropy using an integral Why must the second law of thermodynamics be stated in so many different ways? The Kelvin-Planck and Classius statements are really proto versions that were given before entropy was discovered. The definitive statement is that the entropy of a closed system increases.
Microstates vs. Macrostates Left microstate: part of the royal flush macrostate Right microstate: part of the garbage macrostate The most common macrostates are those with the most microstates.
From Warmup Using ideas from both the reading and from the last lecture, explain why heat flows from hot to cold when the process of energy exchange between two objects is "random". (How can you get directed motion of heat, when energy is being exchanged both ways?!) If we think of the heat as little units of heat, each unit is equally likely to be in each region. Therefore, there are more microstates for which the units are split roughly in half. Instead of thinking of heat flowing from one object to the other, think of it as the heat distributing itself evenly between the objects. Building on this idea: each unit of energy has equal probability to move between objects. Since there are more units in the hot object, it is more probable that the entire object has a net loss of energy and that the cool object will have a net gain. This is what we perceive as heat.
Calculating Entropy Entropy is a state variable It doesn t matter what path you use to calculate it. Always use a path that is internally (not totally) reversible This means it is a path on a P-B diagram Only really matters for adiabatic free expansion. Examples (In groups) Change in entropy for an adiabatic compression? Isothermal Compression? Isochoric process? Isobaric process?
From Warmup When two systems A and B can exchange energy, the entropy of system A *always* decreases when system A gives energy to system B. If that's so, why would energy ever spontaneously flow from system A to sytem B? If the total entropy of the system AB is increased while simultaneously lowering that of system A (eg. B entropy is increased more than A decreases) then energy will flow. If the entropy of B increases more than the entropy of A decreases, the energy transfer an be spontaneous. The combined entropy is increasing If the entropy is always increasing how do we get things like stars and planets and life to spontaneously form?
Dice You roll two dice. What are the microstates? (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3) How many microstates are there? How many microstates if we roll 5 dice? What is the most likely macrostate?
Many Dice You roll 10^23 dice with your left hand. How many microstates are there? You roll 10^23 dice with your right hand. How many microstates are there? How many microstates are there in the COMBINED system? This is getting ridiculous
Solution: Use Logarithm Entropy: S = (constant) x log (# microstates) Log = Natural Logarithm Constant has units J/K. Much more manageable numbers Log(10^23) = 53 Combining two systems: 2 nd Law: System moves to macrostates with more microstates
The second law of thermodynamics System in macrostate with most microstates This is a statistical law shouldn t we see unlikely stuff occassionally? Flip 10^23 coins. What is the probability of getting all heads? Do this once a second for the age for the age of the universe (10^15 sec), what is the probability of getting all heads at least once? Warmup review: What is the probability that all the molecules in a mole of gas are in the left 99% of the container?
Entropy and Information Theory Entropy is the number of microstates in a macrostate Entropy is how much information you would need to identify which microstate the system is in. The second law states that the lost information about the state of the universe always grows. (and can never be recovered!) Clicker Poll: Is the 2 nd Law of Thermodynamics a Fundamental Law of Physics A. Yes B. No
Some Philosophical Questions Thermodynamics is driven by Entropy. Entropy depends on the definition of the macrostates. Who, or what, determines the macrostate? Is it entirely subjective? A matter of perspective? Could we choose different macrostates and get a different theory of thermodynamics? Defining different macrostates for the same system: Dice Sum of the numbers Difference of the numbers Number of 1 s Etc.
Maxwell s Demon o Maxwell imagined a microscopic demon that could violate the 2 nd law o Imagine a gas in a container with a partition and a door. o The demon lets fast moving molecules into the right half and vice versa. o Therefore (Maxwell concludes), the 2 nd law is a statistical consequence of our being big (macroscopic)
Maxwell s Demon Resolution o Maxwell s demon must process information about the speeds of approaching molecules o What does it do with this information? Record it? Erase it? o It can t record information indefinitely eventually it must erase some of its information. o Erasing information increases the entropy! o The increase in entropy caused by erasing the information is always more than the entropy decrease caused by sorting the molecules o Maxwell s demon cannot violate the second law!
Macro/Micro theories in Science Science is hierarchical Branches of science related as macro/micro theories Understanding the relation between macro/micro theories remains one of the fundamental problems in science. Information Physics helps explain the relationship between these fields Microtheory Kinetic/Atomic Theory String Theory Quantum Mechanics Chemistry Biology Microeconomics Weather Macrotheory Thermodynamics Quantum Field Theory Chemistry Biology Psychology Macroeconomics Climate
Relevance to current physics: From Science (November 2013): When we observe systems on different scales (different macrostates), there is a loss of information (entropy) about microscopic details. This is a big part of what makes science possible: If we needed to know string theory to do Newtonian physics, Newton would never have discovered his laws. The same principles hold in other branches of science.