Use a computer to produce a table and graph, like those in this section, for two interacting two-state paramagnets, each containing elementary magnetic dipoles. Take a "unit" of energy to be the amount needed to flip a single dipole from the "up" state (parallel to the external field) to the "down" state (antiparallel). Suppose that the total number of units of energy, relative to the state with all dipoles pointing up, is; this energy can be shared in any way between the two paramagnets. What is the most probable macrostate, and what is its probability? What is the least probable macrostate, and what is its probability?
The most likely macrostate is when the energy units are evenly distributed, , with a probability of . The least likely state is when all the energy units are in partition or , or when , with a chance of .
The probability of is,
The overall multiplicity is,
The total multiplicity is ,
Multiplicity of is,
Multiplicity of is,
Use Stirling's approximation to find an approximate formula for the multiplicity of a two-state paramagnet. Simplify this formula in the limit to obtain . This result should look very similar to your answer to Problem ; explain why these two systems, in the limits considered, are essentially the same.
Calculate the number of possible five-card poker hands, dealt from a deck of 52 cards. (The order of cards in a hand does not matter.) A royal flush consists of the five highest-ranking cards (ace, king, queen, jack, 10) of any one of the four suits. What is the probability of being dealt a royal flush (on the first deal)?
For a single large two-state paramagnet, the multiplicity function is very sharply peaked about .
(a) Use Stirling's approximation to estimate the height of the peak in the multiplicity function.
(b) Use the methods of this section to derive a formula for the multiplicity function in the vicinity of the peak, in terms of . Check that your formula agrees with your answer to part (a) when .
(c) How wide is the peak in the multiplicity function?
(d) Suppose you flip coins. Would you be surprised to obtain heads and 499,000 tails? Would you be surprised to obtain 510,000 heads and 490,000 tails? Explain.
Consider again the system of two large, identical Einstein solids treated in Problem .
(a) For the case , compute the entropy of this system (in terms of Boltzmann's constant), assuming that all of the microstates are allowed. (This is the system's entropy over long time scales.)
(b) Compute the entropy again, assuming that the system is in its most likely macro state. (This is the system's entropy over short time scales, except when there is a large and unlikely fluctuation away from the most likely macro state.)
(c) Is the issue of time scales really relevant to the entropy of this system?
(d) Suppose that, at a moment when the system is near its most likely macro state, you suddenly insert a partition between the solids so that they can no longer exchange energy. Now, even over long time scales, the entropy is given by your answer to part (b). Since this number is less than your answer to part (a), you have, in a sense, caused a violation of the second law of thermodynamics. Is this violation significant? Should we lose any sleep over it?
For each of the following irreversible processes, explain how you can tell that the total entropy of the universe has increased.
Stirring salt into a pot of soup. Scrambling an egg. Humpty Dumpty having a great fall. A wave hitting a sand castle. Cutting down a tree.Burning gasoline in an automobile.
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