Math 480, Fall 2016, Assignment 11
From cartan.math.umb.edu
The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.
- - Saint Augustine
Carefully define the following terms, then give one example and one non-example of each:[edit]
- Boltzmann distribution (on the energies of the particles in some physical system).
- Thermalization (of a physical system).
- Temperature (of a physical system which has already thermalized).
Carefully state the following theorems (you need not prove them):[edit]
- Theorem characterizing the most entropic energy distribution consistent with a fixed expected energy.
- Shannon's Noisy Channel Theorem.
Solve the following problems:[edit]
- What energy distribution corresponds to zero temperature? What distribution corresponds to "infinite temperature?"
- (Optional) Our calculation of the maximally entropic distribution consistent with a fixed mean energy was rather cavalier, since we assumed without justification that the maximum would occur in the interior of the set of all distributions consistent with the given mean energy. Give an example of a system and a mean energy for which the maximum occurs on the boundary. (Hint: technically one can do this with a two-level system, but it is really more instructive to work with a three-level system.) Explain why this does not occur for realistic systems (in which the set of possible energies is actually discrete but unbounded).
- Is temperature really just "what a thermometer measures?" Consider the following statement, taken from the Wikipedia article on the "thermosphere" (an outer layer of the Earth's atmosphere): "The highly diluted gas in this layer can reach 2,500 °C (4,530 °F) during the day. Even though the temperature is so high, one would not feel warm in the thermosphere, because it is so near vacuum that there is not enough contact with the few atoms of gas to transfer much heat. A normal thermometer might be significantly below 0 °C (32 °F), at least at night, because the energy lost by thermal radiation would exceed the energy acquired from the atmospheric gas by direct contact." Explain.
