Classical Mechanics

Statistical Thermodynamics

Statistical Thermodynamics: Level 1-2 Challenges


The Earth is protected by a layer of atmosphere that can absorb heat from the Earth. Does this keep the Earth warmer, cooler, or at the same temperature as it would be without the atmosphere?


  • For simplicity, we assume that the atmosphere is transparent to radiation from the Sun. It can only absorb infrared radiation from the Earth, which is created when radiation from the Sun hits the Earth.

You are an astronaut, it is a few hours before you are scheduled to launch for an intergalactic space mission, and you're sitting on the shore of the Atlantic Ocean near Cape Canaveral, savoring the view one last time. This mission is especially trying; because of the relativistic speed of your spaceship, Earth will be one billion years into the future upon your return.

Depressed beyond words, you decide, in one of your weaker moments, to pee into the Atlantic Ocean.

Flash forward: 1 billion years.

You return to Earth, expecting a hero's welcome, but instead you find that all of humanity has vanished. Instead, the Earth is run by a peaceful clan of telekinetic dolphins who made off with the lion's share of Bitcoins that were abandoned by the last humans as they uploaded their souls to the singularity server. In a disillusioned haze, you bend down and fill your astronaut survival cup with refreshing lake water, hoping for some clarity. After drinking the glass, you realize that this far into the future, all the Earth's water has been thoroughly well mixed since the time you took off on your mission.

Approximately how many molecules of your billion year old urine did you just consume?

Assumptions and details

  • You released 1 L\textrm{1 L} of urine into the Atlantic Ocean.
  • Approximate your urine as consisting of pure water.

A system is said to obey detailed balance if every forward process is balanced by its reverse process. In other words, there is no net movement from any state to any other state.

Suppose Sue has 250 books on her shelf, and there are a total of 15,000 books she is potentially interested in buying from booksellers. If Sue's bookshelf is in detailed balance with the world of booksellers, find P(book from Sue’s shelf booksellers)P(book from booksellers Sue’s shelf)\frac{P(\text{book from Sue's shelf}\rightarrow \text{ booksellers})}{P(\text{book from booksellers}\rightarrow \text{ Sue's shelf})}


  • A book might go from Sue's shelf, to the booksellers if Sue sells some of her books to a bookseller, or donates some of them.
  • P(AB)P(A\rightarrow B) represents the probability of a book going from state AA to state BB.

You have a paper cup filled with water and you heat the bottom with the flame from a bunsen burner. What happens to your system?


  • We are thinking about the period in which the cup still contains water (it hasn't all been lost due to evaporation, or some other process).

The shape in the sky is:


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