Books in the balance

Classical Mechanics Level 2

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})}

Details

  • 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 ( A → B ) P(A\rightarrow B) represents the probability of a book going from state A A to state B B .


The answer is 60.

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Josh Silverman by Josh Silverman

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2 solutions

Peter Macgregor
May 7, 2015

I think the problem needs to be stated more clearly.

Suppose each book in Sue's collection has the same probability of going to the booksellers in a given period (let's say one full week). Denote this by P ( Sue’s shelf → Booksellers ) P(\text{Sue's shelf} \rightarrow \text{Booksellers})

Similarly let each of the 15000 books Sue is interested in have the same probability of being purchased by Sue in one full week. Denote this by P ( Booksellers → Sue’s shelf ) P(\text{Booksellers }\rightarrow \text{Sue's shelf})

Then the expected number of books sold or donated by Sue in one week is

250 P ( Sue’s shelf → booksellers ) 250 P(\text{Sue's shelf} \rightarrow \text{booksellers})

and the expected number of books bought by Sue in one week is

15000 P ( booksellers → Sue ) 15000 P(\text{booksellers} \rightarrow \text{Sue})

Equating these to get detailed balance leads easily to

P ( Sue’s shelf → booksellers ) P ( booksellers → Sue ) = 15000 250 = 60 \frac{ P(\text{Sue's shelf} \rightarrow\text{ booksellers})}{P(\text{booksellers} \rightarrow \text{Sue})}=\frac{15000}{250}=\boxed{60}

7 Helpful 3 Interesting 4 Brilliant 2 Confused

The inflow must be equal to the outflow.

However, the flow is the expected number of books flowing, that is, the probability of the flow times the number of books in the source

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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