By Prasenjit Saha

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6) i=1 and this is true for general rational pi . Now choose ni = M/N, assuming N divides M. 7) and among other things it immediately tells us that s(1) = 0. 7) with respect to N, set N = 1 and integrate with the initial condition s(1) = 0 to obtain the unique solution s(N) ∝ ln N. As things stand, s(N) ∝ ln N clearly is a solution, but we still have to prove uniqueness. This is where the monotonicity requirement comes in. 8) for any integer k, n. Now let k, l be any integers ≥ 2. Then for sufficiently large n we can find an integer m such that ln k m+1 m ≤ < , n ln l n or lm ≤ kn < lm+1 .

It always works (though if f has an integrable singularity or the region is infinite, one will need an extra transformation). But it can be very inefficient, especially in several dimensions. A more efficient way, if tractable, is to go via the cumulative distribution function. E. Knuth, The Art of Computer Programming, vol. 2 (Addison-Wesley 1979), which includes the gem “Random numbers should never be generated by a program written at random. ” 30 Monte-Carlo Basics 31 and for a uniform random number r, output F−1 (r); it will be distributed according to f(x).

Thermodynamics is about measuring a few macroscopic properties (such as internal energy, volume) of systems that are microscopically very complex—a gram of water has > 3 × 1022 molecules—and predicting other macroscopic properties (such as temperature, pressure). Given some macroscopic data we assign probabilities p1 , p2 , . . to different microstates using the principle of maximum entropy, and then use the assigned probabilities to predict other macroscopic quantities. (Microstate refers to the detailed state—including position and energy of every molecule—of the system.