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\pi_C(t+1)=\sum_{C'} \pi_{C'}(t) p_{C' \to C}

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Questions and Remarks:

  • Q: I don't understand how to obtain the analytic expression for the internal energy in C.4. This seems to involve a very complicated integral over N variables with the expression for E given in (**) and products of density matrices, which I could not even solve with Mathematica.

  • A: In C.4, "Evaluate numerically" means : use the relation (**) to evaluate ⟨E⟩ within the algorithm. "The analytical result" is the one you have seen in the Lecture on the mean energy of the harmonic oscillator. You can also have a look to part 3.1.1 of the SMAC book and use the relation between the partition function and the mean energy ⟨E⟩.
- Vivien.Lecomte Vivien.Lecomte

  • Q: I don't understand why, in the first part (Q2-3-4), we don't have to include the récursion for c(beta) . Indeed,

    So it depend only on c(beta). And so, if we don't touch to c during our calculation, we will get pi(x=0)=c(beta=8) for all value of Delta_Tau... How can it converge to something ? It's constant.
  • A1: (unofficial answer, since I'm a student) Remember that π(x) is normalized, so in practice you need to divide the density matrix by the partition function,

    In this way the coefficient c(β) cancels out in the expression. As to your second question, we are asked to evaluate the discretization error, which is the difference between the numerical result and the exact result (which you can calculate using (3.38) and (3.40). If you plot the result again n (or Delta tau), you will see that it converges to zero as n becomes large (or Delta tau goes to zero).
  • A2: Your answers are indeed right: π(x) is a probability density, it is normalized, whence the division by the expression of the partition function Z(β) that you describe. The "discretization error" is indeed the difference between the exact result from (3.38) and (3.40) and the result obtained by computing the approximate functions f, g and c using the algorithm described in the problem.
- Vivien.Lecomte Vivien.Lecomte

  • Q: I'm not sure of how can we calculate the error in parts A and B. What we are asked for the error as a function of the $\Delta\tau$?is it just the difference between the value obtained numerically (either for the probability or for the free energy) and the anallytical one following (3.38) and (3.40) respectivelly?
  • A: Yes.
- Vivien.Lecomte Vivien.Lecomte

  • Q: I find a scaling error which scales as Delta tau to the power of ~1.58 for the central density and ~1.0 for the free energy, but it seems to me that the theoretical prediction is a power of ~3. Is anyone having similar results or am I mistaken with the theoretical error?
  • A: We can't give you the answer for the exponent, but you should check your results and your prediction. For the determination of the exponent, you can use a log-log plot.
- Vivien.Lecomte Vivien.Lecomte