MultihistogramAnalysis
Documentation for MultihistogramAnalysis.
At the moment the library will only work with systems that have a discreet energy spectra. Generation of histograms neccesarily means that we bin data, and thus throw away data points if the energy spectra is not discreet. Use with caution on systems with continuous energy spectra.
The multihistogram method solves for the density of states by solving for the partition functions at the temperatures simulated at. The main equation that is solved iteratively is.
\[Z(\beta_k) = \sum_{i}\sum_E \frac{N_i(E)}{\sum_j n_j Z^{-1}(\beta_j) e^{(\beta_k - \beta_j)E}}\]
Where $i,j$ indexes the simulation number, $E$ are the energies sampled in the i-th simulation, $N_i$ is the energy histogram (i.e, number of times an energy occurs in the simulation), and n_i are the number of samples in the i-th histogram. For this library, $n_i = 0$ for all $i$. Since the partition functions depend exponentially on the energy, it is better to use the logarithms ($F(\beta_k) = F_k = \log(Z(\beta_k)$) of the partition functions to avoid float overflow. Thus the equation becomes.
\[F_k =\log\left( \sum_i \sum_E \frac{N_i(E)}{\sum_j \log(-F_k + (\beta_k - \beta )E} \right)\]
To avoid overflow further, we normalize all partition functions by the following factor.
\[A = \frac{1}{\sqrt{Z_{max}Z_{min}}} \implies \log(A) = -\frac{1}{2} (F_{max} + F_{min})\]
To interpolate the free energy we use a natural extension of the above equation.
\[F(\beta) = \log\left( \sum_i \sum_E \frac{N_i(E)}{\sum_j \log(-F_k + (\beta - \beta)E} \right)\]
And the calculate the thermal average we use the following.
\[\langle Q \rangle = \frac{1}{Z(\beta)} \sum_{i, (Q, E)} Q \frac{N_i(Q, E)}{\sum_j A Z^{-1}_j e^{(\beta - \beta_j)E}}\]
Logsum Methods
The logsum trick allows us to calculate the logarithm of a sum of a series of logarithms that we want to exponentiate. Specifically, if $l_i$ are logarithms, and we want to calculate $\log(\sum_i e^{l_i})$ without ever exponentiating the logarithms, this trick lets us do that. If $l_0$ is the largest logarithm, then one can calculate the sum as follows.
\[\log\left(\sum_{i=0} \exp(l_i)\right) = \log\left(e^{l_0} \left[1 + \sum_{i=1} e^{l_i - l_0}\right]\right) = l_0 + \log\left(1 + \sum_{i=1} e^{l_i - l_0}\right)\]
This has the advantage of avoiding numerical overflow and calculate observables for very large systems. A function name that ends with _logsum uses this method. Note that this has also been implemented for ComplexF64 vectors.
Interpolatioon Functions
After importing the library, you can write interpolate_ and try to autocomplete with TAB to see all the interpolation methods. Counterintuatively, the MultihistogramData structs go as the last argument rather than the first. I did not have a lot of time to write this library cleanly, will be fixed in the future.