Canonical Estimates¶
In this tutorial we will discuss how estimates for various mean-field properties of a system can be evaluated in the canonical ensemble at finite temperatures. These estimates are useful for basis set extrapolation as well as comparison to the fully interacting results and are non-trivial to evaluate analytically. See [Malone15] for details.
The input file is fairly simple:
sys = ueg {
nel = 7,
ms = 7,
dim = 3,
cutoff = 10,
rs = 1,
}
canonical_estimates {
sys = sys,
canonical_estimates = {
beta = 1,
nattempts = 10000,
ncycles = 1000,
fermi_temperature = true,
},
}
Here we attempt to generate N particle states making nattempts
attempts and then run
the simulation for ncycles*nattempts
iterations in total. The only other options
available are the inverse temperature desired, which can be scaled by the Fermi
temperature (where appropriate). Here we restrict ourself to the fully spin polarised UEG
in M=389 plane waves, which can be compared to the IP-DMQMC simulation in the DMQMC
tutorial.
Running the input file we find
$ hande.x canonical_estimates.lua > canonical_estimates.out
Inspecting the output
, we
see a number of columns for various estimates including the kinetic, potential, internal,
free energy and entropy - precise definitions of everything can be found in the output
file. The data can be analysed to find the mean and standard error using the
analyse_canonical.py
script in the tools/canonical_energy
subdirectory:
$ analyse_canonical.py canonical_estimates.out
which gives
Beta U_0 U_0_error T_0 T_0_error V_0 V_0_error N_ACC/N_ATT N_ACC/N_ATT_error F_0 F_0_error S_0 S_0_error T_HF T_HF_error U_HF U_HF_error V_HF V_HF_error
1.00000000e+00 3.34489604e+01 7.12207413e-03 3.42505858e+01 7.00598613e-03 -8.01625332e-01 1.43282029e-04 1.67618487e-01 1.16924926e-04 -1.83693842e+01 2.03921704e-03 1.79999929e+01 2.45760105e-03 3.37906337e+01 6.93061658e-03 3.29774580e+01 7.04945348e-03 -8.13175652e-01 1.46382195e-04
In particular, we can compare the values of \(U_0\) and \(U_{\mathrm{HF}}\) to the value of 32.91(4) Ha from the IP-DMQMC tutorial.