Perturbation method to calculate the density of states

Abstract

Monte Carlo switching moves ("perturbations") are defined between two or more classical Hamiltonians sharing a common ground-state energy. The ratio of the density of states (DOS) of one system to that of another is related to the ensemble averages of the microcanonical acceptance probabilities of switching between these Hamiltonians, analogously to the case of Bennett's acceptance ratio method for the canonical ensemble [C. H. Bennett, J. Comput. Phys., 22, 245 (1976)]. Thus, if the DOS of one of the systems is known, one obtains those of the others and, hence, the partition functions. As a simple test case, the vapor pressure of an anharmonic Einstein crystal is computed, using the harmonic Einstein crystal as the reference system in one dimension; an auxiliary calculation is also performed in three dimensions. As a further example of the algorithm, the energy dependence of the ratio of the DOS of the square-well and hard-sphere tetradecamers is determined, from which the temperature dependence of the constant-volume heat capacity of the square-well system is calculated and compared with canonical Metropolis Monte Carlo estimates. For these cases and reference systems, the perturbation calculations exhibit a higher degree of convergence per Monte Carlo cycle than Wang-Landau (WL) sampling, although for the one-dimensional oscillator the WL sampling is ultimately more efficient for long runs. Last, we calculate the vapor pressure of liquid gold using an empirical Sutton-Chen many-body potential and the ideal gas as the reference state. Although this proves the general applicability of the method, by its inherent perturbation approach, the algorithm is suitable for those particular cases where the properties of a related system are well known.

On the inner workings of Monte Carlo codes

“…Equation (126) suggests that if the unknown function vðEÞ can be generated for a wide range of energies then the partition function is known, and therefore any desired thermodynamic property. For example, the canonical partition function is related to the free energy by [241] AðT; VÞ ¼ 2 1 b lnQðT; VÞ ð 127Þ…”

“…The computation of the partition function by numerical approximation is the basis for many DOS methods, [241] such as the reference system equilibration (RSE) method, [242] histogramming, [243,244] multihistogramming, [245,246] the histogram reweighting method, [247,248] the Wang -Landau sampling, [249,250] multicanonical methods, [251 -253] transition matrix methods [254,255] and the nested sampling (NS) algorithm. [256 -259] DOS methods use a finite range of energies in practice, which means that the partition function can only be computed to within a multiplicative constant.…”

Discrete and Weyl density of states for photonic dispersion relation

“…Therefore, one should consider this quantum mechanical finiteness of the energy levels and calculate the density of states in a completely discrete manner. Recently, such an approach has been used, contributing to the limited number of studied on the DOS [12][13][14][15][16][17][18][19][20][21][22], to enumerate the number of states exactly by taking into account the finiteness of the minimum allowable energy interval as dictated by quantum mechanics, namely, the discrete density of states function (DDOS) [23]. This approach has been used for the usual, but important, particle in a box model and has been shown to lead to both bounded and unbounded continua expressions in the appropriate limits [23].…”

Sigma method for the microcanonical entropy or density of states

“…is the phase density, also known as the density of states, where C is a constant that assures ω(E) is dimensionless and δ is Dirac's δ function. Algorithms to evaluate ω(E) (and thus S(E)) abound in the literature [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. They each have their advantages and drawbacks, and an exhaustive review of them all is not possible in this Brief Report.…”