Efficient approaches to solutions of partition function for condensed matters

Abstract

The key problem of statistical physics standing over one hundred years is how to exactly calculate the partition function (or free energy), which severely hinders the theory to be applied to predict the thermodynamic properties of condensed matters. Very recently, we developed a direct integral approach (DIA) to the solutions and achieved ultrahigh computational efficiency and precision. In the present work, the background and the limitations of DIA were examined in details, and another method with the same efficiency was established to overcome the shortage of DIA for condensed system with lower density. The two methods were demonstrated with empirical potentials for solid and liquid cooper, solid argon and C60 molecules by comparing the derived internal energy or pressure with the results of vast molecular dynamics simulations, showing that the precision is about ten times higher than previous methods in a temperature range up to melting point. The ultrahigh efficiency enables the two methods to be performed with ab initio calculations and the experimental equation of state of solid copper up to ∼600 GPa was well reproduced, for the first time, from the partition function via density functional theory implemented.

Thermal contributions to the structural phase transitions of crystalline aluminum under ultrahigh pressures

“…As formulated in statistical mechanics, the FE can be readily obtained without empirical parameters as long as the partition function (PF) is solved while unfortunately the exact solution to the PF of condensed matters is a long-standing problem because of the complexity of the high dimensional configurational integral [13,40,41], so that state-of-the-art numerical algorithm for PF can hardly afford a system consisting of more than several hundred particles even using empirical interatomic force field and the determined transition pressures for Al were far from the experimental results [42]. Very recently, we put forward a direct integral approach (DIA) to the PF of condensed state systems with ultrahigh efficiency and precision [43][44][45][46], and has been successfully applied to investigate the phase transitions of vanadium [47], the EOS of copper [43] and the optimum growth condition for 2-D materials [44] combined with density functional theory (DFT). Compared with quasi-harmonic phonon model, DIA was examined to be applicable to much wider realm with much higher precision [46].…”

“…For the system of crystalline Al with the atoms locating at the lattice sites, Q N , and with the corresponding total potential energy, U 0 (Q N ), the DIA [43] firstly introduces a transformation as…”

“…( 5) and ( 6) depends on the geometric symmetry of the investigated structures. For the FCC and BCC structures, as the same procedures applied to copper [43] and vanadium [47] in our previous works, Eq. ( 5) is used to calculate the configurational integrals because the effective lengths along the three axes, L x,y,z , of any atom in a conventional cubic supercell of FCC or BCC are of a same value, while, in terms of the HCP structure, it is obvious that the values of L x,y,z are not equivalent to each other due to the asymmetric geometry along the three axes, and thus, Eq.…”

Pressure-induced structural phase transition of vanadium: A revisit from the perspective of ensemble theory

“…Very recently, we put forward a direct integral approach (DIA) to the PF of condensed state systems with ultrahigh efficiency and precision [25][26][27][28], and has been successfully applied to reproduce the equation of state (EOS) for solid copper [25], argon [27] and 2-D materials [26] obtained from experiments or molecular dynamics simulations. Compared with phonon model based on harmonic or quasi-harmonic approximations, which is currently applied to produce EOS, DIA is applicable to much wider realm with much higher precision [28].…”

“…According to our proposed DIA [25], for a singlecomponent crystal with N atoms placed in their lattice sites Q N and with the total potential energy U 0 (Q N ), we firstly introduce a transformation,…”