Comparison of Two Efficient Methods for Calculating Partition Functions

Gong, Le-Cheng; Ning, Bo-Yuan; Weng, Tsu‐Chien; Ning, Xi-Jing

doi:10.3390/e21111050

Cited by 12 publications

(13 citation statements)

References 40 publications

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“…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].…”

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confidence: 99%

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

Ning¹,

Zhang²

2022

Preprint

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The ab initio computation of thermodynamic properties of condensed matters under extreme conditions is a long-time pursuit. In this work, the pressure induced structural phase transition of crystalline aluminum up to 600 GPa at room temperature is investigated by the criterion of free energy derived directly from the partition function formulated in ensemble theory with the interatomic interactions characterized by density functional theory computations. The transition pressures of the FCC→HCP→BCC phase transition are determined at 194 and 361 GPa, the axial ratio of the HCP structure is found to be equal to 1.62 and the discontinuities in the equations of states are confirmed to be associated with −0.674% and −0.897% volume changes, which all validate the accuracy of measurements by one of the recent experiments but disagree with the observations by others. Compared with the results obtained by the widely used criterion of enthalpy at 0K, this work shows that the thermal contributions play a nontrivial role for the structural stability of aluminum even under a room-temperature circumstance.

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confidence: 99%

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

Ning¹,

Zhang²

2022

Preprint

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“…The real computer time of DIA to calculate the PF of 4000 Cu atoms characterized by the manybody TB potential [38] is about 5 minutes with full use of a desktop 8-core AMD Ryzen 1800X CPU (3.6GHz per core), and, for the 32000 Ar atoms characterized by the pairwise L-J potential [6], is about 2 minutes using one physical core of the CPU. For a solid argon system of 500 atoms described by L-J potential at temperatures ranging from 80K to 300K, we ran a NS algorithm with N c ∼ 10 9 and DIA with N c ∼ 10 4 (real computer time is about 4 hours for NS and about 5 seconds for DIA respectively by using one physical core of the CPU) to calculate internal energy and pressure, which were compared to the MD simulations using the same potential function, demonstrating that the precision of DIA is about 10 times higher [50]. The accuracy of DIA has also been proved by calculating the internal energy of graphene or γ-graphyne materials on Cu substrate using Brener potential function [51], and silicene on Ag substrate using Tersoff potential function [52].…”

Section: For Solid Argonmentioning

confidence: 99%

Solution to the key problem of statistical physics -- calculations of partition function of many-body systems

Ning,

Gong,

Weng

et al. 2019

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The key problem of statistical physics standing over one hundred years is how to exactly calculate the partition function (or free energy) of many-body interaction systems, which severely hinders application of the theory for realistic systems. Here we present a novel approach that works at least four orders faster than state-of-the-art algorithms to the problem and can be applied to predict thermal properties of large molecules or macroscopic condensed matters via ab initio calculations. The method was demonstrated by C60 molecules, solid and liquid copper (up to ∼ 600GPa), solid argon, graphene and silicene on substrate, and the derived internal energy or pressure is in a good agreement with the results of vast molecular dynamics simulations in a temperature range up to 2500K, achieving a precision at least one order higher than previous methods. And, for the first time, the realistic isochoric equation of state for solid argon was reproduced directly from the partition function. II. DIRECT INTEGRAL APPROACH TO PARTITION FUNCTIONThe original sense of one-fold (1D) integral I 1D = a1 0 f (x)dx is interpreted as the sum of infinite number of rectangles with area A i = f (x i )∆x, and I 1D =

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“…Very recently, we put forward a direct integral approach (DIA) to the PF of condensed state systems with ultrahigh efficiency and precision [29][30][31][32], and has been successfully applied to reproduce the equation of state (EOS) for solid copper [29], argon [31] and 2D materials [30] 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 [32].…”

Section: Introductionmentioning

confidence: 99%

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

Ning¹,

Ning²

2022

J. Phys.: Condens. Matter

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For realistic crystals, the free energy strictly formulated in ensemble theory can hardly be obtained because of the diﬃculty in solving the high-dimension integral of the partition function, the dilemma of which makes it even a doubt if the rigorous ensemble theory is applicable to phase transitions of condensed matters. In the present work, the partition function of crystal vanadium under compression up to 320 GPa at room temperature is solved by an approach developed very recently, and the derived equation of state is in a good agreement with all the experimental measurements, especially the latest one covering the widest pressure range up to 300 GPa. Furthermore, the derived Gibbs free energy proves the very argument to understand most of the experiments reported in the past decade on the pressure-induced phase transition, and, especially, a novel phase transition sequence concerning three diﬀerent phases observed very recently and the measured angles of two phases agree with our theoretical results excellently.

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Comparison of Two Efficient Methods for Calculating Partition Functions

Cited by 12 publications

References 40 publications

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

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

Solution to the key problem of statistical physics -- calculations of partition function of many-body systems

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

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