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Register a new userComparison of two efficient methods for calculating partition functions
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In the long-time pursuit of the solution to calculate the partition function (or free energy) of condensed matter, Monte-Carlo-based nested sampling should be the state-of-the-art method, and very recently, we established a direct integral approach that works at least four orders faster. In present work, the above two methods were applied to solid argon at temperatures up to $300$K, and the derived internal energy and pressure were compared with the molecular dynamics simulation as well as experimental measurements, showing that the calculation precision of our approach is about 10 times higher than that of the nested sampling method.
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First-order phase transitions that proceed via nucleation of bubbles are described by the homogeneous nucleation theory of Langer. The nucleation rate is one of the most interesting parameters of these transitions. In previous works we have computed nucleation rates with two different methods: (A) employing coarse-grained potentials; (B) by means of a saddle-point approximation, using dimensionally regularized field theory. In this article we compare the results of the two approaches in order to test their reliability and to determine the regions of applicability. We find a very good agreement.
Vibrational spectra and wavefunctions of polyatomic molecules can be calculated at low memory cost using low-rank sum-of-product (SOP) decompositions to represent basis functions generated using an iterative eigensolver. Using a SOP tensor format does not determine the iterative eigensolver. The choice of the interative eigensolver is limited by the need to restrict the rank of the SOP basis functions at every stage of the calculation. We have adapted, implemented and compared different reduced-rank algorithms based on standard iterative methods (block-Davidson algorithm, Chebyshev iteration) to calculate vibrational energy levels and wavefunctions of the 12-dimensional acetonitrile molecule. The effect of using low-rank SOP basis functions on the different methods is analyzed and the numerical results are compared with those obtained with the reduced rank block power method introduced in J. Chem. Phys. 140, 174111 (2014). Relative merits of the different algorithms are presented, showing that the advantage of using a more sophisticated method, although mitigated by the use of reduced-rank sum-of-product functions, is noticeable in terms of CPU time.
We derive an integral-free thermodynamic perturbation series expansion for quantum partition functions which enables an analytical term-by-term calculation of the series. The expansion is carried out around the partition function of the classical component of the Hamiltonian with the expansion parameter being the strength of the off-diagonal, or quantum, portion. To demonstrate the usefulness of the technique we analytically compute to third order the partition functions of the 1D Ising model with longitudinal and transverse fields, and the quantum 1D Heisenberg model.
Efficient implementations of the classical molecular dynamics (MD) method for Lennard-Jones particle systems are considered. Not only general algorithms but also techniques that are efficient for some specific CPU architectures are also explained. A simple spatial-decomposition-based strategy is adopted for parallelization. By utilizing the developed code, benchmark simulations are performed on a HITACHI SR16000/J2 system consisting of IBM POWER6 processors which are 4.7 GHz at the National Institute for Fusion Science (NIFS) and an SGI Altix ICE 8400EX system consisting of Intel Xeon processors which are 2.93 GHz at the Institute for Solid State Physics (ISSP), the University of Tokyo. The parallelization efficiency of the largest run, consisting of 4.1 billion particles with 8192 MPI processes, is about 73% relative to that of the smallest run with 128 MPI processes at NIFS, and it is about 66% relative to that of the smallest run with 4 MPI processes at ISSP. The factors causing the parallel overhead are investigated. It is found that fluctuations of the execution time of each process degrade the parallel efficiency. These fluctuations may be due to the interference of the operating system, which is known as OS Jitter.
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 emph{ab initio} calculations.The method was demonstrated by C$_{60}$ molecules, solid and liquid copper (up to $sim 600$GPa), 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 $2500$K, 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.
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