福田 育夫

<p>分子シミュレーション法は，原子間の力をできるだけ正確にモデル化することで，原子からなる多体系（生体分子や材料など）の性質をミクロな立場から調べることのできる重要な計算手法である．ただし，この分子シミュレーション法で最も厄介なのが長距離相互作用の扱いである．というのも，古典系では基本的に自由度数の二乗に比例する計算コストがかかり，また境界条件についても悩ましい問題があるからである．さらに，クーロン静電相互作用は，長距離相互作用の中でも取り扱いが自明ではない．距離の関数としてのクーロン関数の減衰が遅いため，最も簡単な計算スキームである単純カットオフ（ある所まで距離が離れたら力をゼロとする）が許されないからである．但し，クーロン相互作用は，重力（万有引力）と同様な関数形を持ちながらも，正負の符号がある点が異なり，これが物理現象の本質を捉える上でも，また効率的な計算スキームを考える上でも非常に重要になる．</p><p>我々は，この正負の符号によって生じる相互作用のキャンセルという物理的アイデアを，数学的に定式化することで，「零多重極子和法」（Zero-Multipole summation Method; ZMM）という静電相互作用計算法を作った．この計算法では，クーロン関数の原子ペア毎の和の代わりに，クーロン関数を変形して得られたある関数の原子ペア毎のカットオフ形式での和を採用する．この変形は，静電相互作用がキャンセルされるという「中性条件」を満たした配置からの寄与を効果的に取り入れるために導入される．カットオフ形式での原子ペア毎の和で相互作用エネルギーが定義されるため，大規模系では系のスケールに比例する計算コストで済む．また，周期境界条件は必須では無くなるため，本来非周期的な系への周期境界条件の適用による不自然な問題は生じない．さらに，波数空間部分の計算も不要なため，高速フーリエ変換を用いた際の通信等の問題も回避でき，並列計算時等における大幅な計算時間短縮につながる．これらの点は，従来のエバルト法に基づいた方法と異なる．また，運動方程式を考えた時のエネルギー及び全運動量の保存則を壊すことも無い．</p><p>ZMMを完全な対称性を持つイオン結晶系，及びその対称性が熱揺らぎにより崩れた液体イオン系に適用して高精度なエネルギーを得た．その際に得られた精度の，ZMMの持つパラメタへの依存性は理論的に説明できるものであった．さらに，ZMMを水分子系に適用し，エネルギー及び諸物理量を測定して，高精度な結果を得た．個々の水分子は永久双極子のため双極子中性条件を満たさないにも拘らず，良好な結果が得られた解釈として，常温・常圧の水分子系ではランダムに永久双極子が配向して，相互作用のキャンセルが起こるためだと考えることができる．しかし，そのようなことが起こらないはずの強誘電性結晶に適用しても良い結果が得られた．その完全な理解には到達していないが，相互作用の相殺以外の概念から導かれた静電相互作用計算手法とZMMとの関連を考察することが有効であろう．</p>

The following article appeared in J. Chem. Phys. 140, 194307 (2014) and may be found at http://scitation.aip.org/content/aip/journal/jcp/140/19/10.1063/1.4875693In the preceding paper [I. Fukuda, J. Chem. Phys.139, 174107 (2013)], the zero-multipole (ZM) summation method was proposed for efficiently evaluating the electrostatic Coulombic interactions of a classical point charge system. The summation takes a simple pairwise form, but prevents the electrically non-neutral multipole states that may artificially be generated by a simple cutoff truncation, which often causes large energetic noises and significant artifacts. The purpose of this paper is to judge the ability of the ZM method by investigating the accuracy, parameter dependencies, and stability in applications to liquid systems. To conduct this, first, the energy-functional error was divided into three terms and each term was analyzed by a theoretical error-bound estimation. This estimation gave us a clear basis of the discussions on the numerical investigations. It also gave a new viewpoint between the excess energy error and the damping effect by the damping parameter. Second, with the aid of these analyses, the ZM method was evaluated based on molecular dynamics (MD) simulations of two fundamental liquid systems, a molten sodium-chlorine ion system and a pure water molecule system. In the ion system, the energy accuracy, compared with the Ewald summation, was better for a larger value of multipole moment l currently induced until l ≲ 3 on average. This accuracy improvement with increasing l is due to the enhancement of the excess-energy accuracy. However, this improvement is wholly effective in the total accuracy if the theoretical moment l is smaller than or equal to a system intrinsic moment L. The simulation results thus indicate L ∼ 3 in this system, and we observed less accuracy in l = 4. We demonstrated the origins of parameter dependencies appearing in the crossing behavior and the oscillations of the energy error curves. With raising the moment l we observed, smaller values of the damping parameter provided more accurate results and smoother behaviors with respect to cutoff length were obtained. These features can be explained, on the basis of the theoretical error analyses, such that the excess energy accuracy is improved with increasing l and that the total accuracy improvement within l ⩽ L is facilitated by a small damping parameter. Although the accuracy was fundamentally similar to the ion system, the bulk water system exhibited distinguishable quantitative behaviors. A smaller damping parameter was effective in all the practical cutoff distance, and this fact can be interpreted by the reduction of the excess subset. A lower moment was advantageous in the energy accuracy, where l = 1 was slightly superior to l = 2 in this system. However, the method with l = 2 (viz., the zero-quadrupole sum) gave accurate results for the radial distribution function. We confirmed the stability in the numerical integration for MD simulations employing the ZM scheme. This result is supported by the sufficient smoothness of the energy function. Along with the smoothness, the pairwise feature and the allowance of the atom-based cutoff mode on the energy formula lead to the exact zero total-force, ensuring the total-momentum conservations for typical MD equations of motion.

The following article appeared in J. Chem. Phys. 134, 164107 (2011) and may be found at http://scitation.aip.org/content/aip/journal/jcp/134/16/10.1063/1.3582791We propose a novel idea, zero-dipole summation, for evaluating the electrostatic energy of a classical particle system, and have composed an algorithm for effectively utilizing the idea for molecular dynamics. It conceptually prevents the nonzero-charge and nonzero-dipole states artificially generated by a simple cutoff truncation. The resulting energy formula is nevertheless represented by a simple pairwise function sum, which enables facile application to high-performance computation. By following a heuristic approach to derive the current electrostatic energy formula, we developed an axiomatic approach to construct the method consistently. Explorations of the theoretical details of our method revealed the structure of the generated error, and we analyzed it by comparisons with other methods. A numerical simulation using liquidsodium chloride confirmed that the current method with a small damping factor yielded sufficient accuracy with a practical cutoff distance region. The current energy function also conducts stable numerical integration in a liquidMD simulation. Our method is an extension of the charge neutralized summation developed by Wolf et al. [J. Chem. Phys.110, 8254 (1999)]. Furthermore, we found that the current method becomes a generalization of the preaveraged potential method proposed by Yakub and Ronchi [J. Chem. Phys.119, 11556 (2003)], which is based on a viewpoint different from the neutrality. The current study presents these relationships and suggests possibilities for their further applications.

我々はTsallis dynamics (TD)を導出し,指数1以上でフルエネルギー型のTsallis分布に従う系の分子動力学計算を可能とすることに初めて成功した.さらに,この方程式の関与するエルゴード性及び分布のエネルギーに関する緩やかな減少性が,複雑なエネルギー面を持つ生体分子等の系の状態サンプリングに有効な事を利用し,効率的なボルツマン分布の再構成法を構築してきた.これらの話題について紹介する.

この論文は国立情報学研究所の電子図書館事業により電子化されました。

A new method for solving optimization problems is proposed in terms of a dynamical system. This method aims for compatibility, which has often been problem in solutions, between two requirements : searching with a high probability for the most likely candidates of the optimal points, and searching quickly in the region. The proposed dynamical system realizes setting vlues of the visiting-weight and speed of the orbits to the system, with an assumption of ergodicity, by values of long-time limit. High values can be set for the visiting-weight to areas where the objective function takes low (high) values. Further, independently, a high-value can be set for the speed of the orbits. Consequently, the two requirements can be satisfied. Accordingly, a trade-off between a searching-weight and a searching-speed does not exist. In a numerical simulation assuming an objective function, validity within a finite time for the system formulated with the long-time limit was confirmed.