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非平衡系におけるHyperuniformityの理論的研究

松山, 洋道 名古屋大学

2022.06.02

概要

熱平衡系では、物理量の揺らぎは長距離の相関を持たない。しかし、臨界点近傍では、粒子の運動が協同的となり、相関長が増大するために、長距離の揺らぎが発散する。近年の研究で、このような密度揺らぎの増大とは対照的に、相関長は増大するものの長距離の密度揺らぎが抑制される現象が報告された。この現象は、Hyperuniformity と呼ばれている。

鳥の視細胞分布から初期宇宙の密度揺らぎに至るまで、 Hyperuniformity は幅広い非平衡系で報告されており、新しいタイプの長距離揺らぎとして注目を浴びている。しかし、非平衡系における Hyperuniformity の本質的な理解は十分には得られていない。そこで、本研究では、3 つの非平衡系を選び、それらの系に対して、分子動力学法を用いた数値計算を行った。

1 つ目は、自発的な運動性を持つ粒子で構成された Active Matter 系の研究を行った。本研 究では、 Active Matter の理論模型の一つである Generalized ActiveOrnstein-Uhlenbeck Particle (GAOUP)模型を用い、研究を行った。我々は、広いパラメータ領域を探索することで、GAOUP 系の相図を完成させた。その結果、密度揺らぎの増大で特徴付けられる凝集相と、それとは対照的な密度揺らぎの減衰を示すHyperuniformity 相が広い領域に渡って存在することを示した。さらに、Hyperuniformityを特徴付ける量である Hyperuniformity 指数が非自明な値を持つことを示した。また、密度場だけでなく、速度場にも長距離相関が生じることを示した。

2 つ目は、Jamming 系における Hyperuniformity について解析を行った。粒子を系に詰めていくとき、低密度では流動的であるが、密度を上げていくと、系はある密度で弾性的に振舞う Jamming 相へと転移する。この Jamming 転移点近傍では、Hyperuniformityが生じることが知られている。これまで、Jamming 転移点での Hyperuniformity は空間次元に依存しないとされていたが、我々は数値実験や過去のデータの再解析から、空間次元依存性を明らかにした。また、この依存性は、Jamming 転移の上部臨界次元が 2 であると仮定すれば、自然に導かれるものであることを示した。Jamming 転移点は、系に力学的アニールを施すことで増大することが知られている。我々は、この Jamming 転移点の増加に伴って、Hyperuniformity 指数が増大することを示した。

3 つ目は、高密度かつ有限温度である、ガラス系に着目した。ガラス系においても、熱揺らぎの背後に Hyperuniformity が存在することが報告されている。我々は、Hyperuniformity の特性長を定量化し、低温領域でこの特性長が急激に増大することを確認した。さらに、緩和ダイナミクスとエネルギー地形の解析も行うことで、エネルギー地形の定性的な変化に伴って、この特性長が増大していることを示した。この結果は、ランダムに見えるガラスの粒子配置に一種の秩序が存在し、それが温度の低下と共に成長していくことを示している。

以上の 3 つの異なる系の結果は、系のダイナミクスや生成手法、エネルギー地形などに強く依存して、 Hyperuniformity の性質が変化することを示している。

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