콜라츠 추측의 동역학을 Cellular Automata에 전역적인 무작위 요소를 추가한 모형으로 이해 = Understanding Collatz Dynamics through a Cellular Automata Model with Global Stochastic Elements

다국어 초록 (Multilingual Abstract)

The Collatz conjecture is the proposition that for any positive integer, repeated application of the rule "divide by 2 if even, multiply by 3 and add 1 if odd" will eventually reach 1. In this study, we analyzed the dynamical system generated by the Collatz map from a statistical physics perspective. This study observed that the binary representation of Collatz trajectories exhibits dynamical characteristics similar to sandpile models and analyzed them using a cluster-based approach. We defined clusters as consecutive sequences of 1s or 0s in binary representation and confirmed that the 3x operation functions through mechanisms of cluster collapse and coalescence/annihilation. In particular, the coalescence /annihilation of alternating patterns of 1s and 0s depends on the carry from the previous step, thereby acting as a quasi-random element determined by the global context. By analyzing trajectories of approximately 250,000 steps starting from seeds on the order of 10^10000, we confirmed that the cluster size distribution follows a geometric distribution. Previous studies have primarily presented probabilistic analyses showing that trailing zeros follow a geometric distribution; however, this study proposes a mechanism for the mixing of 1s and 0s from the perspective of dynamical equilibrium, through dynamics exhibiting behavior similar to sandpile models. This study does not aim to directly prove the Collatz conjecture but rather contributes to the understanding of Collatz dynamics by providing a structural mechanism underlying probabilistic observations. This suggests that the deterministic cellular automaton exhibits quasi-random behavior through global interactions, operating as a self-regulating mechanism analogous to self-organized criticality.

목차 (Table of Contents)

• 1. 서론 1
• 1.1 콜라츠 추측 1
• 1.2 콜라츠 추측에 대한 통계물리학적 접근에서의 기존 연구 1
• 1.3 세포자동자와 모래더미 모형 3
• 1.4 연구 동기 및 목적 5
• 2. 방법론 6
• 2.1 이진수 변환 및 CA 대응 6
• 2.2 클러스터의 정의 7
• 2.3 3x 연산의 클러스터 동역학 8
• 2.4 모래더미 모형과의 비교 10
• 2.5 +1 연산과 전역적 효과 11
• 3. 실험 결과 12
• 3.1 데이터 수집 12
• 3.2 클러스터 크기 분포 12
• 3.3 최대 클러스터 크기 16
• 3.4 동역학적 특성 18
• 4. 논의 18
• 4.1 기하분포의 메커니즘적 설명 18
• 4.2 모래더미 모형과의 비교 19
• 4.3 준-무작위성의 의미 21
• 4.4 음의 표류 21
• 5. 결론 21
• 참고문헌 23