Mixing of underdamped Langevin dynamics: from cut-off to Erying?Kramers formula = Underdamped 랑쥐뱅 동역학에서의 섞임 현상: 컷오프 현상과 Erying-Kramers 공식

다국어 초록 (Multilingual Abstract)

The underdamped Langevin dynamics is a stochastic model describing evolution of thermostated molecular dynamics. In this thesis, we discuss mixing behavior of the underdamped Langevin dynamics in the low temperature regime. We observe the cut-off phenonmenon when there is only one stable equilibrium, while observe the metastability when there are multiple stable equilibria. In one stable equilibrium case, we provide detailed analysis of the long-time behavior of the underdamped Langevin dynamics. We first provide a necessary condition guaranteeing that the zero-noise dynamical system converges to its unique attractor. We also observed that this condition is sharp for a large class of linear models. We then prove the so-called cut-off phenomenon in the small-noise regime under this condition. In double-well case, We derive the low-temperature asymptotic of its mean-transition time between basins of attraction. This asymptotic is called Eyring-Kramers law. The main difficulty of problem is the degeneracy of the generator associated with the underdamped Langevin dynamics.

국문 초록 (Abstract)

Underdamped 랑쥐뱅 동역학은 물리학, 화학등의 여러 파트에서 어떠한 형태의 장에 영향을 받는 작은 입자의 운동을 나타내는 것으로 많은 연구가 되어 왔다. 이 학위논문에서는 낮은 온도의 underdamped 랑쥐뱅 동역학에서의 입자의 형태를 고려한다. 실험적으로 낮은 온도에서 입자에 작용하는 힘의 안정점이 하나인 경우, cut-off 현상이 관찰되고, 안정점이 하나가 아닌 2개 이상일 경우에는, 해당 안장점에서 다른 안장점으로 입자가 움직이는데 매우 오래 걸리는 metastability현상이 관찰되어왔다. 본 학위 논문에서는 각각의 경우에 대하여 수학적으로 엄밀한 접근을 한다. 즉, 안장점이 하나인 경우에는 cut-off 현상이 발생함을 수학적으로 증명하고, 안장점이 2개인 경우에는 입자가 한 안장점에서 다른 안장점으로 가는데 걸리는 시간을 정확하게 추산한 공식인 Erying-Kramers formula를 증명하여 확인한다.

목차 (Table of Contents)

• Abstract i
• 1 Introduction 1
• 1.0.1 Underdamped Langevin dynamics 1
• 1.0.2 Thermalization of underdamped Langevin dynamics 2
• 1.0.3 Cut-off phenomenon 3
• 1.0.4 Cut-off phenomenon for underdamped Langevin dynamics 5
• 1.0.5 Metastability 7
• 1.0.6 Metastability of the overdamped Langevin process 8
• 1.0.7 Metastability of the underdamped Langevin process 10
• 2 Model and Main Result 12
• 2.0.1 Stability of underdamped Langevin dynamics 12
• 2.0.2 Notation 13
• 2.0.3 Asymptotic stability analysis of zero-noise dynamics 15
• 2.0.4 Thermalization of underdamped Langevin dynamics 18
• 2.0.5 Main result: Cut-off phenomenon 20
• 2.0.6 Main result: Eyring-Kramers law 22
• 2.0.7 Sketch of Proof of Eyring-Kramers law 25
• 3 Proof of Cut-off phenomenon 29
• 3.1 Linear Models 29
• 3.1.1 Elementary lemmata 29
• 3.1.2 Asymptotic stability of linear models 32
• 3.2 Lyapunov Function 35
• 3.2.1 Lyapunov function 35
• 3.2.2 Global exponential stability of zero-noise dynamics 37
• 3.2.3 Moment estimates 39
• 3.2.4 Uniqueness of stationary distribution 42
• 3.3 Approximation by a Gaussian Process 44
• 3.3.1 Heuristic explanation 44
• 3.3.2 Gaussian approximation of the underdamped Langevin dynamics 46
• 3.3.3 Matrix equations 47
• 3.3.4 Preliminary moment estimates on Y_t 49
• 3.3.5 Quadratic Gronwall inequality 53
• 3.3.6 Proof of Proposition 3.3.2 56
• 3.4 Covariance Matrix of the Gaussian approximation 62
• 3.4.1 Covariance matrix of Y_t 62
• 3.4.2 Long-time behavior of the covariance matrix 63
• 3.4.3 Small-time behavior of the covariance matrix 67
• 3.4.4 Total variation distance between Gaussian processes 69
• 3.5 Estimate of Total Variation Distance 72
• 3.5.1 Long-time behavior on zero-noise dynamics 72
• 3.5.2 Strategy to estimate total-variation distance 74
• 3.5.3 Proof of Proposition 3.5.3 76
• 3.5.4 Proof of Proposition 3.5.4 81
• 3.6 Proof of Theorem 2.0.8 82
• 4 Proof of Eyring-Kramers formula 86
• 4.1 Preliminary analysis 86
• 4.1.1 Tightness of invariant measure 86
• 4.1.2 Non-explosion and positive recurrence of the underdamped Langevin process 88
• 4.2 Proof of Proposition 2.0.21 90
• 4.2.1 Perturbed underdamped Langevin process 90
• 4.2.2 Distance between X_ε(t) and X_{α,ε}(t) 92
• 4.2.3 Approximation of h^*_{M_ε,S_ε} 94
• 4.2.4 Perturbed version of Proposition 2.0.21 97
• 4.2.5 Proof of Proposition 2.0.21 101
• 4.3 Proof of Lemma 4.2.7 105
• 4.3.1 Preliminary results 106
• 4.3.2 Neighborhood of ∂O_{N,ε} 109
• 4.3.3 Proof of Lemma 4.2.7 114
• 4.4 Proof of Proposition 2.0.22 119
• 4.5 Proof of Proposition 2.0.23 125
• 4.6 Transition density of the process absorbed at a boundary 129
• 4.7 Lyapunov function and zero-noise Langevin process 138
• 4.7.1 First exit time probability 138
• 4.7.2 Distance to the zero-noise underdamped Langevin process 142
• 4.8 Proof of Proposition 4.5.3 145
• 4.8.1 Preliminary results 145
• 4.8.2 Proof of Proposition 4.5.3 152
• 4.9 Proof of Proposition 2.0.24 156
• 4.9.1 Spectrum at the saddle point 157
• 4.9.2 Partitioning the phase space 158
• 4.9.3 Definition of the test function 161
• 4.9.4 Computation of C_ε(M_ε, S_ε) 165
• Abstract(in Korean) 182
• Acknowledgement (in Korean) 183