2020/05/04 by Andreas Bittracher, Christof Schütte, Bittracher, Andreas +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Protein Structure and Dynamics #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2005.01631
openalex publication_date 2020/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a novel characterization of slow variables for continuous Markov\nprocesses that provably preserve the slow timescales. These slow variables are\nknown as reaction coordinates in molecular dynamical applications, where they\nplay a key role in system analysis and coarse graining. The defining\ncharacteristics of these slow variables is that they parametrize a so-called\ntransition manifold, a low-dimensional manifold in a certain density function\nspace that emerges with progressive equilibration of the system's fast\nvariables. The existence of said manifold was previously predicted for certain\nclasses of metastable and slow-fast systems. However, in the original work, the\nexistence of the manifold hinges on the pointwise convergence of the system's\ntransition density functions towards it. We show in this work that a\nconvergence in average with respect to the system's stationary measure is\nsufficient to yield reaction coordinates with the same key qualities. This\nallows one to accurately predict the timescale preservation in systems where\nthe old theory is not applicable or would give overly pessimistic results.\nMoreover, the new characterization is still constructive, in that it allows for\nthe algorithmic identification of a good slow variable. The improved\ncharacterization, the error prediction and the variable construction are\ndemonstrated by a small metastable system.\n