We can, in principle, also make connections to both longer or shorter timescales. We demonstrate our analysis in both stochastic and deterministic systems, and with an application to the movement dynamics of an entire organism, unravelling new insight in long time scale behavioral states directly from measurements of posture dynamics. These operators play a central role, guiding the construction of maximally-predictive short-time states from incomplete measurements and identifying collective modes via eigenvalue decomposition. The statistics of these transitions are captured compactly by transfer operators. Just as a musical piece transitions from one movement to the next, the collective dynamics we infer consists of transitions between macroscopic states, like jumps between metastable states in an effective potential landscape. We introduce a principled framework for learning effective descriptions directly from observations.
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The challenge we address is how to identify collective variables which distinguish structures across such disparate time scales. In musical composition, e.g., sounds and silences combine to form longer time scale structures motifs form passages which in turn form movements. POPULAR SUMMARY Complex structure is often composed from a limited set of relatively simple building blocks such as novels from letters or proteins from amino acids. We demonstrate how sequences simulated from the ensemble evolution capture both fine scale posture dynamics and large scale effective diffusion in the worm’s centroid trajectories and introduce a top-down, operator-based clustering which reveals subtle subdivisions of the “run” behavior. elegans, we derive a “run-and-pirouette” navigation strategy directly from posture dynamics. Applied to the behavior of the nematode worm C. Applicable to both deterministic and stochastic systems, we illustrate our approach through the Langevin dynamics of a particle in a double-well potential and the Lorenz system. The evolution is parameterized by a transition time τ : capturing the source entropy rate at small τ and revealing timescale separation with collective, coherent states through the operator spectrum at larger τ. Trading non-linear trajectories for linear, ensemble evolution, we analyze reconstructed dynamics through transfer operators. We reconstruct a state space by concatenating measurements in time, building a maximum entropy partition of the resulting sequences, and choosing the sequence length to maximize predictive information.
We leverage the interplay between microscopic variability and macroscopic order to connect physical descriptions across scales directly from data, without underlying equations.
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