What question did this study set out to answer?

To present a model-independent method for diagnosing broken time-reversal symmetry using finite trajectory data.

February 12, 20260 citationsOpen Access

Equivariant Window Embeddings: A Symmetry-Based Diagnostic for Microscopic Reversibility

Key Points

To present a model-independent method for diagnosing broken time-reversal symmetry using finite trajectory data.
Introduce Equivariant Window Embeddings (EWE) for feature mapping of trajectories.
Develop a symmetry-based observable G that is odd under time reversal.
Analyze the expectation of G in the context of detailed balance.
Demonstrate that nonzero empirical means of G indicate irreversibility.
Establish that E[G(Φ(ω))] must vanish under detailed balance, providing a new diagnostic.
Show that EWE can reveal nonzero entropy production without needing fitting models.

Abstract

This upload contains an 8-page note introducing Equivariant Window Embeddings (EWE): a model-independent, symmetry-based diagnostic for detecting broken time-reversal symmetry / failure of detailed balance from finite trajectory data. The method maps two-sided trajectory windows into a feature space via an equivariant embedding ΦΦ that commutes with time reversal, Φ (Rω) =JΦ (ω) (R) =J () Φ (Rω) =JΦ (ω), and evaluates an observable GGG that is odd under the induced involution, G (Jv) =−G (v) G (Jv) =-G (v) G (Jv) =−G (v). The core result (Theorem 3. 1) shows that under detailed balance the expectation EG (Φ (ω) ) EG ( () ) EG (Φ (ω) ) must vanish; therefore a statistically significant nonzero empirical mean provides a certificate of irreversibility and (in physical settings) nonzero entropy production without fitting parametric models or estimating forward–reverse likelihood ratios.

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