Dissipation at limited resolutions: power law and detection of hidden dissipative scales

Abstract Non-equilibrium systems, in particular, living organisms, are maintained by irreversible transformations of energy that drive diverse functions. Quantifying their irreversibility, as measured by energy dissipation, is essential for understanding the underlying mechanisms. However, existing techniques usually overlook experimental limitations, either by assuming full information or by employing a coarse-graining method that requires knowledge of the structure behind hidden degrees of freedom. Here, we study the inference of dissipation from finite-resolution measurements by employing a recently developed model-free estimator that considers both the sequence of coarse-grained transitions and the waiting time distributions: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>σ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mn>2</mml:mn> <mml:mi>ℓ</mml:mi> </mml:msubsup> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mn>2</mml:mn> <mml:mi>t</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> . The dominant term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mn>2</mml:mn> <mml:mi>ℓ</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> originates from the sequence of observed transitions. We find that it scales with resolution following a power law. Comparing the scaling exponent with a previous estimator highlights the importance of accounting for flux correlations at lower resolutions. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mn>2</mml:mn> <mml:mi>t</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> comes from asymmetries in waiting time distributions. It is non-monotonic in resolution, with its peak position revealing characteristic scales of the underlying dissipative process, consistent with observations in the actomyosin cortex of starfish oocytes. Alternatively, the characteristic scale can be detected in a crossover of the scaling of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mn>2</mml:mn> <mml:mi>ℓ</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> . This provides a novel perspective for extracting otherwise hidden characteristic dissipative scales directly from dissipation measurements. We illustrate these results in biochemical models as well as complex networks. Overall, this study highlights the significance of resolution considerations in non-equilibrium systems, providing insight into the interplay between experimental resolution, entropy production and underlying complexity.