HomeSciencePhysicsMeasuring Entropy in Active-Matter Systems

# Measuring Entropy in Active-Matter Systems

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A software for estimating the native entropy manufacturing fee of a system permits the visualization and quantification of the out-of-equilibrium areas of an active-matter system.

A film of a molecule jostling round in a fluid at equilibrium seems to be the identical when performed ahead and backward. Such a film has an “entropy production rate”—the parameter used to quantify this symmetry—of zero; most different motion pictures have a nonzero worth, which means the visualized programs are out of equilibrium. Researchers know methods to compute the entropy manufacturing fee of straightforward mannequin programs. But measuring this parameter in experiments is an open downside. Now Sungham Ro of the Technion-Israel Institute of Technology, Buming Guo of New York University, and colleagues have devised a way for making native measurements of the entropy manufacturing fee [1]. They exhibit the method utilizing simulations and micro organism observations (Fig. 1). The technique, which entails evaluating ahead and time-reversed particle trajectories, may enable researchers to disclose the affect of native entropy manufacturing charges on the worldwide dynamics of complicated residing programs from micro organism to tissues.

Active-matter programs are these composed of energy-consuming brokers that exhibit complicated collective conduct [24]. The classical examples of such brokers are birds and fish, which may respectively type flocks that swoop and faculties that swirl once they work together in giant numbers. Scientists know that these programs break time-reversal symmetry on the single-agent degree due to their fixed power consumption [5], and thus active-matter programs have nonzero entropy manufacturing charges on the native scale [6].

Scientists additionally know that active-matter programs break time-reversal symmetry on the worldwide scale, the place the brokers are noticed to collectively manage into patterns—like flocks or faculties—whose size scales are a lot bigger than the brokers themselves. We don’t, nonetheless, know the connection between the worldwide and native breaking of time-reversal symmetry. And merely realizing that the worldwide system has a nonzero entropy manufacturing doesn’t inform us a lot concerning the system’s conduct. For occasion, it tells us how far the system as a complete is from equilibrium, however not the place the system is out of equilibrium. Also, we stay in the dead of night about variations within the system’s thermodynamic properties on intermediate scales. Knowing this info may enable scientists to hyperlink the native dynamics of an active-matter system to its world sample formation, permitting them to make use of one to foretell the opposite.

To acquire this info, scientists have advised computing native entropy manufacturing charges utilizing field-theoretical approaches, however the concept has solely been examined for just a few particular fashions [79]. Capturing the entropy manufacturing fee this fashion entails amassing “stationary” trajectories (film frames) from the ahead film after which evaluating these frames to ones from the time-reversed film. This comparability ought to enable us to quantify areas of the system the place nonequilibrium conduct is probably the most marked. However, precisely quantifying the distinction between a stationary trajectory and a time-reversed one is difficult [10]. The work of Ro, Guo, and colleagues takes on this problem.

For the research, the group considers a generic active-matter system consisting of self-propelled particles that transfer randomly in area, performing a “persistent” random stroll—a selected form of random stroll the place particles transfer in roughly straight traces. They discretize the system on a grid, associating every level of the grid with an integer quantity that takes on one in every of a finite set of values, relying on whether or not there’s a particle at that location (Fig. 2). The worth assigned to every level varies in time, giving it a selected time sequence. For the time-reversed film, the group takes every of those sequences and performs it in reverse: the final worth turns into the primary. The mannequin takes the sequences within the two motion pictures and compares them utilizing a parameter referred to as the cross-parsing size, which is utilized in info idea to quantify the variety of shared patterns in two sequences [11]. The extra shared patterns within the ahead and backward motion pictures, the extra symmetrical they’re in time and the decrease the entropy manufacturing fee. Thus, the cross-parsing size is linked to the native entropy manufacturing fee.

To benchmark their technique, Ro, Guo, and colleagues utilized it, for instance, to numerical simulations of lively Brownian particles present process so-called motility induced part separation—the place particles transfer slower in denser clusters, making a system with dense and dilute areas—and to experiments with E. coli micro organism which can be directed into a selected area through a funnel. For the Brownian particle simulations, they discovered that the entropy manufacturing fee is highest on the boundaries between the dense and dilute areas and lowest of their facilities. For the E. coli experiments, they discovered that the entropy manufacturing is highest across the funnels.

The computational protocol launched by Ro, Guo, and their colleagues will enable researchers to begin exploring in additional element the nonequilibrium properties of each pure and fabricated active-matter programs. In precept, that would result in research of how nonequilibrium options unfold in residing programs and the way these options connect with the patterns the programs type. For instance, it may enable scientists to create nonequilibrium “atlases” for cells and to higher perceive the collective conduct of residing organisms that haven’t any equilibrium counterpart.

## References

1. S. Ro et al., “Model-free measurement of local entropy production and extractable work in active matter,” Phys. Rev. Lett. 129, 220601 (2022).
2. M. C. Marchetti et al., “Hydrodynamics of soft active matter,” Rev. Mod. Phys. 85, 1143 (2013).
3. C. Bechinger et al., “Active particles in complex and crowded environments,” Rev. Mod. Phys. 88, 045006 (2016).
4. M. J. Bowick et al., “Symmetry, thermodynamics, and topology in active matter,” Phys. Rev. X 12, 010501 (2022).
5. E. Fodor et al., “Irreversibility and biased ensembles in active matter: Insights from stochastic thermodynamics,” Annu. Rev. Condens. Matter Phys. 13, 215 (2022).
6. E. Fodor et al., “How far from equilibrium is active matter?” Phys. Rev. Lett. 117, 038103 (2016).
7. C. Nardini et al., “Entropy production in field theories without time-reversal symmetry: Quantifying the non-equilibrium character of active matter,” Phys. Rev. X 7, 021007 (2017).
8. F. Caballero and M. E. Cates, “Stealth entropy production in active field theories near Ising critical points,” Phys. Rev. Lett. 124, 240604 (2020).
9. M. Paoluzzi, “Scaling of the entropy production rate in a
${𝜙}^{4}$

model of active matter,” Phys. Rev. E 105, 044139 (2022).

10. E. Roldán and J. M. R. Parrondo, “Entropy production and Kullback-Leibler divergence between stationary trajectories of discrete systems,” Phys. Rev. E 85, 031129 (2012).
11. J. Ziv and N. Merhav, “A measure of relative entropy between individual sequences with application to universal classification,” IEEE Trans. Inform. Theory 39, 1270 (1993).

Matteo Paoluzzi is an affiliate researcher within the condensed matter physics division of the University of Barcelona, Spain. Paoluzzi acquired his PhD at Roma Tre University, Italy, and had postdoctoral positions at Syracuse University, New York, and Sapienza University of Rome. His analysis exercise is concentrated on understanding collective phenomena in equilibrium and nonequilibrium programs starting from glasses to active-matter supplies. He is presently exploring the affect of lively movement on the glass transition and the universality class of scalar lively matter.

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