The non-equilibrium regions of an active matter system can be seen and measured using a technique to calculate the local entropy production rate of a system. When played back and forth, the film of a molecule writhing in a liquid in equilibrium looks the same. The metric used to measure this symmetry, called the "entropy generation rate," is zero in such a movie; non-zero in most of the other films, indicating that the systems depicted are out of balance.
Researchers can calculate the entropy generation rate of simple model systems. However, measuring this parameter in trials is still a challenge.
Sungham Ro of the Technion-Israel Institute of Technology, Buming Guo of New York University and other researchers have now developed a technique for determining the local entropy production rate. They use simulations and microbiological observations to demonstrate the technique. Researchers can use the technique comparing forward and time-reversed particle trajectories to determine how local entropy production rates affect the overall dynamics of complex living systems such as bacteria and tissues.
Systems composed of energy-consuming agents that exhibit complex group behaviors are known as active substance systems.
Two traditional examples of these agents are fish and birds, which when interacting in large numbers can form spinning flocks and swirling flocks, respectively. Active matter systems exhibit nonzero local entropy production rates because these systems are known to violate time-reverse symmetry at the single-agent level due to their sustained energy consumption.
In addition, active substance systems are known to violate time-reverse symmetry globally.
Items appear to be collectively arranged in swarm or school-like patterns with much larger length scales than the items themselves.
However, we are unaware of the relationship between global and local time-reverse symmetry breaks. Moreover, just knowing that the global system has a non-zero entropy production does not provide us much information about the behavior of the system. For example, it tells us about the overall distance of the system from equilibrium, but not the exact position of the unbalance state. We also do not know how the thermodynamic properties of the system change at intermediate scales.
By linking the local dynamics of an active substance system with global model development, scientists can use one to predict the other.
Scientists have proposed calculating local entropy production rates using field theoretical methods to obtain this information, but this suggestion has only been validated for a few specific models. By collecting "stationary" trajectories from the forward film and comparing these frames with those in the time-reversed film, it is possible to measure the rate of entropy creation in this way.
Thanks to this comparison, we should be able to measure the parts of the system where non-equilibrium behavior is most evident. However, it is difficult to precisely measure the difference between a stationary trajectory and a time-reversed trajectory. This issue is being addressed by the work of Ro, Guo and colleagues.
For the study, the group considers a general active matter system made up of self-propelled particles that move arbitrarily through space while participating in "permanent" random walk, a special type of random walk in which particles essentially move in roughly straight lines. The system is discretized on a grid, and each grid point is assigned an integer that can have one of a limited number of values depending on whether a particle is present or not.
Each point has a unique time sequence due to the variable value assigned to it in time. Each of these sequences is used by the team for the time-reversed movie, which is played backwards so that the last value is the first value.
The cross-decomposition length, a parameter used in information theory to measure the number of shared patterns in two sequences, is used in the model to compare sequences in two films. Forward and backward films are more symmetrical in time and produce less entropy when there are more shared patterns between them. As a result, the cross-decomposition length is related to the rate of local entropy creation.
Ro, Guo, and colleagues used numerical simulations of active Brownian particles, called motion-induced phase separation, as the basis for their methodology.
This leads to systems with dense and dilute regions and studies using E. coli microbes directed to a specific area; because particles move slower in denser clusters.
For the Brownian particle simulations, they discovered that the entropy generation rate was highest at the boundaries between the dense and dilute regions and lowest at their centres. In their E. coli trials, they discovered that entropy production was highest around the funnels.
The calculation protocol by Ro, Guo and colleagues will enable researchers to begin to delve deeper into the non-equilibrium properties of both real and fabricated active material systems. Theoretically, this could lead to research on the distribution of non-equilibrium traits in living systems and their relationship to the patterns produced by these traits. For example, it could lead to the development of unstable "atlases" for cells and a better understanding of the collective behavior of living things that have no counterpart in equilibrium.