Research Article: Oscillatory dynamics of an electrically driven dissipative structure

Date Published: May 29, 2019

Publisher: Public Library of Science

Author(s): Benjamin De Bari, James A. Dixon, Bruce A. Kay, Dilip Kondepudi, Dante R. Chialvo.


Physical systems open to a flow of energy can exhibit spontaneous symmetry breaking and self-organization. These nonequilibrium self-organized systems are known as dissipative structures. We study the oscillatory mode of an electrically driven dissipative structure. Our system consists of aluminum beads in shallow oil, which, when subjected to a high voltage, self-organize into connected ‘tree’ structures. The tree structures serve as pathways for the conduction of charge to ground. This system shows a variety of spatio-temporal behaviors, such as oscillating movement of the tree structures. Utilizing a dynamical systems model of the electromagnetic phenomena, we explore a potential mechanism underlying the system’s behavior and use the model to make additional empirical predictions. The model reproduces the oscillatory behavior observed in the real system, and the behavior of the real system is consistent with predictions from the model under various constraints. From the empirical results and the mathematical model, we observe a tendency for the system to select modes of behavior with increased dissipation, or higher rates of entropy production, in accord with the proposed Maximum Entropy Production (MEP) Principle.

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Developments in non-equilibrium thermodynamics have yielded an account of how physical structures spontaneously emerge from flows of energy and matter [1,2] in non-living systems. It has long been recognized that biological systems are a subset of this larger class of self-organized, nonequilibrium systems [3,4,5]. This account has clear implications for our understanding of morphological structures in biotic systems. However, biological systems also have a functional aspect; they behave in ways that allow them to maintain their own existence. Typically, function is assumed to be a higher-order property of living systems, dependent upon the pre-existence of a variety of supporting structures. We have presented evidence that dissipative structures can be end-directed in a way that is functional [6,7,8]. Specifically, dissipative structures are end-directed towards states that generate higher rates of entropy production. (In the literature the “rate of entropy production” is also often referred to simply as “entropy production”; we shall use both terms interchangeably). This end-directedness is functional for dissipative structures, because increasing the rate of entropy production in these systems seems to increase the stability of the structure, thus allowing it to persist over longer times and stronger perturbations. Our work shows that both the morphology and behavior of dissipative structures will change to increase the rate of entropy production, thus increasing the structure’s own probability of existence. In this way, maximizing the rate of entropy production appears to provide a fundamental form of functionality for dissipative structures.

In order to investigate the nature of the tree’s motions, a minimal case of the tree oscillation was observed wherein we restricted the location of the base bead of the tree (i.e. the bead in contact with the ring electrode), such that it could only move minimally on the ring (Fig 2). This fixes the point of rotation of the tree and allows us to reduce the oscillations to one dimension. A full account of the tree motion would be complex and involve the fluid dynamics of the oil, but by restricting the motion of the tree, some of the fluid-driven motion may be reduced and the charge-driven motion may be studied. The setup is similar to that in Fig 1, with the addition of plastic insulators used to gate the tree (Fig 2). To capture the behavior of the tree, we focus primarily on the terminal bead, which swings back and forth in approximately a one-dimensional trajectory. Under constant voltage, the tree is observed to settle into steady oscillations, over trials as long as eight hours. Given the importance of the current to the systems state-selection, the purpose of this first experiment was to quantify the relationship between bead position and current.

Experiment 1 was concerned primarily with the relationship between the current and the oscillations. Fig 5 shows the data from 300s of a single trial. The blue curve is the normalized displacement of the terminal bead from the source over time, and the orange curve is the normalized current through the system. The minima of the bead displacement curve correspond to the bead being minimally displaced from the source. The maxima correspond to the ends of the trajectory. A peak-picking function finds the maxima and corresponding time indices of each time-series (filtered with a net fourth-order Butterworth low-pass filter), which are then used to calculate the relative phases of the peak events. The average relative phase (current peaks in cycles of position) of the two signals is -2.7921 radians (SD = 0.7881 radians). The two signals are nearly perfectly out of phase; i.e. the current is maximal while the bead is minimally displaced from the source.

Experiment 1 found that the cycles of the bead’s displacement and the current were nearly perfectly out of phase, meaning the current was maximal while the bead is minimally displaced from the source. These results are sensible; we ought to expect that the current is highest while the bead is nearest to the source, a region of greater charge-density. We hypothesized that the motion was driven by the build-up and depletion of charge on the surface of the oil; this mutual constraint of the charge-distribution and the bead is the core feature of the CDM. The bead in the CDM also readily oscillates. That the CDM generates oscillations similar to what is observed in the E-SOFI is a compelling result, showing that the forces on the bead as prescribed in the model capture important variables in the E-SOFI. Further the CDM data corroborate the relative phase results from the E-SOFI; we find that the current is consistently nearly out of phase with the bead reaching its maximum displacement from the source, similar to what was observed in the E-SOFI.

The initial results of the relation between the current and the bead’s position, as well as the experimental confirmation of predictions generated from the CDM, suggest that the model is accurately representing some important features of the real system. The fundamental coupling between the position of the bead and the charge-distribution is well-motivated. Further development of the model should be done to attempt to capture more behaviors observed in the E-SOFI, such as results demonstrating its ability to traverse regions of low concentration of charge in order to reach regions of higher concentration of charge [7]. To improve the accuracy and scope of the model, its spatial-dependent properties should be elaborated, especially adding a second dimension. These and other developments of the CDM will enrich understanding of the behaviors of the E-SOFI system, and may have further implications for developing a principle of “maximum entropy production” or more precisely “maximum rate of entropy production” (MEP). As a formal principle, MEP has been proposed in many, sometimes seemingly contradictory forms [14, 15], but it is not a universally valid principle, applying only to a particular class of systems [14, 15, 16, 17].




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