On variational principles for polarization responses in electromechanical systems

Abstract

Classical electrodynamics uses a dielectric constant to describe the polarization response of electromechanical systems to changes in an electric field. We generalize that description to include a wide variety of responses to changes in the electric field, as found in most systems and applications. Electromechanical systems can be found in many physical and biological applications, such as ion transport in membranes, batteries, and dielectric elastomers. We present a unified, thermodynamically consistent, variational framework for modeling electromechanical systems as they respond to changes in the electric field; that is to say, as they polarize. This framework is motivated and developed using the classical energetic variational approach (EnVarA). The coupling between the electric part and the chemo-mechanical parts of the system is described either by Lagrange multipliers or various energy relaxations.The classical polarization and its dielectrics and dielectric constants appear as outputs of this analysis. The Maxwell equations then become universal conservation laws of charge and current, conjoined to an electromechanical description of polarization Polarization describes the entire electromechanical response to changes in the electric field and can sometimes be approximated as a dielectric constant or dielectric dispersion.

Setting Boundaries for Statistical Mechanics

“…Polarization can be treated as the stress strain relation of charge (see Equations (3.1)-(3.5) of ref. [144]). In its variational form, the stress strain formalism accommodates diffusion and convection that are so important in liquids, yielding the classical Poisson Nernst Planck equations in special cases [53][54][55][56] important in applications ranging from ions in water solutions, ions in protein channels, to ions in gases [145] and plasmas [146][147][148], to holes and electrons that are the quasi-ions of the semiconductors of our computers and smartphones [149][150][151][152][153].…”

“…Maxwell equations-whether core Equations ( 2)-( 5) or classical Figure 2-themselves are not averaged. For example, averaging is usually found in the theories and simulations of polarization, e.g., it occurs in the stress strain theories of the distribution of charge and matter we have discussed [144]. Indeed, if polarization is described in its full complexity of time and field dependence [29,[100][101][102][103][104][105][106][121][122][123], the mathematical structure of the classical Maxwell equations changes.…”

Conformations and Currents Make the Nerve Signal

“…Maxwell's equations guarantee that the total current associated with the movements of these conformation charges is identical to the current flow in the surrounding baths and in the electrodes 3 connected to the baths, independent of any property of matter or polarization whatsoever ( [38][39][40][74][75][76][77][78], crystallized in [79]) because the channel is naturally a one dimensional system [38][39][40] The baths can be made one dimensional conductors with little trouble [62,63] as has been demonstrated by direct measurement [39,40,80].…”

“…This property of Maxwell's equations arises immediately when Ampere's law includes the 'ethereal' current 𝜀 0 𝜕𝐄/𝜕𝑡 that allows light to propagate in a vacuum devoid of charges or matter, once the law is written without mention of polarization, dielectrics or dielectric constants[38][39][40][74][75][76][77][78], crystallized in[79].…”

On a deterministic particle-FEM discretization to micro-macro models of dilute polymeric fluids

“…6,58 This approach was developed from pioneering works of Rayleigh 62 and Onsager, 57,58 and has been successfully applied to build up many mathematical models in physics, chemical engineering and biology. 26,48,70,71 From a numerical perspective, the EnVarA formulation also provides a guide line to develop structure-preserving numerical schemes for different systems. [49][50][51] For an isothermal and closed system, an energy-dissipation law is often given by…”