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Under which circumstances are variational principles based on entropy production rate useful tools for modeling steady states of electric (gas) discharge systems far from equilibrium? It is first shown how various different approaches, as Steenbeck’s minimum voltage and Prigogine’s minimum entropy production rate principles are related to the maximum entropy production rate principle (MEPP). Secondly, three typical examples are discussed, which provide a certain insight in the structure of the models that are candidates for MEPP application. It is then thirdly argued that MEPP, although not being an exact physical law, may provide reasonable

Electric (gas) discharges appear in various natural phenomena like lightning, St. Elmo’s fire, and spark discharges. Moreover, electric discharges are exploited in various technical devices as lamps, circuit breakers, and plasma torches, to mention a few. Despite of the seniority of the “science of discharges” and the today’s increased computational power, the complexity of the involved physical phenomena like radiative transfer, plasma flow (sometimes supersonic and/or turbulent), electrical contact physics,

This article addresses aspects of the pragmatic question, under which circumstances entropy production principles can be helpful for modeling electric discharge phenomena far from equilibrium. “Modeling” will here be associated with relatively simple phenomenological models that provide, for the physical quantities of interest, quantitative predictions, which are probably not exact but serve as estimates with sufficient accuracy for practical purposes. Exhaustive review articles on variational principles for the entropy production rate and similar principles can be found in [

An initial nonequilibrium state is then prepared by charging a battery with energy

When a discharge starts, a constant current

The isolated nonequilibrium system, consisting of a part

For simplicity, we suppose that the “device” Ω is only electrically and thermally coupled to its environment, but does not exchange mass or momentum. This device is described by a phenomenological model

MEPP refers to maximization of

In the former case, MEPP requires that

On the other hand, MEPP is also satisfied for a state that

There are roughly three classes of questions addressed by modeling electric discharge structures: (1) What happens at boundaries and electrodes? (2) What are the values of the variables characterizing the discharge structure? (3) What is the behavior of the discharge structure as a whole in its specific environment? In the following, we illustrate with simple textbook examples how MEPP is answering these questions. In particular, we discuss (1) unipolar charge injection from an electrode, (2) a simple column model for an electric arc, and (3) arc root attachment at the anode in a plasma torch. These examples are not new, but they may give hints for which types of model MEPP can be useful.

Consider a medium, without intrinsic charge carriers, between two parallel metal electrodes separated by a distance

For unipolar injection from an electrode at potential

The integration constant

Space charge influenced electric field (solid), space charge density (dashed;

High current electric arc columns are among the most frequent applications of Steenbeck’s principle [

By elimination of

(a) Air conductivity at

Plasma torches are widely used for plasma spraying, cutting and welding. A sketch of a plasma torch is shown in

One may, alternatively, apply MEPP, e.g., in the form of Steenbeck’s principle by minimizing the voltage drop along the arc [

the arc state and the gas flow properties. Because these are approximately constant upon variation of

Plasma arc torch as described in the text.

These three examples show how values for a priori unknown model parameters can be obtained with Steenbeck’s principle and thus with MEPP. All of them refer to systems with nonlinear current voltage behavior (the arc can even have negative differential resistance, depending on the value of

Entropy production rate principles for steady states are generally valid only near equilibrium. Let us first comment on MEPP for this case. According to

A proof for the general validity of a variational principle beyond linear response does not exist [

The task of MEPP is to determine the values of unknown model parameters,

An often heard statement, tempting for using MEPP, is that

Let us illustrate what “quality of information” means in practice for the examples discussed in the previous section. Assuming local thermal equilibrium and a single component fluid description of the arc plasma with known conductivity, the complete information consists in the three hydrodynamic balance equations (together with all initial conditions, boundary conditions, and Ohm’s law). For the arc model above, we used information on the structure of the arc,

Similarly, the charge injected in the first example is a “weak” mode, because the electrode is able to provide whatever charge is needed. For instance, ohmic contact behavior is exactly reproduced by MEPP in this case. But when the steady state net current becomes of the order of the injection current of the contact, contact physics starts to play a role. In that case, relevant information (

Also the arc attachment location

These experiences suggest the conjecture that the applicability of MEPP requires the presence of such a “weak” mode. Although, unfortunately, “weakness” is not clearly defined, it is correlated to the irrelevance of information. In other words, there are many weak, different physical effects of low relevance that would have to be taken into account in the missing equations for

A different viewpoint on the interpretation of the occasional success of MEPP is based on work by Kohler [

These observations suggest as a second conjecture, that good results obtained from MEPP for seemingly far from equilibrium systems might rely on a hidden linearity of the underlying Boltzmann transport equation for the concerned quantities.

I would like to thank Frank Kassubek for many critical comments and careful reading of the manuscript.

Although not fully correct in certain respects, we will generally identify Prigogine’s principle with the principle of minimum entropy production rate.

_{6}gas viewed from sequential generation of a dc partial discharge

We give a simple example that shows graphically, that the difference between minimum and maximum entropy production rate principles is related to the convexity properties of the optimization problem. Consider in

Illustration of the type of optimum of the solution (black dot). Solid curves: contour lines of