## The nature of the analog models that constitute network thermodynamics

The generality of Network Thermodynamics as modeling tool and theoretical formalism for all of physical systems theory is well established through the modeling relation. (Rosen [18,19]) The modeling relation can be used to define analog models. If the same Formal System is able to form commuting models for two or more Natural Systems, these systems are said to be analogs of each other and each could serve as a formal system for all the others. This is the case among physical systems with electrical networks being the representative Natural System. The fact that electrical networks were the first to be formalized extensively has made electrical network theory the source of models for a broader class of physical systems. To exemplify this analogy it is instructive to look at the constitutive relations in more detail.

3.5. The constitutive laws for all physical systems are analogous to the constitutive laws for electrical networks or can be constructed as the models for electronic elements

There are four possible binary relations among the network observables after the time integrals used to define charge and momentum are included (Oster, Perelson and Katchalsky , Peusner , Mikulecky ). Charge is the time integral of flow (or flow is a rate of change of charge) and the momentum is a time integral of effort. There are four distinct general network elements, each deriving their name from their electrical prototype.

• RESISTANCE relates effort to flow.

• CAPACITANCE relates charge to effort.

• INDUCTANCE relates momentum to flow.

• MEMRISTANCE relates charge to momentum.

Each of these network elements has its own unique interpretation with respect to how it handles energy. Foremost is the resistor, which is an idealization having the purpose to embody all the dissipation that goes on in a locality of the network element. Dissipation is the crux of irreversibility and the second law of thermodynamics. Systems in stationary states away from equilibrium are governed totally by resistance. Transient behavior comes from the time derivatives introduced by capacitance and inductance. In the Langrangian formulation of networks, it is the resistance that represents the non-conservative aspects of the system (Mikulecky, Weigand and Shiner ). Capacitance is a form of energy storage without dissipation and is, therefore, also an idealization. The capacitor is a good model of a reversible isolated system in its behavior as it approaches its equilibrium potential. Since there is always dissipation in real processes, reality requires that there also be a resistor somewhere in series with the capacitor in order for the mathematics to be a faithful description of the system's trajectory. It is the capacitors that provide the dynamics in most models. This is a bit counterintuitive, since it is in equilibrium thermodynamics that these reversible (dissipation-less) energy transfers arise. Inductors are the idealized iner-tial elements and occur mainly in mechanics. The isomorphism between the differential equations for harmonic oscillators and LRC circuits has often been made. In the case of a weight bobbing up and down on a spring fastened to a stationary object at its other end, the elastic spring is a capacitor, friction is the resistor, and the inertial force is the inductor. These seemingly simple analogical identifications link any physical system to the body of formal power resting in electrical network theory. This has been proven beyond a doubt to be a very significant formalism by its results, the myriad of achievements of modern electronics. This leaves the fourth element, the memristor. It also has analog physical realizations but they are rare (Chua ). So far, in all the applications encountered it has not been needed.