## Tellegens theorem and the onsager reciprocal relations ORR

Lars Onsager [130,131] won the Nobel Prize a while back in part for his work on non-equilibrium thermodynamics and in particular, his proof of the "reciprocal relations". His proof used statistical physics in the domain of fluctuations around equilibrium. The decay of those fluctuations was described by the linear phenomenological laws and the use of statistical physics was thought necessary to obtain the proof. Peusner [11, 12] was able to accomplish the proof much more simply, accurately and directly using Tellegen's Theorem which is a result at the most basic level in network thermodynamics. He systematically matched elements of his proof with that of Onsager to show that all the molecular statistics could have been ignored since the necessary and sufficient elements of the proof are topological properties that Onsager had implicitly assumed and which were in no way dependent on molecular statistics.

Tellegen's theorem is, in its simplest form, a statement of power conservation in a complete network. To be complete, the network must have energizing sources or charged capacitors to make it work. The simplest possible example is the series circuit containing a single resistor and a constant source of current or voltage. Tellegen's Theorem is then

For any network the vector of flows and the vector of efforts (forces) are orthogonal:

In other words, the power dissipated by the resistor and the power supplied by the source are equal and by using the appropriate sign convention of opposite sign so they sum to zero. This looks trivial, but it has a very general form for any network that can be derived using Kirchhoff's laws and the network topology, independent of the identity of the circuit elements. For that reason, the quasi power theorem is easily proved as well.

In this version there are two different networks (starred and unstarred) with exactly the same topology. If we take a vector of flows from one and a vector of forces (efforts) from the other, these vectors will always be orthogonal!

These can either be two different networks or the same network at different times. Using this Onsager's reciprocity theorem can be proven without the use of statistical thermodynamics or near equilibrium assumptions other than that the system is still in the linear domain.

It had been know by experimentalists for about a century, that the reciprocal relations (in particular Saxen's relations (Miller [138, 139]) held in non-equilibrium situations far enough from the domain of Onsager's proof (fluctuations around equilibrium) to make his proof seem odd, at best. Peusner's topological proof not only did not need or use statistical physics, but also was valid everywhere in the linear domain far beyond the domain of Onsager's attempt. Thus it was in complete harmony with the myriad of experimental results.

Furthermore there is an intimate link between the ORR and the ability to have a simple, topologically connected network element representing any linear non-equilibrium (Onsager) system. In fact this is the canonical representation that yields the metric in entropy or energy space. This is but one more demonstration that topological (non-mechanistic) considerations underlie many fundamental relations in these systems.