Finally published is my one of my first papers, which I wrote in 2011 at a time where I was not knowledgable on the strategies of scientific publishing. I naïvely believed that, being a crucial piece of science, it would be accepted welcomingly. I submitted it to a much-too-high-impact-factor journal where it was refused without even going through the peer-review process. My delusion was such that I dismissed it. Four years later I picked it up, did some minor modifications and proposed it as a conference proceeding:

• MP, *System/environment duality of nonequilibrium network observables*, in

Mathematical Technology of Networks, Springer Proceedings in Mathematics & Statistics Volume **128**, ed. Delio Mugnolo (Springer, 2015), pp 191-205, arXiv:1106.1280

The idea is the following. A system is said to be in a **nonequilibrium steady state** when its state does not change in time, but still there is internal motion. Take for example convective cells of water in a pan where a temperature difference is kept between the top and the bottom surfaces. Then, the steadiness condition implies that the working substance within the system, water in this case, must flow in **cycles** and there is no net flow, meaning that if one ideally **cuts **the pan in two, there will be no flow from one side to the other. On a network, this is a manifestation of **Kirchhoff’s Current Law**.

Now let’s focus on the environment. As the water flows in cycles, it picks up heat from the hotter reservoir and it gives it away to the colder one, hence creating a net flux of heat across the environment (a manifestation of **Kirchhoff’s Loop Law**). This directed flux does not contain cycles.

So, there is a **dual** behavior of system and environment:** cyclic flow in the system implies directed flow through the environment**, and vice versa, so that in a way **the system is the environment of the environment**.

In fact, this duality can be formulated in rigorous terms, at least when such flows occur on a network. The mathematical technology is that of graphs, their cycle and cut vector spaces, and the well-known notion of a dual of a (planar) graph (from the Wikipedia page):

Although I found this idea quite fascinating, I didn’t advance afterwards, partly because it has a static character that does not adapt well to thermodynamics of master equation systems which is my main interest.

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