I did not find much on 'period three' googling, except the theorem of Li & Yorke that any one-dimensional system which exhibits a regular cycle of period three will also display regular cycles of every other length as well as completely chaotic cycles. Though I am a bit confused how this theorem applies to this example.
the idea is that you don't want to live in a static regime, but you don't want to live in turbulence either. You want to have as much structure as possible, but (to drop some more jargon again) you want to be this side of "period three" in the period-doubling cascade
das monde:
On the other hand, is waste necessarily synonymous to entropy increase? In the simple head conduction example, energy flow could be potentially useful if organized in some way, but that waste is not harmful either, since there is not much organization to break. Are we wasting Sun's energy by letting it escape to the space?
I agree that energy flow could be wasted without any work. Are there any theorems how much work (or complexity, or information) can be maximally created from a given energy and waste budget?
Entropy is heat divided by temperature. Entropy never decreases. If heat flows from a hot source to a cold sink, the heat received by the cold sink is some fraction (between Tcold/Thot and 1: call this fraction x) of the heat given out by the hot source. The entropy received by the cold sink is the fraction 1 ≤ x Thot/Tcold ≤ Thot/Tcold of the entropy given out by the hot source. The more work is extracted the less entropy is created. It is possible (theoretically) to extract maximal work with zero entropy production. Spontaneously, no work is done and maximal entropy is generated.
Normally the efficiency of (ideal) heat engines is quoted as 1 - Tcold/Thot, which for real engines is 1 - x (with the definition of x I gave above). I think it is instructive to consider also the ratio of work to unnecessary heat produced. That would be (1 - x):(x - Tcold/Thot) but we know that people like to reduce things to a single number.
I hope this goes towards answering
Firstly, I am wondering, how necessary is dissipation waste given a constant energy source? Could a system, in principle, recycle anything it produces within finite time, without local entropy increase, so that overall entropy increase would be accounted at the energy source alone? Or does any 'work' requires local dissipation?
Does Carnot's theorem mean that the fraction x of the energy must dissipate? What is the "cold sink" in Earth's biosphere, or in the global economy? What is its "temperature"?
Spontaneously, no work is done and maximal entropy is generated.
How fast are simple undisturbed flows transformed into entropy? There seem to be examples of non-dissipating flows, until they reach dissipating objects.
If you like, you can take the "hot source" for the earth as the Sun's photosphere and the "cold sink" the average temperature of the Earth (more precisely, the temperature of the heat radiated to space by the earth). There is little prospect of changing those temperatures or the amount of energy flowing, so the parameters of the system are fixed. We get to tweak x. Note also that the energy used by the Earth's systems to provide "ecosystem services" is part of the "dissipation". It might be okay to replace the natural cycles with more efficient artificial cycles providing the same services, but disturbing the "services" to divert energy to making "stuff" is a Bad Idea™
In the case of the Earth we don't get to replace cork with copper, but we may be able to rearrange the cork creating denser areas and air pockets and moving parts in such a way as to extract some useful work from the heat flow. When the capital development of a country becomes a by-product of the activities of a casino, the job is likely to be ill-done. — John M. Keynes
Carnot's theorem means that the fraction Tcold/Thot of the energy wasn't convertible into useful work to begin with. It doesn't "dissipate" in that it is inevitable that it will flow out of the system into the "cold sink". I understand "dissipation" as entropy producion inside the system, not across its boundaries. In that sense, x - Tcold/Thot does dissipate, and 1 - x is converted into work.
x flows into the cold sink. Of this, Tcold/Thot flows into the cold sink inevitably, and the rest (x - Tcold/Thot) as a result of not being captured in the bulk. That is definitely wasted.
Now, the remaining 1 - x can either be dissipated in the bulk or captured as "useful work".
Back in the ecological analogy, 1 - x is made up of the energy dissipated by the weather system, and the energy used by the biosphere, of which we're taking an ever large share. When the capital development of a country becomes a by-product of the activities of a casino, the job is likely to be ill-done. — John M. Keynes
I am inclined to see now that some local waste is inevitable. To sort the mess, even a colder sink would be needed (say, the outer space). What does it imply for global warming by the greenhouse effect? Are we not letting infrared photons (as "waste" from high energy Sun photons) to escape the earth, and so we have to suffer "unexportable" entropy increase throughout all planet's systems?
The fraction 1 - Tcold/Thot is the "high-quality energy" that we can let dissipate, be captured by the ecosphere, or use ourselves.
Some local waste is inevitable, but that's more in the nature of an engineering problem if you like. There's no reason in principle why very close to 100% of 1 - Tcold/Thot cannot be tapped. That's what a Type-1 civilisation does.
As for lowering the cold temperature, if we decided to run large industrial facilities in space they could run at low temperatures and reduce the Tcold. Note that the Earth is warmer than it would be if it had no atmosphere: we already have a substantial greenhouse effect making the Earth habitable and keeping Tcold relatively high. When the capital development of a country becomes a by-product of the activities of a casino, the job is likely to be ill-done. — John M. Keynes
What do you mean with "Some local waste is inevitable..."? Do you mean here that the minimal portion Tcold/Twarm of energy will go to 'warm' the cold sink? That in principle reduces its quality other things being neutral, however minutely, right? Or do you mean that ideally we could actually have no waste, in whatever sense?
It is the part that could be captured but isn't that is wasted (i.e., if more than Tc/Th flows out to the cold sink, the excess can properly be described as wasted - some of this waste will indeed occur because of local dissipation). The part that is not wasted may or may not be exploited "usefully".
My main focus is not whether energy is used most effectively, but how much 'waste' has to be dumped locally. In many texts on complex systems, including on self-organized criticality, I read that dissipation waste is a characteristic feature of complex systems. So I wonder, how necessary is this feature.
Once again, my philosophy here is that the spontaneous arrangement of a stationary flow does no work and dissipates all the energy that could usefully be captured as work. Self-organisation of the complex system arises when the flow is fast enough as to trigger a period-doubling cascade (the stationary flow becomes unstable at the point of the first bifurcation). Then some of the energy is used to drive the (quasi)periodic limit cycle (self-organisation), that is, to do work against local dissipation (inside the system) and less flows out directly. When the capital development of a country becomes a by-product of the activities of a casino, the job is likely to be ill-done. — John M. Keynes
you can see "period three" in the vertical gap towards the right edge of the picture. To the left of that is the "period-doubling cascade" which in this case is mislabelled "chaotic region". Chaos is to the right of the period-three gap: while to the left one finds only periodic limit cycles of arbitrarily long periods, to the right one finds quasiperiodic limit orbits.
To my view, chaotic regions of deterministic models can be fine for complex systems. As the deterministic model is merely an approximation, the actual system might pick up and stabilize several near-cycles of the deterministic model, as basis of its working. It is imaginable that the system would "force" those basic regularized cycles when needed, or could sometimes "relax" and allow a random drift.
On the other hand, being close to the edge of chaos might be the most effective way to switch between functional regimes (which would be based on stable cycles on the "quiet" side) and opportunistic drifts (which would be switched on by crossing the chaos boundary). Did anyone think like that?
The simplest r/K selection model is based on Verhulst's logistic equation. Did anyone look in natural ecosystems for exact bifurcation proportions that follow from the same diagram?
If you really want to delve into the relationship between periodic, quasiperiodic and chaotic behaviour, I can suggest no better source than Predrag Cvitanovic's webbook. You should be able to read at least the introduction and some of the introductory chapters before the math becomes impossible :-)
The period three gap is that widest one, closest to the right edge, right? Why the gap is so wide? Does the period 3 occurs at the right edge of the gap, or at its middle? I assume that the "period 6" gap is to the right of the cascade intersection labeled "Chaotic region".
I'd like to send you an e-mail about the thermodynamic heat/work/dissipation/entropy stuff we discussed yesterday. Can you drop me an email at my public address? When the capital development of a country becomes a by-product of the activities of a casino, the job is likely to be ill-done. — John M. Keynes
