Thank you for your very thoughtful note! What is the export of Trinidad to which you refer in your footnote 6 — natural gas?
If we want to compare the carrying capacity of Spaceship Earth under solar power in the future and the past, I think it behooves us to consider the whole sunbeam-to-wheels efficiency, or sunbeam-to-hooves, as the case may be. It's true that photosynthesis is about 1–3% efficient (I think beema and sugarcane can approach 4% at times), but typical crop yields are closer to 0.3 W/m² [1], which is an efficiency of 0.1% if sunlight averages 300 W/m². Also, muscles are not perfectly efficient, and animals can't spend 100% of their energy input on their muscles; they have to spend some of it on digestion, homeostasis, reproduction, and — though this may be redundant — rest. So if draft animals are 20% efficient [0] and food production by photosynthesis is 0.1% efficient, then the total sunlight-to-hooves efficiency of plowing with oxen, horses, or by pulling the plow by hand, is in the neighborhood of 0.02%.
By contrast, low-cost photovoltaic panels are 16% efficient, as you say; electric motors and generators are typically about 90% efficient in combination, and the rest of the electrical system is typically about 90% efficient, as long as no batteries or high-voltage transmission are required. This gives a sunlight-to-wheels efficiency of about 13% for the best realistic case of solar, about 600 times better than corn-fed oxen. (Multijunction cells could push that up to 33% or so, but they're far too expensive with current production techniques.)
However, it might prove difficult to refine silicon in the case of a collapse of the post-1970s industrial infrastructure, so perhaps it is best not to calculate based on photovoltaic panels. A more conservative low-tech case is, as you imply, a CSP-driven steam engine like Frank Shuman's 1912 Sun Power Company power station Solar Engine One, which I mentioned in https://news.ycombinator.com/item?id=22363678 and which reputedly achieved about 4% efficiency [2]. If applied to Watt's 3%-efficient vacuum engine driving a generator, a solar thermal collector with an easily accessible efficiency of 50% would produce sunlight-to-wheels efficiency of about 1.2% — not great, but still 60 times better than oxen or corn-fed farm boys.
So there are, I think, excellent reasons to believe that solar energy could sustain current levels of human population on Spaceship Earth, even after a hypothetical systemic collapse.
As for proposed energy technologies with efficiencies below 0, these would be perpetual-motion machines: if applied to a 100-watt load, an engine of efficiency -10% would supply the load with 100 watts while consuming -10 watts, which is to say, producing 10 watts in some other form, perhaps electricity. I do not expect to see many of these. Perhaps this was not what you meant?
I will respond separately to your comments about materials recycling.
________________________________
[0] https://www.forbes.com/sites/kevinmurnane/2016/07/18/fueling... Tour de France riders eat 6000 to 8000 kcal per day (25–35 MJ in modern units, 300–400 W), 30% of which is just their basal metabolism, while racing 6 hours per day at about 400 watts of power output. That's about 30% cornbread-to-wheels efficiency, but it omits the power usage of reproduction, growth, and homeostasis — racers can't train year-round at Tour-de-France intensity levels without dying, nor can they do it while pregnant or growing out of childhood. Even during the race, most days are less than 6 hours of racing, and there are two official "rest" days with no racing at all.
[1]
http://www.worldofcorn.com/#us-average-corn-yield-metric
says US corn yield is 11 tonnes/hectare, presumably per year. If corn is the standard 15.5% moisture and the rest carbohydrate and protein at 4 kcal/g — not precisely correct, but close enough — that's about 0.5 W/m².
https://apps.fas.usda.gov/psdonline/circulars/production.pdf
table 04 says the world produced 1079.9 million metric tons of corn in the 2017/18 year on 192.12 million hectares, for a yield of 5.62 tonnes/hectare, so that seems reasonable, but that's only ¼ W/m².
[2] https://www.scientificamerican.com/article/power-from-sunshi... is an article Shuman wrote before building it. The actually built plant was reported to produce only about 50 to 60 horsepower when the sun was shining; according to M. Ragheb's Solar Thermal Power and Energy Storage Historical Perspective, it had five 62 m × 4 m troughs at 7.6 m spacing, totaling 248 m² of collector. However, this would collect 248 kW of insolation, implying an efficiency of 15–18%, which is implausibly high for a steam engine. Ragheb also, contradictorily, reports that it was 4% efficient. I suspect a unit conversion error.
Trinidad exports: Actually, ammonia for agricultural fertiliser, though I believe natural gas is also directly exported. Ammonia is easier to handle (which ... says things), so much as crops represent effective water exports, ammonia is a virtual export of natural gas.
On net ag efficiency: I've ... looked into this somewhat, with Vaclav Smil and Howard & Eugene Odum being generally recommended sources.
One aspect of the ecological approach is to look at plants not so much as inefficient, but as the end result of about 3.5 billion years of process refinement, optimising for numerous characteristics, not simply stored carbohydrate/lipids energy, including diseases, weather, pest, and other tolerances. Much of human ag selective breeding borrows energy from those plant services in favour of food productivity. The result is plants less able to thrive on their own. Optimising fertiliser and watering quantities and schedules also allows greater productivity. How much of that is specifically reliant on additional energy inputs is harder to pin down, though fertilisers pre Haber-Bosch were reliant on accumulated deposits generally moved by sailing ships. We could (if necessary) revert to sail, but those deposits are largely gone.
Looking at annual areal output is a good metric. US productivity actually lags Europe if I recall -- Holland is especially efficient.
On animal energy output levels, athletes are best considered as demonstrating a maximum possible short-term output, not a long-term population average. Particularly in the face of suboptimal nutrition, disease, and injury.
Smil's got an impressive set of tables and charts (making excellent use of logarithmic scales) showing output of humans, various draught and domesticated animals, etc. One factoid I've stored away is that a blue whale has roughly the metabolic output of a tractor-trailer rig. And yes: sustained animal output is pretty sharply limited, across scales: per day, sustained over weeks or months throughout the year, or even over lifetimes. Overexertion or overuse severely curtails output, and peak outputs are not sustainable over long periods. (Something Frederick Taylor rather famously omitted from his "scientific" studies.) Figuring about 25% of a typical human's 2,500 - 3,500 calories gives a range of about 30-45 watts of continuous output.
Note that the acre was originally unit of area derived from a measure of work: the amount of land a farmer and team of oxen could plough in a day. In German: Tagwerk, literally "days' work". The unit was variable (condition of land, soil, oxen, and plough determined tillable area), and represented less than a full day's work as the animals had to be rested and pastured.
(Smil notes that a large waterwheel or windmill, and many early Watt steam engines, delivered about 5-20 horsepower, about 3.7 - 15 kW, of power. And that made a huge difference.)
On carrying capacity net net, one of the most highly recommended sources I've run across (though not yet read) is Joel E. Cohen's How Many People Can the Earth
Support? (https://www.worldcat.org/title/how-many-people-can-the-earth...). My understanding is that he explores the basis and implications of values ranging from < 1 billion to > 1,000 billion, which includes most credible (and possibly some less than) estimates.
One point most serious discussions hammer is that "how many" goes along with "how much", in terms of resources:
I = P * A * T
Environmental impact is a function of population, affluence (resource consumption), and technology (as an inverse).
On energy systems and conversion mechanisms: the interesting thing is how little the story's changed in 50, 75, or even 100 years. We've added fission, and PV's gotten remarkably better. Batteries have improved tremendously. Fusion's still a pipe dream. Otherwise: plants, sun, wind, water, geothermal, waves. Flux per unit area is a very good analysis metric, see the late David MacCay's Renewable Energy Without the Hot Air (http://withouthotair.com)
My suspicion is that direct solar thermal (avoids conversion losses) and CSP power (simple and robust) are likely mainstays. There's been some interesting research into low-tech silicon-fabrication processes, with the Global Village Construction Kit (now apparently rolled into Open Source Ecology / Appropedia) doing some work. Purity and process control are major limitations.
NB: Technologies with a less than zero efficiency factor remove useful energy from the system. A brake would be an example. Perpetual motion technologies have efficiencies >= 1. Which seems unlikely in practice.
If we want to compare the carrying capacity of Spaceship Earth under solar power in the future and the past, I think it behooves us to consider the whole sunbeam-to-wheels efficiency, or sunbeam-to-hooves, as the case may be. It's true that photosynthesis is about 1–3% efficient (I think beema and sugarcane can approach 4% at times), but typical crop yields are closer to 0.3 W/m² [1], which is an efficiency of 0.1% if sunlight averages 300 W/m². Also, muscles are not perfectly efficient, and animals can't spend 100% of their energy input on their muscles; they have to spend some of it on digestion, homeostasis, reproduction, and — though this may be redundant — rest. So if draft animals are 20% efficient [0] and food production by photosynthesis is 0.1% efficient, then the total sunlight-to-hooves efficiency of plowing with oxen, horses, or by pulling the plow by hand, is in the neighborhood of 0.02%.
By contrast, low-cost photovoltaic panels are 16% efficient, as you say; electric motors and generators are typically about 90% efficient in combination, and the rest of the electrical system is typically about 90% efficient, as long as no batteries or high-voltage transmission are required. This gives a sunlight-to-wheels efficiency of about 13% for the best realistic case of solar, about 600 times better than corn-fed oxen. (Multijunction cells could push that up to 33% or so, but they're far too expensive with current production techniques.)
However, it might prove difficult to refine silicon in the case of a collapse of the post-1970s industrial infrastructure, so perhaps it is best not to calculate based on photovoltaic panels. A more conservative low-tech case is, as you imply, a CSP-driven steam engine like Frank Shuman's 1912 Sun Power Company power station Solar Engine One, which I mentioned in https://news.ycombinator.com/item?id=22363678 and which reputedly achieved about 4% efficiency [2]. If applied to Watt's 3%-efficient vacuum engine driving a generator, a solar thermal collector with an easily accessible efficiency of 50% would produce sunlight-to-wheels efficiency of about 1.2% — not great, but still 60 times better than oxen or corn-fed farm boys.
So there are, I think, excellent reasons to believe that solar energy could sustain current levels of human population on Spaceship Earth, even after a hypothetical systemic collapse.
As for proposed energy technologies with efficiencies below 0, these would be perpetual-motion machines: if applied to a 100-watt load, an engine of efficiency -10% would supply the load with 100 watts while consuming -10 watts, which is to say, producing 10 watts in some other form, perhaps electricity. I do not expect to see many of these. Perhaps this was not what you meant?
I will respond separately to your comments about materials recycling.
________________________________
[0] https://www.forbes.com/sites/kevinmurnane/2016/07/18/fueling... Tour de France riders eat 6000 to 8000 kcal per day (25–35 MJ in modern units, 300–400 W), 30% of which is just their basal metabolism, while racing 6 hours per day at about 400 watts of power output. That's about 30% cornbread-to-wheels efficiency, but it omits the power usage of reproduction, growth, and homeostasis — racers can't train year-round at Tour-de-France intensity levels without dying, nor can they do it while pregnant or growing out of childhood. Even during the race, most days are less than 6 hours of racing, and there are two official "rest" days with no racing at all.
[1] http://www.worldofcorn.com/#us-average-corn-yield-metric says US corn yield is 11 tonnes/hectare, presumably per year. If corn is the standard 15.5% moisture and the rest carbohydrate and protein at 4 kcal/g — not precisely correct, but close enough — that's about 0.5 W/m². https://apps.fas.usda.gov/psdonline/circulars/production.pdf table 04 says the world produced 1079.9 million metric tons of corn in the 2017/18 year on 192.12 million hectares, for a yield of 5.62 tonnes/hectare, so that seems reasonable, but that's only ¼ W/m².
[2] https://www.scientificamerican.com/article/power-from-sunshi... is an article Shuman wrote before building it. The actually built plant was reported to produce only about 50 to 60 horsepower when the sun was shining; according to M. Ragheb's Solar Thermal Power and Energy Storage Historical Perspective, it had five 62 m × 4 m troughs at 7.6 m spacing, totaling 248 m² of collector. However, this would collect 248 kW of insolation, implying an efficiency of 15–18%, which is implausibly high for a steam engine. Ragheb also, contradictorily, reports that it was 4% efficient. I suspect a unit conversion error.