Ancient Roman industrial revolution followed by collapse to Barbarian invasion

Why not? Industrial revolutions are intensely dislocating experiences, redistributing power away from old regions/elites into new regions. Given the relative scarcity of coal in the traditional Roman heartlands of Italy and the Balkans (Spain has some coal but it's on the Biscay coast), a coal-fired industrial revolution will see power distributed to Britain, Germany and Northern France - periphery areas. That itself would incentivize more frequent and more effective attempts at breaking away from Rome, ultimately sparking the repeated civil wars that OTL brought down the Romans. The spread of technology into Germania and Persia would, of course, also increase their capabilities in the long-run - but the main threat to Rome is the civil wars, not the 'barbarians'.
 
Why assume a coal-fired industrial revolution?
I'd like to know your alternative industrial revolution scenario: it seems to me that coal (or variants, like peat or charcoal)'s advantage in industrial revolutions is that it generates a lot of energy compared to the input needed to extract/harness it (unlike human labor), it's relatively common and easily transportable (unlike water), and it integrates well with more advanced metalworking processes that, one assumes, an industrial revolution would eventually base itself upon.

Even in an alternative industrial revolution, redistribution of power from regions and from rural-to-urban will still occur: the question is whether those redistributions would seriously affect the foundations the Empire was built upon (Mediterranean manpower, Eastern + African productivity, and a lack of comparable challengers in the Empire's peripheries).
 
I'd like to know your alternative industrial revolution scenario: it seems to me that coal (or variants, like peat or charcoal)'s advantage in industrial revolutions is that it generates a lot of energy compared to the input needed to extract/harness it (unlike human labor), it's relatively common and easily transportable (unlike water), and it integrates well with more advanced metalworking processes that, one assumes, an industrial revolution would eventually base itself upon.

Even in an alternative industrial revolution, redistribution of power from regions and from rural-to-urban will still occur: the question is whether those redistributions would seriously affect the foundations the Empire was built upon (Mediterranean manpower, Eastern + African productivity, and a lack of comparable challengers in the Empire's peripheries).

Given that the historical industrial revolution started with better use of manpower and waterpower, and only shifted to steam power after a solid understanding of industrialization was established, I don't see any reason that this would be particularly different here. I'll note that one of the advantages you list about coal - its portability - means that its less serious for a large empire to only have deposits in the periphery. The coal can simply be shipped to major population areas as needed. And shipping is important, because if we assume steam power and a unified Rome, then the sheer volume of trade that would be going through the Mediterranean will make it an economic linchpin.
 

Hecatee

Donor
Industrialization depends on need, need depends on market, market depends on transportation, so industrialization depends on transports. But transports equals communication too, and so ability to react, including by moving troops fast enough.
The Barbarians fighting against Rome were, at least in the west, using rather few cavalry and were not very fast so improved roman transports thanks to new technologies (trains, steam barges, ...) means better ability to crush either barbarian raids or revolts. One has to remember that a lot of usurper were on very shaky grounds in the first few weeks of their attempts, with little troops under their direct command, and they needed weeks if not months to purge their command of loyalists and establish their own administration, time which they won't have anymore once railroad comes.

And you can bet, given Rome's love of roads, that railroads would be the focus of a lot of emperors as soon as they are invented... The first state lines would probably be between supporting towns and the frontline bases, with a number of networks which would then interconnect with both the other supply networks and the commercial, privates networks that would appear on the rear between other cities. Of course all the nodes of those networks would have garrisons to prevent illegal use of the network so any rebel general would have to conquer each little railroad citadel to be able to move given that trains can't go cross-country. So that's a plus for the stability of the empire.

So you would actually have less revolts and less damages from barbarians.
 
Given that the historical industrial revolution started with better use of manpower and waterpower, and only shifted to steam power after a solid understanding of industrialization was established, I don't see any reason that this would be particularly different here. I'll note that one of the advantages you list about coal - its portability - means that its less serious for a large empire to only have deposits in the periphery. The coal can simply be shipped to major population areas as needed. And shipping is important, because if we assume steam power and a unified Rome, then the sheer volume of trade that would be going through the Mediterranean will make it an economic linchpin.

My concern is that realistically the Romans don't have great options for bulk-shipping, which would make resource extraction for population centres difficult. Horse-drawn canals are a possibility, but the metallurgy to enable trains, as well as large-hold ocean-going vessels means that British Coal is going to be harder to export.

Britain IMO still seems the best candidate for any Roman industrial revolution. If we can combine the ability to drain marshes, build dams, with the rainfall of Britain, Roman aqueducts, and examples already in Britain, it is primed for an obscene quantity of water-powered industry compared to the rest of the Empire.

So I'd probably be looking at a Wool-production PoD, something involving an increase in demand, perhaps the Roman Legions in Britain start using an early form of Gambeson.

Spike in demand can create the need to increase production, and the interest in using manpower and waterpower in Britain better. (It is still miserable and unlike to attract much immigration from Italy proper).

That leads to at least some situation that could cause a form of industrialisation. Now I don't know how much land was used for farming in Britain, but I expect the conversion of land to rearing more sheep would mean there were some more unemployed to work at producing wool in Roman urban areas, increase concentration of wealth, more likely to be able to afford to research more, and the process continues.

All of which can still work within the idea of this OP. Britain industrialising could still leave the rest of the Empire overrun. The only difference is that the next rulers of Britain have a huge industry in place for arming themselves - and potentially the Dux Britanniarum has a cheaper way to preserve a large standing force, or rebuild one rather than rely on Saxons.

There are other areas that can likely have localised industrialisation like suggested here (In fact I'm pretty confident the Romans had plenty of examples), but I see regional industrial networks to be more likely since we're looking at bulk resource use.

If you look at Waterpower in the Med however, then you've got a totally different ball game. Take Roman Illyria. Loads of waterways (and later loads of coal). You still need the reason to move to water power, but you have the benefit of being in the Med, and able to work with Mediterranean ships for bulk-movement of goods. Importing iron from elsewhere and then working it using water-powered bellows.

There is a lot of potential, but I'm not sure what was already being done BY the Romans at the time.
 
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