Soup – Pressure Cooker to Stovetop conversion

There really are a couple separate issues that come together here:

What is collagen to gelatin conversion?

When collagen converts to gelatin, it is not melting (which is the same type of molecule, just as ice and liquid water are the same thing).

Instead, it is being hydrated which is a chemical conversion process, where water is actually being added into the overall structure of the protein molecule, converting it into a different protein molecule.

This does not happen just because of temperature, but rather because a water molecule with the right level of energy to power the process (that is, moving fast enough) hits the collagen molecule at the exactly proper spot to interact with it and become part of the molecule.

The new molecule is called gelatin.

Why does a pressure cooker facilitate effectively higher temperatures than an oven?

Most foods contain a great deal of water. On of the fundamental properties of water is that it takes a relatively large amount of energy to convert it from liquid water just at the boiling point (100 C / 212 F at sea level pressure) to vapor just at the same temperature. This is called the enthalpy of vaporization.

When a food containing water is heated in air at normal pressure, even if the air temperature is much hotter than the boiling point of water, the surface of the food cannot get hotter than the boiling point, because any additional energy goes into converting the water into steam, and drying the surface.

Only once the surface is dry can browning and other processes that happen above 100 C begin.

However, in the inside of the food, which is still wet, the temperature continues to never be able to exceed the boiling point. Very few foods are normally cooked to the point where the interiors are dry enough to get hotter than this.

In a pressure cooker, the boiling point of water is higher once the pressure is achieved (for brevity, I will not discuss why this is so). For example, at 15 bar (typical of a pressure cooker, one additional atmosphere of pressure above normal sea level pressure), water will not boil until about 250 F / 121 C.

This permits both the surface and the interior of the food to reach higher temperatures than at normal pressure. Some cooking processes are accelerated due to this difference.

Why does collagen convert faster?

Collagen conversion to gelatin is a time/temperature dependent process.

That is, the higher the temperature (within reasonable limits, before it burns or decomposes otherwise), the faster the conversion.

Collagen will convert to gelatin at 140 F, but it will take literally several days. At 170 - 180 F (typical internal temperatures in sea level braising), it takes several hours.

In a pressure cooker, this time can be reduced, because the internal temperature can go higher than it can at sea level temperature.

The reason for this is that collagen to gelatin conversion is a stochastic process. What this means is that it is essentially random. As a gross over simplification, imagine that the collagen molecule is a giant molecule (it is) with a button on it.

All of the water molecules are moving around randomly, bouncing off one another. The higher the temperature, the faster they move on average. That is, at a low temperature, most molecules are moving relatively slowly, but some are almost stopped, and a very few are very fast. At a higher temperature, they move faster on average, and a slightly larger number are moving relatively very fast.

Now imagine that the button will not be pressed until a water molecule happens to hit it while going fast enough to hit the button hard enough. The higher the temperature, the less time it will take for this to happen on average, because more of the water molecules are moving fast.

Take this process over many, many, many collagen molecules, and you have the time/temperature conversion curve: the hotter the temperature, the faster the conversion overall.

Conclusion

Collagen to gelatin conversion is faster in a pressure cooker because the internal temperature of the food gets higher than is possible at atmospheric pressure, and this higher temperature speeds the stochastic gelatinization process.

There are several advantages to using a pressure cooker other than speed, the first of which address your concerns about skimming. If operated correctly the water in a pressure cooker will never come to the boil resulting in a clearer stock than one made by convention means. From Modernist Cuisine (2-291):

The liquid inside the pressure cooker will not boil, despite the elevated temperature, unless you let the heat get out of hand. A liquid boils when its vapor pressure exceeds the ambient pressure around it. Inside a sealed pressure cooker, as liquid water vaporizes, it raises the ambient pressure, which in turn increases the boiling point. So long as water vapor is not escaping the pressure cooker, the pressure inside will stay high enough to keep liquid water from boiling. Never reaching a boil is important because it keeps the stock clear. Turbulence from boiling emulsifies oils and small food particles from the ingredients into a stock, thereby making it murky.

A telltale sign that the stock is at a rolling boil inside a pressure cooker is a jet of steam or fog from the overextended pressure valve. This jet means that the pressure cooker is overpressurized, and for safety's sake the valve is relieving the pressure.

You can further increase the clarity by adding some raw meat to your ingredients to act as a filtering agent. From Modernist Cuisine (2-295):

Dispersing a small amount of raw ground meat along with the other ingredients when making a pressure-cooked stock yields a stock with the clarity of a traditional consommé. This works because protein extracted from the uncooked ground meat acts just like protein in egg white: it sponges up tiny light scattering fat globules and other small particles.

Another benefit comes from the sealed nature of the cooking environment, which prevent volatile aromas from evaporating into the air. Heston Blumenthal cites this as one of the reasons why The Fat Duck began using pressure cookers for their stock in this article:

Now, this may seem obvious, but when you smell those wonderful odours while you're cooking, it's a sign that you're losing flavours through those volatile elements that disappear in the air. A pressure cooker, however, keeps the aromas and flavour molecules sealed in the pot...In a normal stockpot, by contrast, water evaporates at boiling point, taking flavour with it.

Modernist Cuisine (2-292) concurs:

When the the pressure-cooking step is done, you must let the cooker cool before removing the lid...Cooling first means that volatile aromas in the vapor above the liquid will condense back into the liquid rather than escape into the kitchen.

However, not all pressure cookers are created equal and some will release vapor to maintain pressure resulting in a poorer quality stock. Dave Arnold and Nils Norén from Cooking Issues have done extensive experiments into this phenomenon here: in blind tastings vented pressure cookers produced inferior results to both conventionally cooked stocks and those made with more advanced spring valved cookers; however, the non-vented vessels produced the best results out of all three methods.

Finally, the elevated temperature increases the rate of Maillard reactions resulting in a more flavourful stock. In the Cooking Issues article above the resulting browning was significant enough that it was easy to tell which stock was pressure cooked and which wasn't just by visual inspection:

The aroma of the pressure cooked stock was clearly superior. The color was deeper as well (because of this all future tests were done actually blind –with our eyes closed). Unfortunately the conventional stock tasted better. It had a stronger chicken flavor and was better balanced overall.

...

I took 4 liters of conventional chicken stock from the restaurant and pressure-cooked half while the other half simmered on the stove. This time, I didn’t use the school’s pressure cooker, I used my own. When I compared the two stocks side by side the pressure cooked one was far browner. I hadn’t thought the pressure cooker would change the color of a pre-made stock. When we tasted them the pressure-cooker won.