tolkiens legendarium - How did the One Ring abandon Gollum?

I'm sorry I have to mix a little movie line with The Hobbit the novel.

In Peter Jackson's The Fellowship of the Ring, Lady Galadriel, while narrating the beginning of the movie, mentions that the the One Ring "abandoned Gollum" and the ring is shown bouncing down a slight incline, as if propelling itself to get away. I realize this is not canon.

In The Hobbit, here's how Bilbo finds the ring; it's really rather mundane!

[He] crawled along for a good way, till suddenly his hand met what felt like a tiny ring of cold metal lying on the floor of the tunnel. It was a turning point in his career, but he did not know it. He put the ring in his pocket almost without thinking; certainly it did not seem of any particular use at the moment.

_{The Hobbit -- page 68 - Del Rey Paperbacks - chapter 5, Riddles in the Dark}

Later in the book, Gollum has his freakout:

Suddenly, [Bilbo] heard a screech. It sent a shiver down his back. Gollum was cursing and wailing away in the gloom, not very far off by the sound of it. He was on his island, scrabbling here and there, searching and seeking in vain.

'Where iss it? Where iss it?' Bilbo heard him crying. "Losst it is, my precious, lost, lost! Curse us and crush us, my precious is lost!'

_{The Hobbit - page 82 - Del Rey Paperbooks - chapter 5, Riddles in the Dark}

Did Gollum himself lose the ring in the cave? Did the One Ring, hearing the call from its master Sauron, manage to move itself through Gollum's lake to the land?

How did the One Ring physically abandon Gollum?

Answer

From The Lord of the Rings book 1, Chapter 2: The Shadow of the Past:

"The Ring of Power looks after itself, Frodo. It may slip off treacherously, but its keeper never abandons it... It was not Gollum, Frodo, but the Ring itself that decided things. The Ring left him...

"There was more than one power at work, Frodo. The Ring was trying to get back to its master. It had slipped from Isildur's hand and betrayed him; then when a chance came it caught poor Deagol, and he was murdered and after that Gollum, and it had devoured him. It could make no further use of him: he was too small and mean; and as long as it stayed with him he would never leave his deep pool again. So now, when its master was awake once more and sending out his dark thought from Mirkwood, it abandoned Gollum."

So there's the motivation for why the ring left Gollum: Sauron was calling, and Gollum was useless. As for how, we know from the Hobbit that Gollum occasionally left his dark pool, wearing the ring, so that he could sneak up on orcs (called "goblins" in my copy) and kill them to eat. Most probably it was on one of these excursions that the ring left Gollum, which is why Bilbo found it so far from the pool.

Regarding the mechanism by which the ring abandoned Gollum, from the same chapter:

"[Bilbo] had found out that the [ring] needed looking after; it did not seem always of the same size or weight; it shrank or expanded in an odd way, and might suddenly slip off a finger where it had been tight."

This was how the ring originally betrayed Isildur, and it would have been how it ultimately left Gollum: by expanding at the opportune moment and falling right off his finger.