Heat transfer happens proportionally to the temperature difference; the bigger the temperature difference, the faster heat moves. Cold is just the absence of heat, so it follows the same principles.
You can think of heat like people at a house party. If someone opens up an empty room for the guests, they will very quickly flood in and spread themselves out over the new space. Whereas if you just open a door between a mostly occupied room and a full room, you might have a few people move over to the less occupied room, but it’ll be pretty slow.
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