What is the scientific/engineering basis for the break-in/burn-in effects, esp for different components, like batteries, or cables or circuits or speakers? What's happening when beak-in occurs? 
There is no proven basis for many "break-in" claims.
Mechanical items do break-in. This is well-accepted. So, speakers may well be subject to break in. I've seen measurements of drives during break-in and the T/S parameters can change.
Things like a turntable, CD or DVD transport, and RtR tape deck probably show some break-in effects. Whether it's audible or not is debatable. Still, if someone said their turntable smoothed out after 100 hours I'd give them the benefit of the doubt. There's a basis in fact.
Some electronic components like capacitors need to "form" the first time they're used (and after long periods of disuse). That's not really break-in, and it doesn't take very long.
Other than that, there isn't any real evidence for "break-in" of electronic components. There are a lot of people who try to explain it one way or another, but there isn't a factual basis to any of it, and they can never show measurements that illustrate their claims. They can't even show that they can actually hear a difference when subjected to a controlled test.
So, things like wire "break-in" are faith-based, like a number of other things about high-end audio.
It's up to you whether you choose to believe in it or not.
