Muscle wire works through heating and normally takes about half an amp. This is way too much for the Arduino's output so you need a relay or transistor to switch it. You also need an external power supply to give that sort of current.
The same is true of a potentiometer, most will not be able to stand so much current. A carbon pot will just vaporise if you have so much current flowing through the end track. Of course you can get wire wound pots but these are bulky and expensive. I did have several 1KW pots for controlling theatre lights years ago they were over two feet long.