i don't know that arduino can power something like that from the board itself or that there is a device made specifically for arduino boards. anyone?
i did stumble upon this
http://www.ebay.com/itm/Ultrasonic-Mist-Maker-Fogger-Water-Fountain-Mini-Air-Humidifier-Atomizer-12-LED-/161278307828?hash=item258cefadf4
which seems maybe like what you mean and has a picture of it in action.
however, it runs on 17V 800mA, so thats 13.6W
the adapter that supplies that runs on 110-250V, 50/60Hz
maybe you can use a small inverter like a tweaked car plug adaptor from ebay to run off a 12v battery
if you want to activate it maybe it can be done by manual switch with the LEDs you put on it being arduino controlled.
if you wan't to control the powering of it by arduino, i don't know, i am hoping to know about running higher power devices myself.
i hope someone more knowledgeable come along to help