The problem is, again, thermodynamics. You are going to get entropy, i.e. conversion of useful energy into unusable energy without getting work out of that process.
This means you are going to need to pump more energy in than you produce. Which begs the question GP asked.
Put another way, perpetual motion machines are impossible. What you propose is that.
Put some nice nuclear batteries on this and you might have a chance. But net power in is greater than net power out, again due to thermodynamics. So ...
Solar + methane itself fed back into the loop.