I see no laws of thermodynamics violated in Atkins diet.

When we are on a high carb diet we use glucose for fuel. We don't waste any of this fuel and if we have more of it we store it as body fat.

When we are on a low carb diet, we use ketones (which are carbon fragments) for fuel. Some of the ketones are lost without being used and none could be stored as body fat.

So if we are to use engines for comparison, we should be comparing a gasoline engine with an engine which runs on coal and leak some of its fuel! Who could expect the two engines to be equally efficient? Or in other words, who could expect the two engines to produce the same output energy when both get equal amounts of input energy?