There's been a lot of work on this subject and the papers of Mézard and collaborators are relevant here as are the papers of Montanari; in fact Mézard and Montanari have written a book on the subject. Cf. http://lptms.u-psud.fr/membres/mezard/  and http://web.stanford.edu/~montanar/ for instance. Of course the story goes much, much,  further back. 

One such story that is perhaps interesting to consider available on RG...

Article Thermodynamic depth of causal states: Objective complexity v...

Many thanks for yours interesting responses. I'm not convinced that yours suggestions are exactly what I'm looking for, but in my short question I did not write everything what I mean. Unfortunately there is not enough space to explain everything but I'm looking for a new view of computer processing in terms of energy transformation. Typically when we try analyze algorithms we usually refer to mathematical approach where we use typical mathematical tools. For example, in analysis of sorting algorithms we support our efforts by such terms like inversions, sequences, replacement, dominant operations, etc. That is OK, but let's note that any kind of computer processing is done thanks to algorithms but on physical machines (computers). When we analyze behavior of any algorithms (Turing machines) we should have in mind that finally their work is done on physical devices and these machines need energy for work; in other words they worked like engines transforming one kind of energy (typically electric energy) into useful work - computations. I think having this it will be interesting for example to determine not only the total amount of energy needed for algorithm behavior in pessimistic case (computational complexity by O notation gives us this answer) but how much energy is needed for each processed element from input set. Computational complexity analysis doesn't allow to  answer this proposal if we assume the independence between input set properties and algorithm.

Well that's one of the topics of quantum computing-how fundamental physical processes set limits to information processing. So this, http://iqim.caltech.edu/ might be useful, to start with. 

Yeah, but what about for non-quantum computations ;)

Thanks for your link.

I'd suggest Feynman's ``Lectures on computation'', http://books.google.fr/books/about/Feynman_Lectures_on_Computation.html?id=NCJQq1_ZKgEC&redir_esc=y

I think there are too many degrees of freedom in a real computing system to allow any exact measure of energy usage.

Probably this is true but maybe there are several topics when this is possible.

If you can restrict yourself to something like a "cpu". But even to define a "cpu" (a logical entity) nowadays is not so easy :-)

A well bounded physical entity like a chip is more promising. Some interesting research has been done in the crypto field to measure exact energy usages under specific scenarious. Many cryptochips which essentially contain crypto-algorithms with keys have been cracked with the helpof this method.

OK, I understand your way of argumentation but what about classical approach when we describe the difficulty of problems solving in terms of dominant operations assuming that this is some kind of universal measure. Computational complexity expresses how the problem is difficult to solve it and in my humble opinion this should have some relation to the amount of necessary energy. Or maybe it is not important how much energy we need during processing - the only important fact is how many dominant operations we should perform. But this 'performing' needs energy because is done on physical machine.

I hope I understand you right. I would follow the path of "dominant OP" (how ever you can define it). The energy seems to me too dependant on current technology. OPs on a seperate controller with a badly programmed interface will use much energy without it being neccessary becáuse of the type of OPs. The design and technology define your real energy use more than the algorithms. Probably you can use the concept of a minimal machine (automat with a minimum number of internal states which solves a given problem) --> much more abstract. After my opinion the "energy" leads into a dead end.

I understand your way of thinking and many thanks for your opinions. Despite your doubts I still will try to find some solutions in this field. Maybe this is not a dead end :)

Regards

P.S. In your response I've found something like this ' --> '. My surname in Polish means arrow ;)

Funny, Mr. Arrow :-)

Good luck with your research; I'll follow it a bit.

Cheers,

Wolfgang