# Do calculations require energy?

A typical computer takes some energy "from the wall" while operating. I know that energy is spent on:

• Heating various components (psu/vrm/cpu/ram/ssd, so on)
• Fan/air motion to remove that heat (also vibrations, absorbed by case and surroundings)
• Magnetic emission from various chokes and long +5/+12v wires.

Let's consider that PC is disconnected from the network and has no attached devices and wires other than power cord. So that electric energy has no other ways out/wires to go. Then, all those waste-factors (mentioned above) are parasitic, in the sense that we want typically to avoid/lower them.

But the computer also does some meaningful process called calculation. For example, calculating a list of a prime numbers, or splitting very big number to prime factors - that is not an easy tasks - and that cannot just be done "instantly".

So the question is: is there some other kind of "compute" energy (other than mentioned above)? Does the energy go somewhere else, other than mentioned above?

Or would I get exactly what we took out of the wall? (If we sum up heat + motion + emi waste, mentioned above).

I can even reword it: if we consider some theoretical recuperation device that would absorb every wasted watt of heat, emi and vibration from working PC (with 100% efficiency), and convert it back to electricity (also with 100% efficiency), would calculation run forever "for free"? Or is there some other energy leak spot?

## 2 Answers

Landauer's principle relates to the smallest theoretical amount of energy required for computation.

The principle asserts that the minimum energy required to erase one bit of information is:

$$k T \ln 2$$

where k is the Boltzmann constant (~1.38×10−23 J/K), T is the temperature of the environment, and ln 2 is the natural logarithm of 2 (~0.69315).

The derivation comes from the second law of thermodynamics and entropy.

From Wikipedia, the following paragraph illustrates how this compares with contemporary computers:

At 20 °C (room temperature, or 293.15 K), the Landauer limit represents an energy of approximately 0.0172 eV, or 2.75 zJ. Theoretically, room‑temperature computer memory operating at the Landauer limit could be changed at a rate of one billion bits per second with energy being converted to heat in the memory media at the rate of only 2.85 trillionths of a watt (that is, at a rate of only 2.85 pJ/s). Modern computers use millions of times as much energy per second.

I suggest reading the whole article for more detail.

• Thanks alot, I think that modern humanity has no exact answer on "where does extra stuff goes". It's good that you've pointed me directly on how is this called. – xakepp35 Jun 3 at 16:37
• @xakepp35 The energy that Landauer's principle talks about is lost as heat, but modern computers produce a lot more waste heat than that theoretical minimum limit. – PM 2Ring Jun 3 at 16:46
• @PM2Ring So, in a free space, (with very low ambient T), in order to "erase a bit" we have to dissipate some energy, that would be emitted as some "small portion of microwave radiation".. and that is our computation price. correct? – xakepp35 Jun 3 at 16:49
• @PM2Ring What if we have second computer, that has all bits initially zeroed out. To set them up it has to increase entropy.. So may it take that heat from first computer for its operation? – xakepp35 Jun 3 at 16:54
• – PM 2Ring Jun 3 at 16:57

Computer do cause a decrease in the amount of usable energy, which is consistent with the second law of thermodynamics, but there is no violation of the first law; the total energy is constant, and can only be converted to other forms. There might be some energy stored as potential energy within the computer (for instance, the "1" state of a memory unit might hold more potential energy than the "0" state), but there is no energy destroyed by the computer.

• so a "pure computation" process (which only exist "on paper") - is totally energy-free process? and energy consumption is only due to imperfection? – xakepp35 Jun 3 at 15:50
• I mean is creating some order is "free", but actually arranging something (1s or 0s, whatever it's represented in) in that order - costs energy? – xakepp35 Jun 3 at 15:54
• There is minimum, given by the Landauer Principle. It is not free. – rghome Jun 3 at 15:57
• @xakepp35 If we're talking about total energy, rather than usable energy, then the first law of thermodynamics says that nothing "uses up" energy. – Acccumulation Jun 3 at 16:12
• @xakepp35 I think that is another question about thermodynamics and entropy and it applies to any device (e.g. you can't take the exhaust heat of a car engine and power another car engine of the same size). I am not qualified to answer that well, except to say that the waste heat has a higher entropy than the fuel and so is not as useful. – rghome Jun 3 at 16:33