Why Every Computation Has an Unavoidable Energy Cost

Every time your computer performs a computation, it consumes energy. We all know this intuitively, your laptop battery drains as you use it. But that raises a deeper question: is this merely an engineering limitation, or does physics itself require computation to consume energy?

To answer that question, we need one idea from information theory: entropy.

Information theory tells us that:

1. The total entropy of a system can only go up or stay constant. It never goes down [unless you supply energy into the system] That's why things rot when you leave them, and that's why cold things gradually and spontaneously warm up if you leave them alone. But a warm thing will never spontaneously cool down after it has reached thermal equilibrium, because a system with less thermal energy usually has fewer accessible microstates, hence lower entropy.

2. The higher the possible number of possible microstates of a system, the higher the entropy.

This is where information theory becomes surprisingly useful...

Remember, logic is just state manipulation. We take some input state and transform them to some desired output state.

Every computation can ultimately be broken down into billions of these tiny state transitions happening inside logic gates.

Let's examine how the state evolves over the course of an AND operation, one of the simplest logical operations.

From truth tables, we know that two input bits produce one output bit.

AND Truth Table

$P$	$Q$	$P \land Q$
0	0	0
0	1	0
1	0	0
1	1	1

So we have 4 possible input states (00, 01, 10, 11), which the AND operation shrinks to 2 possible output states (0 or 1).

Looking at that in isolation, you might say, what?! In this closed system, you had 4 possible input states and this "AND operation" maps 4 possible input states onto only 2 output states, and we know that entropy increases as the number of accessible microstates increases, so does this "AND operation" thing, violate the laws of thermodynamics by somehow decreasing entropy? At first glance, this seems impossible.

...but of course, we do not violate any physical laws.

Because the 4 input states collapse into 2 outputs, 1 bit of logical information is permanently erased (notice that from the output alone, you can no longer tell whether the input was 00, 01, or 10. That information has been irreversibly lost.). Landauer's principle dictates that this lost information must physically manifest as an increase in the entropy of the gate's environment, which we measure as waste heat.

Therefore, every time your CPU performs an irreversible computation, it must dissipate at least a tiny amount of energy as heat. Modern processors are still millions to billions of times above this theoretical minimum because of real-world electrical losses, but Landauer’s principle tells us that even a perfectly engineered computer could never reduce that cost to zero for irreversible computation. That’s one of the fundamental reasons computation isn’t free, and ultimately why your CPU gets hot.

Landauer's principle is how we calculate that theoretical minimum amount of energy required for irreversible computation or bit erasure. It is stated in the formula:

$$E \geq k_{B}T\ln2$$

Where, $k_{B}$ is the Boltzmann constant, and $T$ is the absolute temperature

Numerically, the Landauer limit represents an energy value of about $0.018eV or\ 2.9\times10^{-21} Joules$ .

This is the minimum amount of energy required to irreversibly erase one bit of information. Any irreversible computation must incur at least this cost, although real computers consume vastly more energy for other physical reasons.

More information about it here:
landauer's theory wikipedia

Bonus:

Strictly, Landauer's limit applies to irreversible logic operations (such as AND, OR, etc), and not reversible ones (such as CNOT, CCX gates). So if you had a computer built out of reversible logic gates, you could theoretically avoid that energy cost completely.

Reversible logic operations are computation steps where the input states can be uniquely retrieved from the output states, meaning no information is erased, and energy loss is avoided.

(Note: This theoretical limit assumes zero leakage and infinitely slow operations, but the principle stands).

Conclusion

What fascinates me most is that a simple question about why computers consume power ultimately leads to information theory, thermodynamics, and statistical mechanics. It’s one of those rare moments where two seemingly unrelated fields turn out to be describing the same underlying reality.

Also, fun fact, your brain runs on a continuous 20w power draw to run your entire life, Let's do the math: Landauer per bit operation is $2.9 \times 10^{-21} J$ . The brain does ~ $10^{15}$ synaptic operations per second. $20W / 10^{15} \approx 2 \times 10^{-14} J/op$ . That is ~7,000,000 times higher than the theoretical Landauer limit, which is is remarkably efficient compared to modern silicon (which is ~ $10^{9}$ times over the limit)

Further reading:
Landauer Principle and Thermodynamics of Computation by Pritam Chattopadhyay, Avijit Misra, Tanmoy Pandit, and Goutam Paul - Cryptology and Security Research Unit, Indian Statistical Institute, Kolkata 700108, India
https://arxiv.org/pdf/2506.10876v2