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# Computation that computes
- URL: https://www.slightlyacidic.com/2026/09/25/computation-that-computes/
- Published: 2026-09-25T15:20:00.000Z
- Updated: 2026-09-29T02:16:37.000Z
- Author: VM
- Tags: physical systems, computing, equilibrium state, free energy, graphics processing unit, Hooke's law, John Hopfield, linear algebra, loss function, machine learning, non-equilibrium state, optimisation problem, parameter space, physical laws, stiffness

While I was recently browsing the web for interesting physics papers, I came across one in *Nature* titled [‘A thermodynamically favoured molecular computer’](https://www.nature.com/articles/s41586-026-10996-5?ref=slightlyacidic.com). The work itself seemed interesting but what caught my eye was a pair of lines in the abstract: “… computation can be embedded in a system relaxing to a thermodynamically favoured equilibrium state. Machine learning and search algorithms use this idea, although executed on non-equilibrium architectures at enormous energy cost.”

I have come across versions of this statement when writing about the connections between machine learning and statistical physics — a feat the Nobel Prize for physics recognised in 2024\. But then I thought that in the spirit of really learning how this was and to have some reference text I could come back to, I should write about it. So here goes.

The central idea seems to be something physicists and computer scientists are very familiar with but which nonetheless landed, to me, as a wonderful surprise when I first stumbled across it in the context of quantum computing: in the right conditions, the natural tendency of a physical system to move towards equilibrium can perform a computation.

This is quite unlike the computer processor we are more familiar with, which computes by manipulating small electric currents passing through semiconductors. Put another way, it was easier to grasp the ability of a processor to be capable of computing because it had been outwardly apparently engineered by humans to do so. On the other hand, that a natural phenomenon could perform calculations whose outputs are valuable specifically to certain humans seemed bizarre.

Of course, once you think about it, it becomes clear that that happens more often than we explicitly acknowledge. The sun’s position in the sky at a fixed time on every day of the year marks out a figure called the analemma, and people can use it to determine sunrise and sunset times every day as well as the directions of due east and due west. The essential idea is to map objects’ properties and their interactions with each other to information of value to people.

That said, the idea that systems simply abiding by the natural physical laws can somehow solve sophisticated maths problems or help discover new drugs boggles the mind. Sure, this sentence is reductive, but you get the point: how do we go, rather did we go, from an ordinary tendency of the universe’s matter all the way to computing?

The classical textbook example in physics of a system moving towards equilibrium starts with a ball at the top of a hill. Unless something is holding it back, the ball will roll downhill and eventually come to rest in a valley. But this is not the physical system of interest *per se*. It is the one centred on this system’s free energy. The ball rolled down and into the valley because doing so lowered its potential energy, and thus its appropriate free energy. The journey to equilibrium is thus a system’s pursuit of lower free energy, using quantities like kinetic/potential energy, entropy, temperature, interactions between its components (e.g. molecules), and so on.

To take advantage of this phenomenon for the purposes of computing, scientists design physical systems so that their ‘valleys’ correspond to the solutions of a problem while its starting position corresponds to… well, the starting position. Then, by mapping the first and final states to parts of the problem itself, scientists can use the system’s evolution to arrive at the answer.

Let us consider an example. Say you need to solve the following equations for *x* and *y*:

3*x* – *y* \= 1

–*x* \+ 2*y* \= 0

And to do that, you set up this mechanical system:

![](https://storage.ghost.io/c/4f/29/4f2939bc-a4a6-402d-b122-db86d3e7e547/content/images/2026/09/1xwck.jpg)

In words: the distances of the blocks X and Y from the wall stand for the value of the variables *x* and *y*. X is attached to the wall by a spring with a stiffness of 2 N/m and Y is attached to the wall with a spring of stiffness 1 N/m. And X and Y are connected to each other by a separate spring of stiffness 1 N/m. (The cylinders attached to the springs are viscous dampeners: they ensure the blocks stop moving cleanly instead of oscillating back and forth first).

Now, using a load weighing around 102 g hanging at the end of a pulley, you apply a force of 1 N towards the right on block X. The moment the load is applied, the system will find itself in a non-equilibrium state. X will begin moving towards the right, pulling Y along, even as X’s and Y’s springs pull them towards the wall. The system’s components will begin exchanging kinetic energy (from the blocks), elastic energy (from the springs), and potential energy (from the load) — even as the dampeners turn some of that mix into heat. Eventually the system will come to rest, i.e. equilibrium.

By definition, in this state, the total force acting on X and Y must vanish. The forces acting on X will have been:

(i) The wall spring pulling to the left at 2 N/m times *x* (by Hooke’s law, force equals stiffness times distance),

(ii) The connecting spring pulling in the direction of X at 1 N/m times *x* – *y*, and

(iii) The load pulling to the right at 1 N.

At equilibrium, then: 1 – 2*x* – (*x* – *y*) = 0\. If you rearrange this, you get: 3*x* – *y* \= 1, which is the first equation. Similarly, if you tally the forces acting on Y to zero, you get the second equation: –*x* \+ 2*y* \= 0\. In (other) words, you have confirmed that the physical system correctly instantiated the two equations, which represents the mapping.

Now, all you need to do is measure the distances of X and Y from the wall: the two blocks would not have stopped at some random position; instead, thanks to the mapping, they would have come to rest at a position that would allow both blocks to have zero net forces acting on them. Thus you will have your answer: 40 cm and 20 cm, i.e. *x* \= 0.4 and *y* \= 0.2.

But here is the funky part. Just as you worked out the forces acting on each block, you can also work out the formula for the system’s potential energy:

![](https://storage.ghost.io/c/4f/29/4f2939bc-a4a6-402d-b122-db86d3e7e547/content/images/2026/09/e1.png)

The first three terms represent the energy stored in the various springs and the fourth represents the potential energy in the load. If you work it out, the equation becomes:

![](https://storage.ghost.io/c/4f/29/4f2939bc-a4a6-402d-b122-db86d3e7e547/content/images/2026/09/e2.png)

This is an algebraic equation whereas the system was purely mechanical. Obviously the system does not ‘know’ algebra — but thanks to the way the two equations were mapped to the forces acting on the blocks, the system’s evolution according to simple laws of physics corresponded to an evolution of this equation to its minimum value. And at that value, 3*x* – *y* will equal 1 and –*x* \+ 2*y* will equal 0.

In much the same way, if you can set up a highly complicated system with hundreds or even thousands of coupled elements, its physical properties can be mapped to (or mapped to mappings of) way-more-complicated problems in, say, optimisation and linear algebra. Then you pull the system to a non-equilibrium state and release it.

Scientists have applied the same technique, in its fundamentals at least, to machine learning because machine learning is often essentially an optimisation problem. For example, training a neural network to perform a specific task can be tantamount to searching through a large parameter space for values that minimise a loss function. Imagine this space to be a strange land filled with hills and valleys: the neural network’s act of learning becomes, in a loose yet illustrative mathematical sense, journeying downhill through this landscape, to that point where the loss function hits its lowest possible value.

The advantage here is that the computer is allowed to focus on the maths that captures multiple variables evolving simultaneously instead of having to simulate or develop the several numerical operations representing that complex evolution from scratch, in turn consuming less energy to solve the problem.

That said, there is no free lunch. There many, many instances where machine learning and many other kinds of problem-solving are not synonymous with optimisation problems. More specifically, the technique using the natural behaviour of physical systems to move towards equilibrium is more readily suited for problems whose properties can be (relatively) straightforwardly mapped to an energy landscape whose minima correspond to useful answers. Examples of problems that resist this kind of mapping include [symbolic algebra](https://maths-from-the-past.org/symbolic-algebra/?ref=slightlyacidic.com) and [cryptographic hashing](https://www.geeksforgeeks.org/dsa/cryptography-hash-functions/?ref=slightlyacidic.com).

All this said, the computers we use in our daily lives, from the one you check your email on to the ones running your AI prompt, implement the ‘mapping’ concept only in an abstract way. For example, a GPU running a neural network does not actually build a system with a specific energy landscape and then allow it to move towards equilibrium. Instead, it uses billions of transistors plus clocks, memory banks, data connections, and electronic circuits to, in a manner of speaking, simulate journeys on this landscape — and that too only using math. This is what the *Nature* paper meant when it said “executed on non-equilibrium architectures at enormous energy cost”.

The high-dimensional optimisation problems today’s state-of-the-art AI models grapple with require extraordinary amounts of computation. It is practically computation that computes.