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Every computing system, biological or synthetic, from cells to brains to laptops, has a price. It is not the cost that is easy to recognize, but the energy costs associated with the work required to run a program and the heat dissipated in the process.

Researchers at the Santa Fe Institute and elsewhere have spent decades developing a thermodynamic theory of computation, but previous work on the cost of energy has focused on basic symbolic computations—such as erasing a single bit—that cannot easily be transferred to less predictable , real-world computing scenarios.

In an article published in Physical examination X, a quartet of physicists and computer scientists extends the modern theory of computational thermodynamics. By combining approaches from statistical physics and computer science, the researchers introduced mathematical equations that reveal the minimum and maximum predicted energy cost of computational processes that depend on randomness, a powerful tool in modern computing.

In particular, the framework offers insights into how to calculate bounds on energy costs for computation processes with an unpredictable end. For example: A coin toss simulator can be instructed to stop tossing once it reaches 10 heads. In biology, a cell can stop producing a protein after eliciting a certain response from another cell. The “stop times” of these processes, or the time it takes to reach the goal the first time, can vary from trial to trial. The new framework offers a simple way to calculate energy cost lower bounds for these situations.

The research was conducted by SFI Professor David Wolpert, Gonzalo Manzano (Institute for Interdisciplinary Physics and Complex Systems, Spain), Edgar Roldan (Institute for Theoretical Physics, Italy), and SFI graduate associate Gulce Cardes (CU Boulder). The study reveals a way to reduce the energy costs of arbitrary computing processes. For example: an algorithm that searches for a person’s first or last name in a database may stop working if it finds one of the two, but we don’t know which it found.

“Many computing machines, when viewed as dynamical systems, have this property where if you jump from one state to another, you really can’t get back to the initial state in just one step,” Cardes says.

Wolpert began exploring ways to apply ideas from non-equilibrium statistical physics to the theory of computation about a decade ago. Computers, he says, are a system out of equilibrium, and stochastic thermodynamics gives physicists a way to study non-equilibrium systems. “If you put those two together, it looked like all kinds of fireworks would come out, in the spirit of SFI,” he says.

In recent studies that laid the groundwork for this new paper, Wolpert and colleagues introduced the idea of ​​”discrepancy cost,” or a measure of how much the cost of computation exceeds the Landauer bound. Proposed in 1961 by physicist Rolf Landauer, this limit defines the minimum amount of heat required to change information in a computer. Knowing the cost of the mismatch, Wolpert says, can inform strategies to reduce a system’s overall energy costs.

On the other side of the Atlantic, co-authors Manzano and Roldán developed a tool from the mathematics of finance – martingale theory – to address the thermodynamic behavior of small fluctuating systems at standstill. Roldan et al. “Martingales for Physicists” by Dr. helped pave the way for successful applications of such a martingale approach in thermodynamics.

Wolpert, Cardes, Roldán, and Manzano extend these tools from stochastic thermodynamics to the calculation of mismatch costs to general computational problems in their PRX paper.

Taken together, their research points to a new path to finding the lowest energy required for computation in any system, no matter how it is implemented. “It opens up a huge new set of problems,” Wolpert says.

It could also have a very practical application, pointing to new ways to make computers more energy efficient. The National Science Foundation estimates that computers use between 5% and 9% of global energy generation, but at current growth rates this could reach 20% by 2030.

But previous work by SFI researchers suggests that modern computers are grossly inefficient: Biological systems, by contrast, are about 100,000 times more energy efficient than man-made computers. Wolpert says one of the main motivations for a general thermodynamic theory of computing is to find new ways to reduce the energy consumption of real-world machines.

For example, a better understanding of how algorithms and devices use energy to perform certain tasks can point to more efficient computer chip architectures. Currently, Wolpert says, there is no clear way to make physical chips that can perform computational tasks using less energy.

“These types of techniques can provide a flashlight in the darkness,” he says.