The Kelvin Limit

I have been a small part of the chorus of voices critiquing the AI bubble that I described in Portents Of Doom. But what if we're wrong? What if the overwhelming demand for AI sin't the result of the AI platforms massively subsidizing their products, but because the world needs more and more non-consensual sex images, slop web pages, agentic ransomware attacks, students cheating on exams, hallucinated lawsuits, and all the other benefits of this transformative technology?

Please suspend disbelief and follow me below the fold as I look into a fascinating examination of the implications of the exponential growth in the data centers needed to provide these benefits.

The Future's So Bright, I Gotta Wear Shades

Timbuk3

Not that I ever remember listening to Timbuk3 song, but I distinctly remember, shortly before the Black Monday stock market crash, Scott McNealy celebrating Sun's exponential growth with his version of the title.

Back in 2011, I used Future's So Bright, We Gotta Wear Shades as the title of a post skeptical of the exponential growth behind Moore's and Kryder's Laws. I cited UCSD Prof. Tom Murphy's estimate that, at a 0.023 yr^-1 growth rate the earth would emit as much energy as the sun in 3410.

Nachtrieb & Smith Fig. 2

Robert T. Nachtrieb and Steven J. Smith's AI Hastens Limits to Exponential Growth is a fascinating exploration of the long-term effects of exponential growth. They point out that:

While AI electricity consumption was only 1.5% of the global total in 2024, its power demand has grown at a rate of 0.127 yr⁻¹ since 2015, accelerating to 0.15 yr⁻¹ over the last five years. Projecting from this 2024 baseline, AI’s electricity demand is on track to achieve parity with the rest of the world’s combined consumption by approximately 2050.

All systems that grow exponentially end up running into limits that prevent further growth. Nachtrieb and Smith identify five such limits to the growth of AI data centers:

1. Non-renewable resource depletion
2. The Kelvin Limit
3. The renewable resource limit
4. The Dyson Limit
5. The Asimov Limit

Each of these limits is interesting, but here I only look into my favorite, the Kelvin Limit. They define this limit thus:

Even with an infinite energy source, the laws of physics dictate every unit of energy used eventually becomes waste heat. On a planetary scale, this heat must be radiated into space to maintain a stable environment.

Earth stays at a life-sustaining temperature by balancing received solar energy with infrared radiation emitted back into the cosmos. Human energy consumption from “terrestrial” sources, such as nuclear fusion or fossil fuels, adds “new” heat to this balance. Because this energy was not already part of the solar-to-earth flow, the planet must reach a higher temperature to increase its radiative cooling capacity and shed the additional load.

The “Kelvin Limit” defines the point where this added waste heat pushes Earth’s surface temperature to 373 K (100 ◦C), the boiling point of water. At this threshold, the planet becomes physically uninhabitable.

They build a model to predict when the Kelvin Limit would be reached:

The solar power arriving at the Earth’s cross-section is Lα ≈ 5474 × 10³ EJ yr⁻¹. A portion of this energy is immediately reflected into space by the Earth’s albedo (a), which represents the planet’s reflectivity. Based on NASA data, Earth’s albedo is approximately 0.30, meaning 30% of incoming light reflects away while the remaining 70% is absorbed as heat (P_a).

The model assumes an initial equilibrium where absorbed solar power (P_a) equals the power radiated at the baseline temperature (T₀).
...
The limit occurs at time t₂, when the combined heat of the Sun (P_a) and human energy demand (D) requires the Earth to reach the boiling point (T₂) to maintain equilibrium.
...
This thermal wall represents a hard physical limit. Technological efficiency cannot bypass it; higher energy use to drive AI or industry simply accelerates the transition toward this planetary boiling point.

Table VIII

For each of their cases they compute k, the rate coefficient. The time to hit the limit is t = k/r, where r is the rate of increase of demand. For example, r over the last five years is 0.15. Their results are summarized in Table Viii, showing that in the Kelvin case k is 10.0 for all r. Note that the renewable-only case is the only case where k is less than the Kelvin case, and only by 9%. This demonstrates the very fundamental nature of the Kelvin Limit.

Table IX

Table IX summarizes their resulting t for each case for their ranges of r and k values. For the Kelvin case at 0.15, the recent value of r, they write:

These figures represent a fundamental shift in the prospects of civilization under AI. Under the 15% growth rates currently demonstrated by AI infrastructure, we quickly accelerate past all projected limits. The thermal “Kelvin Limit” (k ≈ 10), which would normally take ten centuries to reach, suddenly appears in just 67 yr, well within a single human lifetime.

This 67 year estimate is, of course, an upper bound. The Earth becomes uninhabitable for humans long before the surface reaches 373 Kelvin. As we see in Texas, the current r = 0.15 is not actually fueled by renewables, and is thus contributing to much faster heating. As I understand it, their demand D is just the demand for running the data centers. At r = 0.15 there is significant extra demand for building 15% more data centers and 15% more Nvidia racks each year than the previous year.

Nachtrieb and Smith's Figure 2 above estimates that at r = 0.15 running the data centers would take half of the "world’s combined consumption by approximately 2050". In 2025 the Gross World Product according to the IMF was:

forecast to be around $208.96 trillion, $11.04 trillion up compared to $197.91 trillion in 2024.

Assuming this 5.6% growth rate continued, in 2050, GWP would be 3.7 times higher at $773T. Presumably, half of this would be generated by the data centers, or about $387T. Thanks to the economic mechanism described by W. Brian Arthur in Increasing Returns and Path Dependence in the Economy, it is likely that only one company would dominate the AI market. At 20 times earnings, it would be worth around $7.7 quadrillion.

I think you can understand why investors are pouring money into AI companies.