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Equation 4 · After Moore: The Adaptive Radiation of Compute

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kBTln⁡2=(1.380649×10−23 JK)(300 K)(0.6931)≈2.87×10−21 J≈2.9 zJk_B T \ln 2 = \left(1.380649\times10^{-23}\ \tfrac{\text{J}}{\text{K}}\right)(300\ \text{K})(0.6931) \approx 2.87\times10^{-21}\ \text{J} \approx 2.9\ \text{zJ}

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kBk_B

Symbol k_B

kBk_B is part of the quantity the equation computes from the expression on the right.

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TT

Symbol T

the absolute temperature.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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≈

≈

Approximately equal to; the equality is not exact.

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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superscript

superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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where kBk_B is Boltzmann’s constant. At room temperature — 300 kelvin, a reasonable stand-in for a chip’s operating temperature — plugging in Boltzmann’s constant’s exact SI value gives: kBTln⁡2=(1.380649×10−23 JK)(300 K)(0.6931)≈2.87×10−21 J≈2.9 zJk_B T \ln 2 = \left(1.380649\times10^{-23}\ \tfrac{\text{J}}{\text{K}}\right)(300\ \text{K})(0.6931) \approx 2.87\times10^{-21}\ \text{J} \approx 2.9\ \text{zJ}. That bound is not a rough estimate awaiting refinement; it has been directly tested. A 2012 experiment reported in Nature, using a colloidal silica bead trapped in an optical potential rather than an electronic circuit, measured the heat dissipated while erasing one bit of information stored in the bead’s position and found it consistent with the Landauer bound within experimental uncertainty in the low-erasure-rate limit, the first experiment to test the bound this directly at the…
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where kBk_B is Boltzmann’s constant. At room temperature — 300 kelvin, a reasonable stand-in for a chip’s operating temperature — plugging in Boltzmann’s constant’s exact SI value gives: kBTln⁡2=(1.380649×10−23 JK)(300 K)(0.6931)≈2.87×10−21 J≈2.9 zJk_B T \ln 2 = \left(1.380649\times10^{-23}\ \tfrac{\text{J}}{\text{K}}\right)(300\ \text{K})(0.6931) \approx 2.87\times10^{-21}\ \text{J} \approx 2.9\ \text{zJ}. That bound is not a rough estimate awaiting refinement; it has been directly tested. A 2012 experiment reported in Nature, using a colloidal silica bead trapped in an optical potential rather than an electronic circuit, measured the heat dissipated while erasing one bit of information stored in the bead’s position and found it consistent with the Landauer bound within experimental uncertainty in the low-erasure-rate limit, the first experiment to test the bound this directly at the single-bit scale [ 2 ] . The bound is real, it has been measured, and every irreversible logic technology — which is to say every lineage covered in this article except the one this section is actually about — has to dissipate at least that much heat every time it erases a bit, no matter how exotic its switching mechanism is.

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