I am skeptical that steady state direct current flow attenuation is the entirety of the story (and indeed it seems to underestimate actual coax cable wire energy of ~1e^-21 to 5e^-21 J/bit/nm by a few OOM).

For coax cable the transmission is through a transverse (AC) wave that must accelerate a quantity of electrons linearly proportional to the length of the cable. These electrons rather rapidly dissipate this additional drift velocity energy through collisions (resistance), and the entirety of the wave energy is ultimately dissipated.

This seems different than sending continuous DC power through the wire where the electrons have a steady state drift velocity and the only energy required is that to maintain the drift velocity against resistance. For wave propagation the electrons are instead accelerated up from a drift velocity of zero for each bit sent. It's the difference between the energy required to accelerate a car up to cruising speed and the power required to maintain that speed against friction.

transverse wave

If we take the bit energy to be , then there is a natural EM wavelength of $E_{b} = \frac{h c}{λ}$ , so $λ = \frac{h c}{E_{B}}$ , which works out to ~1um for ~1eV. Notice that using a lower frequency / longer wavelength seems to allow one to arbitrarily decrease the bit energy distance scale, but it turns out this just increases the dissipative loss.

So an initial estimate of the characteristic bit energy distance scale here is ~1eV/bit/um or ~1e-22 J/bit/nm. But this is obviously an underestimate as it doesn't yet include the effect of resistance (and skin effect) during wave propagation.

The bit energy of one wavelength is implemented through electron peak drift velocity on order $E_{b} = \frac{1}{2} N_{e} m_{e} v_{d}^{2}$ , where $N_{e}$ is the number of carrier electrons in one wavelength wire section. The relaxation time $τ$ or mean time between thermal collisions with a room temp thermal velocity of around ~1e5 m/s and the mean free path of ~40 nm in copper is $τ$ ~ 4e-13s. Meanwhile the inverse frequency or timespan of one wavelength is around 3e-14 s for an optical frequency 1eV wave, and is ~1e-9 s for a more typical (much higher amplitude) gigahertz frequency wave. So it would seem that resistance is quite significant on these timescales.

Very roughly the gigahertz 1e-9s period wave requires about 5 oom more energy per wavelength due to dissipation which cancels out the 5 oom larger distance scale. Each wavelength section loses about half of the invested energy every $τ$ ~ 4e-13 seconds, so maintaining the bit energy of $E_{b}$ requires roughly input power of ~ $E_{b} / τ$ for $f^{- 1}$ seconds which cancels out the effect of the longer wavelength distance, resulting in a constant bit energy distance scale independent of wavelength/frequency (naturally there are many other complex effects that are wavelength/frequency dependent but they can’t improve the bit energy distance scale )

For a low frequency (long wavelength) with $f^{- 1}$ << $τ$ :

$E_{b} / d \approx \frac{E_{b} f^{- 1}}{τ λ} = \frac{E_{b}}{τ f λ}$

$λ = \frac{c}{f}$

$E_{b} / d \approx \frac{E_{b}}{τ f λ}$

$E_{b} / d \approx \frac{E_{b}}{τ c}$ ~ 1eV / 10um ~ 1e-23 J/bit/nm

If you take the bit energy down to the minimal landauer limit of ~0.01 eV this ends up about equivalent to your lower limit, but I don't think that would realistically propagate.

A real wave propagation probably can’t perfectly transfer the bit energy over longer distances and has other losses (dielectric loss, skin effect, etc), so vaguely guesstimating around 100x loss would result in ~1e-21 J/bit/nm. The skin effect alone perhaps increases resistance by roughly 10x at gigahertz frequencies. Coax devices also seem constrained to use specific lower gigahertz frequences and then boost the bitrate through analog encoding, so for example 10-bit analog increases bitrate by 10x at the same frequency but requires about 1024X more power, so that is 2 OOM less efficient per bit.

Notice that the basic energy distance scale of $\frac{E_{b}}{τ c}$ is derived from the mean free path, via the relaxation time $τ$ from $τ = ℓ / V_{n}$ , where $ℓ$ is the mean free path and $V_{n}$ is the thermal noise velocity (around ~1e5 m/s for room temp electrons).

Coax cable doesn't seem to have any fundamental advantage over waveguide optical, so I didn't consider it at all in brain efficiency. It requires wires of about the same width several OOM larger than minimal nanoscale RC interconnect and largish sending/receiving devices as in optics/photonics.

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Entropy and Temperature

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Entropy and Temperature