Use partition noise to learn beyond conductance

For noninteracting elastic transport at zero temperature, occupied incoming states inside the bias window provide the available attempts. Fermi statistics suppress the occupation fluctuations that an unconstrained classical Poisson arrival model would assign to these filled states. A channel with transmission T still randomly partitions each attempt, giving a variance proportional to $T(1-T)$. A closed channel contributes neither current nor partition noise. A perfectly transmitting channel carries current but has no zero-temperature partition noise in this model.

For spin-degenerate orbital channels, conductance is $G_0\sum_nT_n$ with $G_0=2e^2/h$. Noise weights the same channels by $T_n(1-T_n)$, so spin must follow the same counting convention in both sums. If transmissions mix modes, the relevant numbers are the eigenvalues of the transmission-probability matrix, rather than arbitrary diagonal amplitudes in a chosen basis. The ratio of these two sums is the Fano factor, a dimensionless measure of noise relative to the Poisson reference.

A noise spectral density is not a variance in amperes squared until integrated over a measurement bandwidth with its specified convention. Here a one-sided low-frequency spectrum is used, so a Poisson stream of elementary charges has $S_I=2e|I|$. Using a two-sided convention changes factors of two. A quoted numerical noise value without its bandwidth or spectral units can therefore be misleading even when its current and Fano factor are correct. This convention should also match the thermal-noise formula used for comparison.

At finite temperature, a perfectly transmitting conductor still has equilibrium Johnson–Nyquist noise $4k_BTG$ in the classical low-frequency one-sided limit. The simple zero-temperature partition expression is most useful when the bias energy greatly exceeds thermal energy, with negligible heating and otherwise valid scattering assumptions. In the crossover region, the full Fermi-occupation integrals combine thermal and partition contributions. Subtracting an equilibrium background and calling the remainder excess noise does not make every finite-temperature expression equal to its zero-temperature limit.

Two channels with transmissions 0.8 and 0.2 have the same conductance sum as channels with transmissions one and zero. The former partition both streams and make noise; the latter transmit one stream deterministically and reject the other. Thus conductance alone cannot distinguish their channel composition. Noise supplies a second moment-like constraint, but many larger channel sets can still share both sums. Recovering every transmission eigenvalue generally requires additional information, assumptions, or higher current cumulants.

When all transmissions are small, $T(1-T)$ is approximately T, giving nearly Poissonian F near one for weak independent tunneling. Strongly transmitting channels suppress noise. Interactions and correlated transfer can produce super-Poissonian noise, so an observed F above one is evidence against this simple partition model rather than a violation of charge quantization. Coulomb blockade, dynamical trapping, and avalanche processes each require their own transport description; the elementary formula cannot identify one uniquely from a single excess-noise ratio.