  The heat channel is defined by a linear time-varying (LTV) filter with
additive white Gaussian noise (AWGN) at the filter output. The continuous-time
LTV filter is related to the heat kernel of the quantum mechanical harmonic
oscillator, so the name of the channel. The channel's capacity is given in
closed form by means of the Lambert W function. Also a waterfilling theorem in
the time-frequency plane for the capacity is derived. It relies on a specific
Szego theorem for which an essentially self-contained proof is provided.
Similarly, the rate distortion function for a related nonstationary source is
given in closed form and a (reverse) waterfilling theorem in the time-frequency
plane is derived. Finally, a second closed-form expression for the capacity of
the heat channel based on the detected perturbed filter output signals is
presented. In this context, a precise differential connection between channel
capacity and the normalized optimal detection error (NODE) is revealed. This
C-NODE relationship is compared with the well-known I-MMSE relationship
connecting mutual information with the minimum mean-square error (MMSE) of
estimation theory.
