Aksayskiy Vladimir
Landauer's principle as a consequence of the Shannon-Hartley theorem or communication and information in the world of black bodies
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Communication and information in the world of black bodies. The message, in contrast to the
information, obeys the Landauer principle, according to which any generator, transmitter, and
receiver of a message is a source of thermal radiation, which in the case of electromagnetic
communication is conveniently considered in terms of the Planck black body model. In this
case, the Shannon - Hartley theorem leads to the well-known formulation of the Landauer
principle.
According to the Shannon - Hartley theorem [1], the data transmission rate over
a communication channel or channel capacity is:
C = B*log_2(1 + S/N) , (1)
where
C is the channel capacity in bits per second, a theoretical upper bound on the net bit rate;
B is the bandwidth of the channel in hertz;
S is the average received signal power, watts (or volts squared);
N is the average power of the noise, watts (or volts squared);
S/N is the signal-to-noise ratio (SNR).
Wien's displacement law [2] allows us to represent the bandwidth in the form of:
B = f_max = (k*T*a)/h , (2)
where
f_max is the frequency corresponding to the maximum of the black body radiation curve, Hz;
k is the Boltzmann constant, J/K;
h is the Planck constant, J*s;
T is temperature, K;
a=2.821439 is a constant.
Substituting (2) into (1) and multiplying both sides of the equation by the Planck constant,
we get:
W = k*T*a*log_2(1 + SNR) , (3)
where
W - energy of a bit in the communication channel, J.
Taking in (3) SNR = 0.1856, we obtain the well-known expression of the Landauer principle:
W = k*T*ln(2) , (4)
Planck's blackbody model is based on the Bose - Einstein distribution, but in some cases it is
convenient to use the Fermi - Dirac distribution.
Links
1_ Shannon-Hartley theorem https://en.wikipedia.org/wiki/Shannon%E2%80%93Hartley_theorem
2_ Wien's displacement law https://en.wikipedia.org/wiki/Wien%27s_displacement_law
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