Derivation of Nyquist 4KTBR Relation using Boltzmann 1/2KT Equipartition Theorem

Steps

• Derive the value of the square of the voltage on the capacitor
• Assume the noise voltage on the resistor can be faked up and simulated by an equivalent voltage source driving the circuit with a noiseless resistor.  The ideal source is assumed to have uniform power spectral density.
• Equate the value of the voltage on the capacitor with that given by Boltzmann's equipartition theorem and try not to get hanged in the mean time.

Equipartition Theorem: Each orthogonal degree of freedom gets 1/2KT of energy.  This can be thought of as maximum entropy using uniform distribution over each axis. Look for the quadratic energy terms.  With a capacitor all you have is:  

Now the circuit analysis

P.S.D. == Power Spectral Density

PSD(Capacitor Voltage) = PSD(Ideal Voltage Source) * ( Frequency response of the RC circuit )    …….this is in the frequency domain

Total energy on the capacitor is the integral [0,infinity]

     The integral of this is arctan()      

The value on the left hand side is the previous value we deduced using Boltzmann:

Which yields the relation between noise voltage, bandwidth and resistance.

Research Links

• Wikipedia: Equipartition Theorem
• Equipartition Theorem: Valance
• Several different approaches to deriving the noise relation including the one taken by Nyquist  – Nyquist approach includes black body radiation concept.
• A brief history of the events leading up to the discovery of thermal noise