Thermal Noise

Acknowledgement:
Most of materials are excerpted from Jim hellums's short note and van der Ziel's book.

Brownian motion is created by the impact of the molecules of the fluid surrounding the object due to the fluctuations in the ambient thermal energy(i.e., temperature)

Let's first find it out about equipartition theorem...because I have heard many times from other physics folks including my professor.

The equipartition theorem is a principle of calssical (non-quantum) statistical mechanics which states that the internal energy of a system composed of a large number of particles at thermal equilibrium will distribute itself evenly among each of the quadratic degree of freedom allowed to the particles of the system. (from Wikipedia). The theorem says that the mean internal energy associated with each degree of freedom of a monatomic (single atom) ideal gas is the same.
The equation is ...
(1/2)M*vx^2 = (1/2)KT
where M: the mass of the particle
vx: instantaneous veolcity component in the x-direction
K: Boltzmann's constant
T: absolute temperature

Einstein showed that
xd^2 = 2Dt
where xd: displacement of x in the x-direction
D: diffusion constant of the particle

The termal noise in circuits is nothing more than Brownian motion of electrons due to the ambient temperature.

***** test *****

\int_{-\infty}^{\infty}e^{-x^{2}}\;dx=\sqrt{\pi}

Later Johnson (experimental physicist) and Nyquist (theorectician) in Bell labs found the termal agitation of electricity in conductors produces a random variation in the potential between the ends of the conductors. The electromotive force developed across the end of the conductor due to Thermal noise is unaffected by the presence or absence of direct current. This can be explained by the fact that electron thermal velocities in a conductor are much greater (~10^3 times) than electron drift veolocities. The amplitude distribution of Thermal noise is Gaussian in three dimensions (Central limit theorem), which can be illustrated by a random walk process.

1) van der Ziel's Derivation of thermal noise
Consider a resistor R in parallel with a capacitor C. As a result of the random thermal agitation of the electrons in the resistor, the capacitor will be charged and discharged at random. The average energy stored in the capacitor will be
(1/2)CV^2 = (1/2)KT
or
V^2 = KT/C
where V^2: the mean square value of the voltage fluctuation empressed across the capacitor.



.....INCOMPLETE!