Electronic noise due to thermal vibration within a conductor
Johnson–Nyquist noise (thermal noise, Johnson noise, or Nyquist noise) is the electronic noise generated by the thermal agitation of the charge carriers (usually the electrons) inside an electrical conductor at equilibrium, which happens regardless of any applied voltage. Thermal noise is present in all electrical circuits, and in sensitive electronic equipment (such as radio receivers) can drown out weak signals, and can be the limiting factor on sensitivity of electrical measuring instruments. Thermal noise is proportional to absolute temperature, so some sensitive electronic equipment such as radio telescope receivers are cooled to cryogenic temperatures to improve their signal-to-noise ratio. The generic, statistical physical derivation of this noise is called the fluctuation-dissipation theorem, where generalized impedance or generalized susceptibility is used to characterize the medium.
Thermal noise in an ideal resistor is approximately white, meaning that its power spectral density is nearly constant throughout the frequency spectrum (Figure 2). When limited to a finite bandwidth and viewed in the time domain (as sketched in Figure 1), thermal noise has a nearly Gaussian amplitude distribution. 
For the general case, this definition applies to charge carriers in any type of conducting medium (e.g. ions in an electrolyte), not just resistors. Thermal noise is distinct from shot noise, which consists of additional current fluctuations that occur when a voltage is applied and a macroscopic current starts to flow.
History of thermal noise.
In 1905, in one of Albert Einstein's "Annus mirabilis" papers the theory of Brownian motion was first solved in terms of thermal fluctuations. The following year, in a second paper about Brownian motion, Einstein suggested that the same phenomena could be applied to derive thermally-agitated currents, but did not carry out the calculation as he considered it to be untestable.
Geertruida de Haas-Lorentz, daughter of Hendrik Lorentz, in her doctoral thesis of 1912, expanded on Einstein stochastic theory and first applied it to the study of electrons, deriving a formula for the mean-squared value of the thermal current.
Walter H. Schottky studied the problem in 1918, while studying thermal noise using Einstein's theories, experimentally discovered another kind of noise, the shot noise.
Frits Zernike working in electrical metrology, found unusual random deflections while working with high-sensitive galvanometers. He rejected the idea that the noise was mechanical, and concluded that it was of thermal nature. In 1927, he introduced the idea of autocorrelations to electrical measurements and calculated the time detection limit. His work coincided with de Haas-Lorentz' prediction.
The same year, working independently without any knowledge of Zernike's work, John B. Johnson working in Bell Labs found the same kind of noise in communication systems, but described it in terms of frequencies. He described his findings to Harry Nyquist, also at Bell Labs, who used principles of thermodynamics and statistical mechanics to explain the results, published in 1928.
Noise of ideal resistors for moderate frequencies.
Johnson's experiment (Figure 1) found that the thermal noise from a resistance formula_1 at kelvin temperature formula_2 and bandlimited to a frequency band of bandwidth formula_3 (Figure 3) has a mean square voltage of:
 formula_4
where formula_5 is the Boltzmann constant ( joules per kelvin). While this equation applies to "ideal resistors" (i.e. pure resistances without any frequency-dependence) at non-extreme frequency and temperatures, a more accurate general form accounts for complex impedances and quantum effects. Conventional electronics generally operate over a more limited bandwidth, so Johnson's equation is often satisfactory.
Power spectral density.
The mean square voltage per hertz of bandwidth is formula_6 and may be called the power spectral density (Figure 2). Its square root at room temperature (around 300 K) approximates to 0.13 formula_7 in units of . A 10 kΩ resistor, for example, would have approximately 13  at room temperature.
RMS noise voltage.
The square root of the mean square voltage yields the root mean square (RMS) voltage observed over the bandwidth formula_3:
 formula_9
A resistor with thermal noise can be represented by its Thévenin equivalent circuit (Figure 4B) consisting of a noiseless resistor in series with a gaussian noise voltage source with the above RMS voltage.
Around room temperature, 3 kΩ provides almost one microvolt of RMS noise over 20 kHz (the human hearing range) and 60 Ω·Hz for formula_10 corresponds to almost one nanovolt of RMS noise.
RMS noise current.
A resistor with thermal noise can also be converted into its Norton equivalent circuit (Figure 4C) consisting of a noise-free resistor in parallel with a gaussian noise current source with the following RMS current:
 formula_11 in dBm. At room temperature (300 K), the available noise power can be easily approximated as formula_12 in dBm for a bandwidth in hertz. Some example available noise power in dBm are tabulated below:
Nyquist's derivation of ideal resistor noise.
Nyquist's 1928 paper "Thermal Agitation of Electric Charge in Conductors" used concepts about potential energy and harmonic oscillators from the equipartition law of Boltzmann and Maxwell to explain Johnson's experimental result. Nyquist's thought experiment summed the energy contribution of each standing wave mode of oscillation on a long lossless transmission line between two equal resistors (formula_13). According to the conclusion of Figure 5, the total average power transferred over bandwidth formula_3 from formula_15 and absorbed by formula_16 was determined to be:
 formula_17
Simple application of Ohm's law says the current from formula_18 (the thermal voltage noise of only formula_15) through the combined resistance is formula_20, so the power transferred from formula_15 to formula_16 is the square of this current multiplied by formula_16, which simplifies to:
 formula_24
Setting this formula_25 equal to the earlier average power expression formula_26 allows solving for the average of formula_27 over that bandwidth:
 formula_28
Nyquist used similar reasoning to provide a generalized expression that applies to non-equal and complex impedances too. And while Nyquist above used formula_29 according to classical theory, Nyquist concluded his paper by attempting to use a more involved expression that incorporated the Planck constant formula_30 (from the new theory of quantum mechanics).
Generalized forms.
The formula_6 voltage noise described above is a special case for a purely resistive component for low to moderate frequencies. In general, the thermal electrical noise continues to be related to resistive response in many more generalized electrical cases, as a consequence of the fluctuation-dissipation theorem. Below a variety of generalizations are noted. All of these generalizations share a common limitation, that they only apply in cases where the electrical component under consideration is purely passive and linear.
Complex impedances.
Nyquist's original paper also provided the generalized noise for components having partly reactive response, e.g., sources that contain capacitors or inductors. Such a component can be described by a frequency-dependent complex electrical impedance formula_32. The formula for the power spectral density of the series noise voltage is
formula_33
The function formula_34 is approximately 1, except at very high frequencies or near absolute zero (see below).
The real part of impedance, formula_35, is in general frequency dependent and so the Johnson–Nyquist noise is not white noise. The RMS noise voltage over a span of frequencies formula_36 to formula_37 can be found by taking the square root of integration of the power spectral density:
formula_38.
Alternatively, a parallel noise current can be used to describe Johnson noise, its power spectral density being
formula_39
where formula_40 is the electrical admittance; note that formula_41
Quantum effects at high frequencies or low temperatures.
With proper consideration of quantum effects (which are relevant for very high frequencies or very low temperatures near absolute zero), the multiplying factor formula_34 mentioned earlier is in general given by:
formula_43
At very high frequencies (formula_44), the function formula_34 starts to exponentially decrease to zero. At room temperature this transition occurs in the terahertz, far beyond the capabilities of conventional electronics, and so it is valid to set formula_46 for conventional electronics work.
Relation to Planck's law.
Nyquist's formula is essentially the same as that derived by Planck in 1901 for electromagnetic radiation of a blackbody in one dimension—i.e., it is the one-dimensional version of Planck's law of blackbody radiation. In other words, a hot resistor will create electromagnetic waves on a transmission line just as a hot object will create electromagnetic waves in free space.
In 1946, Robert H. Dicke elaborated on the relationship, and further connected it to properties of antennas, particularly the fact that the average antenna aperture over all different directions cannot be larger than formula_47, where λ is wavelength. This comes from the different frequency dependence of 3D versus 1D Planck's law.
Multiport electrical networks.
Richard Q. Twiss extended Nyquist's formulas to multi-port passive electrical networks, including non-reciprocal devices such as circulators and isolators. 
Thermal noise appears at every port, and can be described as random series voltage sources in series with each port. The random voltages at different ports may be correlated, and their amplitudes and correlations are fully described by a set of cross-spectral density functions relating the different noise voltages,
 formula_48
where the formula_49 are the elements of the impedance matrix formula_50.
Again, an alternative description of the noise is instead in terms of parallel current sources applied at each port. Their cross-spectral density is given by
 formula_51
where formula_52 is the admittance matrix.
Notes.
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References.
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