Johnson-Nyquist noise
Johnson-Nyquist noise, also called thermal noise or Johnson noise, is the random voltage that appears across a conductor at any temperature above absolute zero, whether or not a current flows through it. In a frequency band of width B, the mean square voltage across a resistance R at temperature T is 4kBTRB, where kB is the Boltzmann constant. The result does not depend on the material or the shape of the resistor. J. B. Johnson measured the noise at Bell Telephone Laboratories, his colleague Harry Nyquist worked out the theory, and the two papers appeared one after the other in the July 1928 issue of Physical Review.[1][2]
The noise sets the smallest signal that a measurement through a resistor can see. A radio receiver at 290 K cannot sense power below 4.0 × 10−21 W per hertz of bandwidth, which is −174 dBm/Hz. A feedback resistor of 1 MΩ in a transimpedance amplifier adds 12.8 pA rms to a current measured in a band of 10 kHz, and choosing that resistor is the first decision in the design of a nanopore amplifier.
History
Johnson's paper opens with the statement that "statistical fluctuation of electric charge exists in all conductors". He measured it with a vacuum-tube amplifier and a thermocouple, and the value of Boltzmann's constant that his data gave lay near the accepted one.[1] Nyquist, who worked in the same laboratory, supplied the theory. His version already contained the quantum correction, which the frequencies of Johnson's apparatus were too low to need.[2]
Nyquist published a second paper that year, on how fast signals can be sent through a telegraph channel. Its conclusion, that a channel of bandwidth B can carry at most 2B independent signal elements a second, is the origin of the Nyquist rate in sampling.[3]
The noise is the electrical counterpart of Brownian motion. Einstein's treatment of that motion in 1905 tied the diffusion of a suspended particle to the friction on it, and Nyquist's formula ties the noise of a conductor to its resistance in the same way.[4] In 1951 Callen and Welton proved that both are cases of a general relation between the spontaneous fluctuations of a system in equilibrium and the energy it dissipates when driven, now called the fluctuation-dissipation theorem.[5]
Derivation
Nyquist took two resistors of equal resistance R at the same temperature and joined them with a lossless transmission line matched to both. Each resistor sends power down the line and absorbs power from the other, and at equilibrium the two flows are equal. The line carries standing waves, and in the classical limit each mode holds a mean energy of kBT, so the power one resistor can deliver in a band B is kBTB. A source of open-circuit voltage v and resistance R delivers v2/4R to a matched load, and setting that equal to kBTB gives the result.[2]
Formula
For a band B the mean square voltage and current are
Per root hertz the voltage density is √(4kBTR) and the current density √(4kBT/R). At 25 °C the product 4kBT is 1.65 × 10−20 J, so a resistor of 1 kΩ produces 4.06 nV/√Hz and one of 50 Ω produces 0.91 nV/√Hz. Voltage noise grows with resistance and current noise falls with it. A large resistance is therefore quiet as a path for current and loud as a source of voltage. The spectrum is flat, the amplitude Gaussian, and the noise is the same whatever current the resistor carries.
Limits of the formula
The classical result holds while the photon energy hf is much smaller than kBT. Nyquist's full expression replaces kBT with hf/(ehf/kBT − 1). At 300 K it is 8% below the classical value at 1 THz and falls steeply beyond kBT/h, which is 6.25 THz. At 4.2 K the same corner is at 88 GHz.[2] No nanopore amplifier works at either frequency.
The formula applies to the resistive component of an impedance. A capacitor has none and makes no thermal noise of its own. A resistor shunted by a capacitor C shows a total noise voltage of √(kBT/C) across the pair, whatever the resistance; the resistance only decides how that noise is spread over frequency.[a]
Johnson noise thermometry
Because the noise depends only on resistance and temperature, it can be used to measure temperature or the constant itself. In a determination published in 2017, a 100 Ω resistor in a cell at the triple point of water was compared for 100 days with a synthesized noise source of known power. The result, kB = 1.380 649 7(37) × 10−23 J/K, carried a relative standard uncertainty of 2.7 parts per million.[6] Since May 20, 2019, the constant has been fixed at exactly 1.380 649 × 10−23 J/K, and the kelvin is defined through it.[7]
In nanopore instruments
A current-to-voltage converter with feedback resistor Rf turns a current I into a voltage IRf, and the thermal noise of the resistor reaches the input as a current of density √(4kBT/Rf). For 1 MΩ at 25 °C, 4kBT/Rf is 1.65 × 10−26 A²/Hz. The square root is 128 fA/√Hz, and a band of 10 kHz multiplies it by √10,000 = 100, giving 12.8 pA rms.[b] For 1 GΩ the density is 4.06 fA/√Hz and the band gives 0.41 pA. The two differ by a factor of √1000, about 32. The larger resistor is quieter, and it also clips at a smaller current, because the output of the amplifier cannot swing past its supply.
The electrolyte in the pore is a conductor too. A pore that conducts 30 nS adds 22 fA/√Hz, or 2.2 pA rms in 10 kHz. A seal of 1 GΩ between a pipette and a cell membrane contributes the same 0.41 pA as the resistor, which is why single-channel recording waited for the giga-seal.[8]
The open design of the MR1 Nanopore Reader documents a noise floor below 1.2 pA rms in 10 kHz, and its amplifier schematic draws a feedback resistor of 1 MΩ, whose own thermal noise is 10.7 times larger than that floor.[9] The repository's status line reads "Designed. Simulated. Built and measured: not yet."[10] The two descriptions of the front end are set side by side in the article on the transimpedance amplifier.
See also
Notes
- ^ For C = 2 pF at 25 °C the voltage is 45 µV rms. The feedback network drawn in the schematic of the MR1 Nanopore Reader is 1 MΩ in parallel with 2 pF, whose noise bandwidth is 125 kHz; 128 fA/√Hz over that band gives 45 pA at the input, which is the same 45 µV divided by 1 MΩ.
- ^ At 300 K the same calculation gives 12.9 pA, the figure used in the articles on Nanopore noise and the Patch clamp.
References
- ^ a b Johnson, J. B. (1928). "Thermal agitation of electricity in conductors". Physical Review 32 (1): 97-109. doi:10.1103/PhysRev.32.97.
- ^ a b c d Nyquist, H. (1928). "Thermal agitation of electric charge in conductors". Physical Review 32 (1): 110-113. doi:10.1103/PhysRev.32.110.
- ^ Nyquist, H. (1928). "Certain topics in telegraph transmission theory". Transactions of the American Institute of Electrical Engineers 47 (2): 617-644. doi:10.1109/T-AIEE.1928.5055024.
- ^ Einstein, A. (1905). "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen". Annalen der Physik 322 (8): 549-560. doi:10.1002/andp.19053220806.
- ^ Callen, H. B.; Welton, T. A. (1951). "Irreversibility and generalized noise". Physical Review 83 (1): 34-40. doi:10.1103/PhysRev.83.34.
- ^ Qu, J. F.; et al. (2017). "An improved electronic determination of the Boltzmann constant by Johnson noise thermometry". Metrologia 54 (4): 549-558. doi:10.1088/1681-7575/aa781e.
- ^ Tiesinga, E.; et al. (2021). "CODATA recommended values of the fundamental physical constants: 2018". Reviews of Modern Physics 93 (2): 025010. doi:10.1103/RevModPhys.93.025010.
- ^ Hamill, O. P.; et al. (1981). "Improved patch-clamp techniques for high-resolution current recording from cells and cell-free membrane patches". Pflügers Archiv 391 (2): 85-100. doi:10.1007/BF00656997.
- ^ "High-Speed Streaming DAQ & External Interfacing Specification". GitHub. Molecular Reality Corporation. October 2026.
- ^ "MR1 Nanopore Reader". GitHub. October 16, 2026.