page_title: Fluctuation-dissipation theorem
section_title: Thermal noise in a resistor
hierarchical_section_title: Fluctuation-dissipation theorem / Examples in detail / Thermal noise in a resistor
context_page_description: The fluctuation–dissipation theorem or fluctuation–dissipation relation is a powerful tool in statistical physics for predicting the behavior of systems that obey detailed balance. Given that a system obeys detailed balance, the theorem is a general proof that thermodynamic fluctuations in a physical variable predict the response quantified by the admittance or impedance of the same physical variable, and vice versa. The fluctuation–dissipation theorem applies both to classical and quantum mechanical systems.
  The fluctuation–dissipation theorem was proven by Herbert Callen and Theodore Welton in 1951
  and expanded by Ryogo Kubo. There are antecedents to the general theorem, including Einstein's explanation of Brownian motion
  during his annus mirabilis and Harry Nyquist's explanation in 1928 of Johnson noise in electrical resistors.
context_section_description: In 1928, John B. Johnson discovered and Harry Nyquist explained Johnson–Nyquist noise. With no applied current, the mean-square voltage depends on the resistance , , and the bandwidth  over which the voltage is measured :
  This observation can be understood through the lens of the fluctuation-dissipation theorem. Take, for example, a simple circuit consisting of a resistor with a resistance  and a capacitor with a small capacitance . Kirchhoff's law yields
  and so the response function for this circuit is
  In the low-frequency limit , its imaginary part is simply
  which then can be linked to the auto-correlation function  of the voltage via the fluctuation-dissipation theorem
  The Johnson-Nyquist voltage noise  was observed within a small frequency bandwidth  centered around . Hence
caption: A simple circuit for illustrating Johnson-Nyquist  thermal noise in a resistor.
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