7. Friction and Fluctuations — Introduction to Statistical ...
https://web.stanford.edu/~peastman/.../friction.html
These two effects, fluctuations and dissipation, are inseparable from each other. ... This leads to the following equation, known as the Langevin equation: (2) ..... By using the equipartition theorem, we assumed the system was in equilibrium at
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The fluctuation-dissipation theorem
R. KUBO Department of Physics, University of Tokyo, Japan
Contents
1. Introduction . 2. Einstein relation . 3. Classical Langevin equation and the random force I 4. Generalized Langevin equation . 5. Linear response theory . 6. Correlations and correlation spectra . 7. The fluctuation-dissipation theorem . 8. Force correlations . 9. Correlation matrix formulation . 10. Moments, sum rules and continued fraction expansion 11. Density response, conduction and diffusion , References,
Page . 255 . 257 . 258 . 260 . 263 . 266 . 268 . 270 . 273 . 276 . 278 . 283
Abstract. The linear response theory has given a general proof of the fluctuation- dissipation theorem which states that the linear response of a given system to an external perturbation is expressed in terms of fluctuation properties of the system in thermal equilibrium. This theorem may be represented by a stochastic equation describing the fluctuation, which is a generalization of the familiar Langevin equation in the classical theory of Brownian motion. In this generalized equation the friction force becomes retarded or frequency-dependent and the random force is no more white. They are related to each other by a generalized Nyquist theorem which is in fact another expression of the fluctuation-dissipation theorem. This point of view can be applied to a wide class of irreversible process including collective modes in many-particle systems as has already been shown by Mori. As an illustrative example, the density response problem is briefly discussed.
1. Introduction Let us begin with a very simple example of the Brownian motion of colloidal particles floating in a liquid medium or of a small mirror suspended in a rarefied gas. Under an ultramicroscope one can observe irregular motion of colloidal particles or, by reflection of a light beam, an irregular oscillation of the mirror. Such a random motion of colloidal particles, or of the suspended mirror, is well known as direct evidence of thermal molecular motion, which is, of course, the very basis of the microscopic theory of the structure of matter, because the random force driving the particles or the mirror is apparently due to the impacts exerted by the liquid molecules or the gas molecules. These are classical examples of the Brownian motion which always exists, even in thermal equilibrium, as a fluctuation. Now suppose an external force is applied as a driving force. The Brownian particles, if they are charged, can be driven by an external electric field. The
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https://web.stanford.edu/~peastman/.../friction.html
These two effects, fluctuations and dissipation, are inseparable from each other. ... This leads to the following equation, known as the Langevin equation: (2) ..... By using the equipartition theorem, we assumed the system was in equilibrium at
Stanford University
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