Abstract
The trajectory of a Brownian particle is an erratic curve with the characteristic feature that the observed distance in a given time interval \( \Delta t \) depends on the magnification of the microscope (Fig. 6.1). Thus one cannot differentiate this distance unambiguously with respect to time to obtain a velocity. Instead we have to focus on the displacement of the particle, defined as the shortest distance between two positions of the colloid. How the squared displacement by diffusion grows in time was first figured out by Einstein.
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Notes
- 1.
A. Einstein, Elementare Theorie der Brownschen Bewegung, Z. für Electrochemie 14 (1908), 253–239. Quotations in this Chapter are English translations in A. Einstein, Investigations on the Theory of the Brownian Motion, (Ed. by R. Fürth, Dover 1956).
- 2.
Subtitles are from Einstein, in the paper in note 1.
- 3.
Here Einstein refers to collective or gradient diffusion that equalizes concentration differences.
- 4.
Here and later […] indicates that in a quotation text has been skipped.
- 5.
The diffusion coefficient is defined as the ratio of flux to gradient, see also Sect. 5.2.
- 6.
A gradient, it should be noted, that is of unspecified magnitude.
- 7.
A notable exception is the proton which diffuses much faster.
- 8.
A. Einstein, Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, Ann. D. Physik 17, (1905), pp. 549–560.
- 9.
The derivation is in essence that of Einstein’s 1905 paper; one difference is that we do not invoke the explicit solution of the diffusion equation, which simplifies the treatment.
- 10.
M. P. Langevin, Sur la theorie du mouven Brownien, C. R. Acad. Sci. (Paris) 146, 530–533 (1908).
- 11.
A sequence of observations in time on a single particle will yield the same averages.
- 12.
Make use of the Taylor expansion \( \tau_{\text{m}} (1 - \exp [ - t/\tau_{\text{m}} ] ) ] { } = \, t + (1/2)t^{2} / { }\tau_{\text{m}} + \ldots \)
References
An English translation of Einstein’s papers on Brownian motion can be found in: A. Einstein, Investigations on the Theory of the Brownian Motion, (Ed. by R. Fürth, Dover 1956).
Indispensable scientific biography of Albert Einstein, including an elucidating analysis of Einstein’s papers on Brownian motion: A. Pais, Subtle is the Lord (Oxford University Press, 1982).
More extensive treatments of Brownian motion and other transport properties of colloids can be found in: W. B. Russel, D. A. Saville and W. R. Schowalter, Colloidal Dispersions (Cambridge, 1995), and J. K. G. Dhont, An Introduction to Dynamics of Colloids (Elsevier, Amsterdam, 1996).
For an English translation of Langevin’s 1908 paper see: D. S. Lemons and A. Gythiel Am. J. Phys. 65 (11), November 1997.
Brownian motion in its wider context of stochastic processes is treated in: N. G. van Kampen, Stochastic Processes in Physics and Chemistry, (North Holland, 1981) and S. Chandrasekhar, Stochastic Problems in Physics and Astronomy¸ Rev. Mod. Phys. 15, 1 (1993).
Rotational diffusion is treated by Peter Debye in his classic Polar Molecules (Dover Publication, reprint of the 1929 Reinhold edition).
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Philipse, A.P. (2018). Brownian Displacements. In: Brownian Motion. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-98053-9_6
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