Some remarks about the 1905 Brownian motion paper by Einstein

One of the important articles written by Albert Einstein in 1905 was a scientific paper entitled “On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat” , its original German title was Uber die von der molekularkinetischen Theorie der Wärme gefordete Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen“.

I will draw attention to a couple of typographical errors in the English translation of the paper , and I’ll present a brief  explanation and update of calculations in this paper.

For the English translation and pagination I’ll be referring to the English edition published by Dover Publications.

The first typographical error is found on page 8 of the English translation:

eq01en-aThe letter n should be an exponent ,as shown in the German edition of the paper:

eq01deThis is also obvious by analyzing the equation that follows on the same page which can be written as:


The second error is on page 14 of the English translation:

eq02en-bThe coefficient of the second integral should be ∂f/∂x instead of ∂x/∂f , as can be seen from the German version:

eq02deThere are notes related to the paper at the end of the English edition , however I’d like to add a brief note about the last two equations on page 5.

The entropy is expressed as:

entropy-eqThen the (Helmholtz) free energy is given by:


In a more modern notation ,these equations represent the relationships between the entropy  S , the Helmholtz free energy F , the average internal energy of the system  <E>  , and the partition function Z :


Finally , I’ll give an updated result for the calculation (on page 18) of the mean value of the displacement of the particles in the direction of the X axis in time t , using the latest recommended values of the physical constants.

We have:


Using the values above with T = 290.15 K and t = 1s  we find:

lambda-value1The value of the  viscosity of water at 17 °C is nowadays calculated as less than 1.35 cP  (about 1.081 cP , see this table ) , so in this case we have the value:


I’d like to add a last note about the use of the expression  ‘annus mirabilis ‘ to describe the wealth of  important articles and papers that Einstein published in 1905. This expression is translated as ‘extraordinary year  ‘ or ‘miraculous year’ , however the use of the words ‘miraculous ‘ or ‘miracle’ is contrary to the principles of the scientific method , scientific investigation and reasoning . And if we look at what Einstein had done  , read and studied for years before 1905 , we can see that what he published that year was the  result and culmination  of years of progressive  preparation , reading , (critical) thinking and careful examination of the major problems of physics at the beginning of the twentieth century .

Therefore I think it’s not necessary or appropriate to use the word ‘miracle’ (or ‘miraculous’) to describe scientific achievements which could be explained simply and rationally.The same argument applies to the inventions and discoveries Isaac Newton made in the year 1666.


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