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A major part of Einsteins 1905 light quantum paper isdevoted to arguing that high frequency heat radiationbears the characteristic signature of a microscopic energydistribution of independent, spatially localized

components.

Einstein inferred from the thermal properties of highfrequency heat radiation that it behavesthermodynamically as if it constituted of spatiallylocalized, independent quanta of energy.

However, if we locate Einsteins light quantum paper

against the background of his work in statistical physics,its methods are an inspired variation of ones repeatedlyused and proven effective in other contexts on verysimilar problems.

Einsteins light quantum paper (1905a) has nine parts.The lastThree pertain to empirical vindications of the light

quantum hypothesis.

The bulk of the paper, from the first to the sixth sections,is largely devoted to setting up and stating just oneargument that comes to fruition in the sixth section(Interpretation of the expression for the dependence ofthe entropy of monochromatic radiation on volumeaccording to Boltzmanns Principle).

There Einstein infers that the measured thermodynamicproperties of high frequency heat radiation carry thedistinctive signature of independent, spatially localizedpoints.

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There were two steps in the argument. The first appearedin the fifth section (Molecular-theoretical investigation ofthe dependence of the entropy of gases and dilutesolutions on the volume).

It followed immediately from the independence of thepoints that the probability W that all n points are locatedin some sub-volume V is just W = (V/V0) n(1)In that same section, Einstein had already presented abrief argument for what he called Boltzmanns Principle

(See Section 6 below.) That principle S = k log W(2)used Boltzmanns constant k to make a connectionbetween the probabilities W of microstates and theircorresponding macroscopic entropies S. Applying (2) to(1), Einstein inferred that the entropy change associatedwith the volume change of (1) isS-S0 = k n log (V/V0)

(3)

In a footnote, Einstein then used standardthermodynamic relations to deduce from (3) that the npoints would exert a pressure P at temperature T thatconforms to the ideal gaslaw PV = nkT(4)

The probability W of (1) is the probability that the gasspontaneously fluctuates to a smaller volume V in thelargervolume V0.

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The second step of the argument was developed in thefollowingsection and was anything but standard. Einstein returned

to the measured results concerning heat radiation that hehad reviewed in earlier sections. From these results, heinferred the volume dependence of the entropy of aquantity of high frequency heat radiation.

That is, if a quantity of heat radiation of fixed energy Eand definite frequency occupies volume V or V0, thenthe corresponding entropies S and S0 are related by (2)S - S0= k (E/h) log V/V0

(5)Einstein then used Boltzmanns principle (2) to invert theinference from (1) to (3). It now followed that there is aprobability W satisfying: W = (V/V0)E/h (6)Einstein immediately noted that this probability was justlike the probability of the process of spontaneous volumefluctuations of n independent, spatially localized points,where

n = E/h(7)

That is, it was as if the energy E of the heat radiation hadbeen divided into n independently moving, spatiallylocalized points of size h.

What Einstein achieved with his miraculous argument

was an inference from the macroscopic properties (thevolume dependence of entropy of high frequency thermalradiation) of a thermal system to it microscopicconstitution (independent, spatially localized energyquanta).

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The central idea of Einsteins miraculous argument canbe encapsulated in a single, powerful idea. Sometimes itis possible to see a signature of the microscopicconstitution of a thermal system in its macroscopic

thermodynamic properties.

Einstein had identified just such a signature for systemsconsisting of independent, spatially localizedcomponents. That signature is given as the dependenceof entropy on the logarithm of volume as expressed inequation (3).

If that dependency can be identified, then we have astrong indication that the system consists of independent,spatially localized components; the number ofcomponents n can be read directly from the constant ofproportionality of (3), which is just kn.