Themes > Science > Physics > Statistical Mechanics > Statistical Mechanics Topics > Entropy Revisited

In the study of heat, and of thermodynamics in general, there was no need to know the microscopic nature of the substance which was being heated or cooled and so on. Instead, we reasoned using general principles such as temperature, work, and energy conservation and explained a number of important phenomenon. We then introduced the idea of entropy, which is needed for explaining the physically observed irreversibility of the world around us. The Second Law of Thermodynamics was then shown to imply a number of new results, including the fact that there can never be a perpetual motion machine. The idea of energy is well understood from mechanics. To explain thermodynamics the new and profound idea of entropy had to be invented. It is the idea of entropy that is central and unique to the study of heat and thermodynamics, and the branch of physics called statistical mechanics has resulted from the attempt to understand entropy from a microscopic point of view. Recall from (9.41) we have a microscopic definition of entropy $S$ given by
$\displaystyle \mathrm{Entropy}$ $\textstyle =$ $\displaystyle k \ln \mbox{\rm {(Number of configurations)}}$  
$\displaystyle S$ $\textstyle =$ $\displaystyle k \ln \Gamma$  

The microscopic definition unavoidably led to assumptions as to what matter is made out of, namely what is the microscopic composition of matter. For example, in applying the equation for entropy to the case of an ideal gas, we had to take into account the microscopic nature of the gas, in particular, that it is made out of an enormous collection of microscopic objects that we identified with atoms. The entire field of statistical mechanics was founded by Boltzmann in the late nineteenth century. As a historical aside, it is worth recording that it was in order to understand the concept of entropy from a microscopic point of view, that Boltzmann had postulated the existence of atoms well before their discovery in the twentieth century. In sum, the challenge posed by thermodynamics was the following: how can we reconcile ideas such as temperature, entropy and so on with the ideas of (Newtonian) mechanics? In particular, if any sample of matter that we observe in daily life is made out of an inordinately large number of atoms, approximately $N_{\mathrm{Avogardo}} \simeq 10^{23}$, how can we apply the laws of mechanics to this large collection of particles? Clearly, it is hopeless to try and describe how every single particle is moving, as this would involve specifying, at each instant, $N_{\mathrm{Avogardo}}$ number of positions and velocities. So what is the way out of this impasse?


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