# What Statistical Mechanics does in classical regime

In a book of Dipankar Home, "Foundations of Quantum Mechanics", he has mentioned that

A newer theory should not only predict all the results that are already predicted by it's predecessor where it is valid but should also give new results in the same domain of the previous theory which cannot be explained by the previous theory.

Keeping this philosophy in mind: What are the macroscopic predictions of statistical mechanics that contradicts or atleast completes a example from thermodynamics?

I know that Thermodynamics is essentially a emperical theory, so it is very hard to contradict it. But still is there any such circumstance?

• I'm not sure where you'd want to draw the line, but I believe the concept of "negative temperature" was only arrived at after we sufficiently understood the relationship between temperature and entropy, and had microscopic models for entropy. That's not to say that some sufficiently clever thermodynamacist couldn't have come up with the same idea, though. – Jahan Claes Mar 9 '16 at 5:02

I think Gibbs gave a good motivation in the preface of his 1902 book where he coined the term "Statistical Mechanics":

The laws of thermodynamics, as empirically determined, express the approximate and probable behavior of systems of a great number of particles, or, more precisely, they express the laws of mechanics for such systems as they appear to beings who have not the fineness of perception to enable them to appreciate quantities of the order of magnitude of those which relate to single particles, and who cannot repeat their experiments often enough to obtain any but the most probable results. The laws of statistical mechanics apply to conservative systems of any number of degrees of freedom, and are exact. This does not make them more difficult to establish than the approximate laws for systems of a great many degrees of freedom, or for limited classes of such systems. The reverse is rather the case, for our attention is not diverted from what is essential by the peculiarities of the system considered, and we are not obliged to satisfy ourselves that the effect of the quantities and circumstances neglected will be negligible in the result. The laws of thermodynamics may be easily obtained from the principles of statistical mechanics, of which they are the incomplete expression, but they make a somewhat blind guide in our search for those laws. This is perhaps the principal cause of the slow progress of rational thermodynamics, as contrasted with the rapid deduction of the consequences of its laws as empirically established. To this must be added that the rational foundation of thermodynamics lay in a branch of mechanics of which the fundamental notions and principles, and the characteristic operations, were alike unfamiliar to students of mechanics.

(emphasis is mine)

From the point of view of a nanotechnologist like myself, the most significant thing is that statistical mechanics extends thermodynamics to small systems where fluctuation phenomena become serious (systems of "few degrees of freedom" in Gibbs' terminology). It provides the tools we need to describe precise microscopic mechanics of real world systems where nothing is in a pure state. And most importantly for semiconductor physics (and to answer your specific question), stat mech lets us cleverly break down the behaviour of a noninteracting macroscopic electron gas in terms of its microscopic parts.

But stat mech is not just a justification for thermodynamics, in fact it is a generalization of normal conservative mechanics. Simply ask yourself: how do I exactly describe the evolution of a mechanical system where the initial state is incompletely specified (i.e., there is a probability distribution of states). The answer is the fundamental equation of statistical mechanics -- the Liouville theorem -- for which ordinary mechanics is a special case. In Gibbs' words:

But although, as a matter of history, statistical mechanics owes its origin to investigations in thermodynamics, it seems eminently worthy of an independent development, both on account of the elegance and simplicity of its principles, and because it yields new results and places old truths in a new light in departments quite outside of thermodynamics. Moreover, the separate study of this branch of mechanics seems to afford the best foundation for the study of rational thermodynamics and molecular mechanics.

Unfortunately Gibbs' deeply elegant point of view is often lost in undergraduate textbooks on Stat Mech, which seem to only discuss the thermodynamic applications. I think the biggest misunderstanding many people have is that statistical mechanics is somehow an approximate theory, since their teacher justified it using Stirling's approximation.

• There exist a thermodynamics for small systems, sometimes named nanothermodynamics. – juanrga Jul 31 '16 at 13:26

The description (and quantification) of entropy is hard to fathom without statistical mechanics - but essential for a proper description of thermodynamics.

Once you have entropy, irreversibility can be explained - as can the first law of thermodynamics. The law existed empirically, but with statistical mechanics it can be explained.

One would first start by defining what one means by statistical mechanics and by thermodynamics. If we mean equilibrium statistical mechanics and classical thermodynamics, then statistical mechanics complements thermodynamics by providing explicit values to some coefficients that thermodynamics cannot obtain by itself like, for instance, $C_V$. It is not a one-way route, however, because statistical mechanics also requires of thermodynamics for identifying some parameters. E.g. using standard methods, equilibrium statistical mechanics yields formulas with undetermined parameters such as $\beta$. It is only after comparing the statistical mechanics expressions with thermodynamic formulas known a priori that we relate those parameters with thermodynamic quantities: e.g., $k_\mathrm{B}\beta = 1/T$. This two-way relationship is the reason why some academics prefer to use the term statistical thermodynamics, emphasizing the marriage of thermodynamic and statistical mechanics.

Now if we mean non-equilibrium statistical mechanics and thermodynamics of irreversible processes, the situation is more complex and thermodynamics, more concretely its second law, is used as a guiding principle to derive correct statistical mechanics equations.

The relationship between statistical mechanics and thermodynamics is much more complex when we move to quantum thermodynamics. In some formulations quantum thermodynamics provides an extension of ordinary non-equilibrium statistical mechanics to ensembles of sizes well below the thermodynamic limit, whereas in other formulations quantum thermodynamics is a proper nonlinear extension of quantum mechanics

What is Quantum Thermodynamics?