Abstract
In this chapter, we provide the formalism and the elementary background in statistical physics. We first define the basic postulates of statistical mechanics, and then define various ensembles. Finally, we shall derive some of the thermodynamic potentials and their properties, as well as the relationships among them. The important laws of thermodynamics will also be pointed out.
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Notes
- 1.
This is a result of the energy conservation law along with the fact that probability mass behaves like an incompressible fluid in the sense that whatever mass that flows into a certain region from some direction must be equal to the outgoing flow from some other direction. This is reflected in the equation of continuity, which was demonstrated earlier.
- 2.
For example, in the case of an ideal gas, \(\mathcal{E}({{\varvec{x}}})=\sum _{i=1}^N\Vert \vec {p}_i\Vert ^2/(2m)\), where m is the mass of each molecule, namely, it accounts for the contribution of the kinetic energies only. In more complicated situations, there might be additional contributions of potential energy, which depend on the positions.
- 3.
In the example of the ideal gas, since the particles are mobile and since they have no colors and no identity certificates, there is no distinction between a state where particle no. 15 has position \(\vec {r}\) and momentum \(\vec {p}\) while particle no. 437 has position \(\vec {r}'\) and momentum \(\vec {p}'\) and a state where these two particles are swapped.
- 4.
This argument works for distinguishable particles. Later on, a more general argument will be presented, that holds for indistinguishable particles too.
- 5.
Here and in the sequel, the reader might wonder about the meaning of taking derivatives of, and with respect to, integer valued variables, like the number of dislocated particles, n. To this end, imagine an approximation where n is interpolated to be a continuous valued variable.
- 6.
Heat is a form of energy that is transferred neither by mechanical work nor by matter. It is the type of energy that flows spontaneously from a system/body at a higher temperature to one with a lower temperature (and this transfer is accompanied by an increase in the total entropy).
- 7.
In this context, it should be understood that the results we derived for the ideal gas hold only for high enough temperatures: since S was found proportional to \(\ln E\) and E is proportional to T, then S is proportional to \(\ln T\), but this cannot be true for small T as it contradicts (among other things) the third law.
- 8.
Equality of chemical potentials is, in fact, the general principle of chemical equilibrium, and not equality of concentrations or densities. In Sect. 1.3, we saw equality of densities, because in the case of the ideal gas, the chemical potential is a function of the density, so equality of chemical potentials happens to be equivalent to equality of densities in this case.
- 9.
The origin of this name comes from the wave–particle de Broglie relation \(\lambda =h/p\) together with the fact that the denominator, \(\sqrt{2\pi mkT}\), can be viewed as a notion of thermal momentum of the ideal gas, given the fact that the average molecular speed is proportional to \(\sqrt{kT/m}\) (see Sect. 1.1).
- 10.
- 11.
Since the term \(\lambda \cdot L({{\varvec{x}}})\) is not considered part of the internal energy (but rather an external energy resource), formally, this ensemble is no longer the canonical ensemble, but a somewhat different ensemble, called the Gibbs ensemble, which will be discussed later on.
- 12.
At this point, there is a distinction between the Helmholtz free energy and the Gibbs free energy. The former is defined as \(F=E-TS\) in general, as mentioned earlier. The latter is defined as \(G=E-TS-\lambda L=-kT\ln Z\), where L is shorthand notation for \(\langle L({{\varvec{x}}}) \rangle \) (the quantity \(H=E-\lambda L\) is called the enthalpy). The physical significance of the Gibbs free energy is similar to that of the Helmholtz free energy, except that it refers to the total work of all other external forces in the system (if there are any), except the work contributed by the force \(\lambda \) (Exercise 2.4 show this!). The passage to the Gibbs ensemble, which replaces a fixed value of \(L({{\varvec{x}}})\) (say, constant volume of a gas) by the control of the conjugate external force \(\lambda \), (say, pressure in the example of a gas) can be carried out by another Legendre–Fenchel transform (see, e.g., [5, Sect. 1.14]) as well as Sect. 2.2.3 in the sequel.
- 13.
More precisely, the one–dimensional Legendre–Fenchel transform of a real function f(x) is defined as \(g(y)=\sup _x[xy-f(x)]\). If f is convex, it can readily be shown that: (i) The inverse transform has the very same form, i.e., \(f(x)=\sup _y[xy-g(y)]\), and (ii) The derivatives \(f'(x)\) and \(g'(y)\) are inverses of each other.
- 14.
Note that the expected number of ‘activated’ particles \(\langle n \rangle =NP(1)=Ne^{-\beta \epsilon _0}/(1+e^{-\beta \epsilon _0})=N/(e^{\beta \epsilon _0}+1)\), in agreement with the result of Example 2.1 (Eq. (2.2.17)). This demonstrates the ensemble equivalence principle.
- 15.
The best way to understand this is in analogy to the derivation of \(\epsilon ^*\) as the minimizer of the free energy in the canonical ensemble, except that now the ‘big’ extensive variable is V rather than N, so that \(z^NZ_N(\beta ,V)\) is roughly exponential in V for a given fixed \(\rho =N/V\). The exponential coefficient depends on \(\rho \), and the ‘dominant’ \(\rho ^*\) maximizes this coefficient. Finally, the ‘dominant’ N is \(N^*=\rho ^*V\).
- 16.
Exercise 2.5 Write explicitly the Legendre–Fenchel relation (and its inverse) between the Gibbs partition function and the canonical partition function.
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Merhav, N. (2018). Elementary Statistical Physics. In: Statistical Physics for Electrical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-62063-3_2
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