Abstract
As emphasized in the Introduction, a macroscopic system consists of 1019 — 1023 particles and correspondingly has an energy spectrum with spacings of ΔE ~ e—N. The attempt to find a detailed solution to the microscopic equations of motion of such a system is hopeless; furthermore, the required initial conditions or quantum numbers cannot even be specified. Fortunately, knowledge of the time development of such a microstate is also superfluous, since in each observation of the system (both of macroscopic quantities and of microscopic properties, e.g. the density correlation function, particle diffusion, etc.), one averages over a finite time interval. No system can be strictly isolated from its environment, and as a result it will undergo transitions into many different microstates during the measurement process. Fig. 2.1 illustrates schematically how the system moves between various phase-space trajectories. Thus, a many-body system cannot be characterized by a single microstate, but rather by an ensemble of microstates. This statistical ensemble of microstates represents the macrostate which is specified by the macroscopic state variables E, V, N,...1 (see Fig. 2.1).
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A different justification of the statistical description is based on the ergodic theorem: nearly every microstate approaches arbitrarily closely to all the states of the corresponding ensemble in the course of time. This led Boltzmann to postulate that the time average for an isolated system is equal to the average over the states in the microcanonical ensemble (see Sect. 10.5.2).
The surface area of the energy shell Ω (E) depends not only on the energy E but also on the spatial volume V and the number of particles N. For our present considerations, only its dependence on E is of interest; therefore, for clarity and brevity, we omit the other variables. We use a similar abbreviated notation for the partition functions which will be introduced in later sections, also. The complete dependences are collected in Table 2.1.
The derivation of (2.2.13) will be given at the end of this section.
N.G. de Bruijn, Asymptotic Methods in Analysis,(North Holland, 1970); P. M. Morse and H. Feshbach, Methods of Theoretical Physics,p. 434, (McGraw Hill, New York, 1953).
In the literature of magnetism, it is usual to denote the magnetic field by H or H. To distinguish it from the Hamiltonian in the case of magnetic phenomena, we use the symbol ∆∼ for the latter.
See e.g. G. Baym, Lectures on Quantum Mechanics (W.A. Benjamin, New York, Amsterdam 1969), p. 393
A heat bath (or thermal reservoir) is a system which is so large that adding or subtracting a finite amount of energy to it does not change its temperature.
Exceptions are photons and bosonic quasiparticles such as phonons and rotons in superfluid helium, for which the particle number is not fixed (Chap. 4).
For the reasons mentioned at the end of the preceding section, we replace E and N in (2.7.16) and (2.7.17) by E and N.
F. Schwabl, Advanced Quantum Mechanics (QM II),Springer Berlin, Heidelberg, New York 1999. This text will be cited in the rest of this book as QM II.
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Schwabl, F. (2002). Equilibrium Ensembles. In: Statistical Mechanics. Advanced Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04702-6_2
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DOI: https://doi.org/10.1007/978-3-662-04702-6_2
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