Abstract
The chapter discusses the statistical physics approach to study of interacting particle systems. We start with a brief review of thermodynamics and then introduce the concept of statistical ensembles, discussing in turn the three major statistical ensembles, the microcanonical, the canonical, and the grand canonical ensemble. We recapitulate the notion of major thermodynamic potentials and how they may be used to define phase transitions. We next discuss equivalence of statistical ensembles, using the Legendre–Fenchel transformation. We work out for a model system of interacting classical Heisenberg spins the properties in canonical equilibrium, and discuss how ensemble equivalence allows to infer properties in microcanonical equilibrium.
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Notes
- 1.
Typical \(\Delta t\)’s would be of the order of seconds, while particles in typical thermodynamic systems move on a timescale of order of milliseconds or less. This suggests that the system would indeed pass through really a very large number of microstates during the duration \(\Delta t\).
- 2.
This is only an a priori guess, and its correctness in a given situation may only be checked a posteriori by validating theoretical results based on this guess against experimental findings.
- 3.
We have already remarked that we have \(E_1 \propto N_1\) and \(E_2 \propto N_2\), which implies that we have \(E \propto N\). Computation of entropy for an arbitrary system is not an easy task because of the difficulties in computing the constrained integral (4.43) defining the quantity \(\Gamma (N,V,E)\). Nevertheless, at least for an ideal gas, one can compute \(\Gamma \) exactly and thence, show that one has \(S \propto N\) [3].
- 4.
Note that system 1 and system 2 are parts of the same system, and so the quantities \(H_1\) and \(H_2\) have the same mathematical form; the subscripts 1 and 2 are being added here for ease of comprehension by the uninitiated.
- 5.
Customarily, while defining Z, one divides the right hand side of Eq. (4.72) by \(N! \,h^{3N} \), which would only add a constant term in F. The factor \(h^{3N}\) makes Z dimensionless and the factor of N!, known as the correct Boltzmann counting, takes into account the indistinguishability of particles in quantum mechanics. However, there is no way to derive this within the ambit of classical mechanics, in which no consistent way exists to regard the particles indistinguishable. We remark that these constant factors would prove irrelevant in what concerns observable properties of the system, which would all be defined in terms of derivatives of the free energy, in much the same way as the constant factor \(S_0\) in the definition of the entropy in Eq. (4.47) had no relevance.
- 6.
At a second-order phase transition point, one observes a scaling \(\left( \sqrt{\langle {H^2}\rangle - \langle {H}\rangle ^2}\right) / \langle {H}\rangle \propto N^{-\alpha }\) with \(1/2<\alpha <1\). Thus, the relative fluctuation in energy is still negligibly small for \(N \gg 1\).
- 7.
Since we are dealing with short-range interacting systems, energy and hence, free energy will grow linearly in N. For long-range interacting systems, such a linear dependence will be guaranteed once the Kac’s trick is invoked, see Sect. 3.3.
- 8.
The infimum of a set A is its greatest lower bound, defined as a quantity m such that there exists no member of the set with a value less than m. When the infimum belongs to A, it is considered a minimum of A.
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Das, D., Gupta, S. (2023). Statistical Mechanics. In: Facets of Noise. Fundamental Theories of Physics, vol 214. Springer, Cham. https://doi.org/10.1007/978-3-031-45312-0_4
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