Abstract
In this chapter, we discuss systems with interacting particles. As we shall see, when the interactions among the particles are significant, the system exhibits a certain collective behavior that, in the thermodynamic limit, may be subjected to phase transitions, that is, abrupt changes in the behavior of the system in the presence of a gradual change in an external control parameter, like temperature, pressure, or magnetic field.
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- 1.
Another way to understand the dependence is to observe that occupation numbers \(\{N_r\}\) are dependent via the constraint on their sum. This is different from the grand–canonical ensemble, where they are independent.
- 2.
The reader can find the derivations in any textbook on elementary statistical mechanics, for example, [2, Chap. 9].
- 3.
Here we use the principle of ensemble equivalence.
- 4.
This is a caricature of the Lennard–Jones potential function \(\phi (\xi )\propto [(d/\xi )^{12}-(d/\xi )^6]\), which begins from \(+\infty \), decreases down to a negative minimum, and finally increases and tends to zero.
- 5.
Note, in particular, that for \(J=0\) (i.i.d. spins) we get paramagnetic characteristics \(m(\beta ,B)=\tanh (\beta B)\), in agreement with the result pointed out in the example of two–level systems, in the comment that follows Example 2.3.
- 6.
Once again, for \(J=0\), we are back to non–interacting spins and then this equation gives the paramagnetic behavior \(m=\tanh (\beta B)\).
- 7.
In a nutshell, annealing means slow cooling, whereas quenching means fast cooling, that causes the material to freeze without enough time to settle in an ordered structure. The result is then a disordered structure, modeled by frozen (fixed) random parameters, \({\varvec{J}}\).
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Merhav, N. (2018). Interacting Particle Systems and Phase Transitions. In: Statistical Physics for Electrical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-62063-3_5
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DOI: https://doi.org/10.1007/978-3-319-62063-3_5
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