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The theory of ensembles can be viewed from many points of view (we don't need to imagine any representative systems, for example). There are many more ensembles commonly used in physics than the microcanonical, canonical, and grand-canonical. For example there's the angular-momentum ensemble, described by Gibbs himself (Elementary Principles in Statistical ...
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Since it seems that you're dealing with this in a statistical mechanics context, I'll use the Ising model as an example.
Spontaneous symmetry breaking is a phenomenon in which the Hamiltonian or Lagrangian of your system has a certain symmetry, but some relevant state of your system does not. Take the nearest neighbor 2D Ising model
$$ H=-J\sum_{\langle i,j\...
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The best way to understand the idea is to take some simple example and then generalize it to prove the general result. Here I'll try to give the idea of representation point as you said:
This idea of the motion of representative points is something that confuses me.
The most simple example that every familiar with is of Simple pendulum but I'm involving ...
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Here is the mistake:
Now, for a grand canonical ensemble, the particle number is not conserved due to its contact with a particle reservoir,
[𝐻̂,𝑁̂]≠0.
The mistake is the same as in the following parallel logic:
Now, for a grand canonical ensemble, the energy is not conserved due to its contact with a particle (and heat) reservoir, so
[𝐻̂,𝐻̂]≠0.
...
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I will try a simple explanation. Try putting a cylindrical pencil standing upright on a ball. Eventually, for a second, you will succeed, but then the pencil will probably fall.
What tells the pencil in which direction to fall? At the start (pencil upright on the ball) there is cylindrical symmetry to the whole thing. The answer would be that some ...
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There are a few finer details that one has to keep in mind here:
A single harmonic oscillator is really not a thermodynamic system, so the postulate doesn't apply to it.
All accessible states in case of a constant energy (i.e., when using a microcanonical ensemble) means all the possible states with the same energy. For a single oscillator there is only one ...
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Time translation symmetry does not preclude an arrow of time. The former, which assumes that the laws of physics do not change over time (not that time must move forward), is assumed in both Lagrangian mechanics and thermodynamics and underlies the law of conservation of energy.
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Yes the Rayleigh-Jeans law gives an infinite energy density. This is the ultra-violet catastrophe. The solution requires quantum mechanics and photons.
The business with "perfect standing waves" however is much simpler. The set of such waves form a complete orthonormal basis for the solutions and so any E&M wave bouncing round the cavity can ...
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I think you forgot to take into account $g(E)$ in the calculation for the $1\mathrm{D}$ case. Actually, $g(E) \propto 1/\sqrt{E}$ in that case, which will lead to a different (and hopefully correct) answer.
The reason why you get a different answer by using $v_x$ as your integration variable is because the integrand $\propto \exp \left(-\frac{1mv^2}{2kT} \...
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