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nonequilibrium
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Imagine an ideal two-state paramagnet. I mean a solid consisting of miniscule magnetic dipoles which can only align upward or downward (I have no idea why this latter restriction turns out to be physically feasible -- I am aware the spin of an electron is either up or down, but I was under the impression that the atom bearing the electron can rotate whichever way, but I haven't had any quantum mechanics yet, so this will not be my question). Say the material consists of N dipoles (and I think there is the unmentioned assumption that other intermolecular interactions are negligble). According to the Statistical Mechanical definiton of Entropy:
[tex]S = k ln\left( \stackrel{N}{N_{\uparrow}} \right)[/tex] where [tex]N\uparrow[/tex] is the total number of dipoles pointing up.
From electromagnetism, we can say [tex]E = - B\mu (N\uparrow)[/tex] if we say B is pointing up.
Now using [tex]1/T = \frac{\partial S}{\partial E} [/tex] and a Sterling approximation (or more rigidly using a Boltzmann distribution) we get:
[tex]\frac{N\uparrow}{N} = \frac{1}{2} \left( 1 + tanh \left( \frac{\mu B}{kT} \right) \right)[/tex] (the details aren't important for this discussion, the main fact is that this formula follows from the stated assumptions/details of the system according to SM)
Does this mean that we can know the fraction of the dipoles pointing upward simply by measuring the temperature? That seems quite amazing!
So say we surround this ideal paramagnet with an ideal gas at 300K. Does it matter that in essence the temperature of the resevoir and the system are of a different species? I mean, the gas-temperature is a measure for the average kinetic energy, while that of the paramagnet is a measure of (positive, B-wise) magnetization. But SM-wise, they are defined by the same principle, namely [tex]\frac{\partial S}{\partial E}[/tex] so the gas and the paramagnet reach equilibrium and you know right away the fraction of dipoles pointing up? (taking the ideal gas to be a reservoir, thus having constant T)
Now a very interesting thing, is that the temperature of the magnet is NEGATIVE for the macrostates where more than half of the dipoles are pointing downward/against B. Is this simply a sort of (logical) statement that we can never (no matter how hot our reservoir is) make more than half of the dipoles point against the upward magnetic field B by heating the magnet (because something at a negative temperature, interpreting it as the derivate mentioned earlier, will always give off energy to anything at a positive temperature, i.e. the reservoir)?
Now another very interesting thing, is that this derivation indeed turns out to be physically feasible, as it is a sort of "ideal proof" for Curie's law (or a prediction thereof, anyway) which states that the magnetization of a paramagnetic solid (at high T, low 1/T) is proportional to 1/T. But it seems odd: in a physical paramagnet you can imagine the gas molecules (from the reservoir) bumping into the dipoles, making them vibrate more so they give less of a contribution to the total upward magnetization. But in this ideal model, the dipoles are immobile, they can only switch up or down, yet somehow bumping gas molecules deliver work on them. Does this simply mean my intuitive idea of a real paramagnet is wrong and that Curie's law is much more quantummechanical than I think?
Open to all reactions,
mr. vodka
[tex]S = k ln\left( \stackrel{N}{N_{\uparrow}} \right)[/tex] where [tex]N\uparrow[/tex] is the total number of dipoles pointing up.
From electromagnetism, we can say [tex]E = - B\mu (N\uparrow)[/tex] if we say B is pointing up.
Now using [tex]1/T = \frac{\partial S}{\partial E} [/tex] and a Sterling approximation (or more rigidly using a Boltzmann distribution) we get:
[tex]\frac{N\uparrow}{N} = \frac{1}{2} \left( 1 + tanh \left( \frac{\mu B}{kT} \right) \right)[/tex] (the details aren't important for this discussion, the main fact is that this formula follows from the stated assumptions/details of the system according to SM)
Does this mean that we can know the fraction of the dipoles pointing upward simply by measuring the temperature? That seems quite amazing!
So say we surround this ideal paramagnet with an ideal gas at 300K. Does it matter that in essence the temperature of the resevoir and the system are of a different species? I mean, the gas-temperature is a measure for the average kinetic energy, while that of the paramagnet is a measure of (positive, B-wise) magnetization. But SM-wise, they are defined by the same principle, namely [tex]\frac{\partial S}{\partial E}[/tex] so the gas and the paramagnet reach equilibrium and you know right away the fraction of dipoles pointing up? (taking the ideal gas to be a reservoir, thus having constant T)
Now a very interesting thing, is that the temperature of the magnet is NEGATIVE for the macrostates where more than half of the dipoles are pointing downward/against B. Is this simply a sort of (logical) statement that we can never (no matter how hot our reservoir is) make more than half of the dipoles point against the upward magnetic field B by heating the magnet (because something at a negative temperature, interpreting it as the derivate mentioned earlier, will always give off energy to anything at a positive temperature, i.e. the reservoir)?
Now another very interesting thing, is that this derivation indeed turns out to be physically feasible, as it is a sort of "ideal proof" for Curie's law (or a prediction thereof, anyway) which states that the magnetization of a paramagnetic solid (at high T, low 1/T) is proportional to 1/T. But it seems odd: in a physical paramagnet you can imagine the gas molecules (from the reservoir) bumping into the dipoles, making them vibrate more so they give less of a contribution to the total upward magnetization. But in this ideal model, the dipoles are immobile, they can only switch up or down, yet somehow bumping gas molecules deliver work on them. Does this simply mean my intuitive idea of a real paramagnet is wrong and that Curie's law is much more quantummechanical than I think?
Open to all reactions,
mr. vodka
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