# Tag Info

4

Mad props for a cool question. I'm going to justify essentially the converse of the statement because it doesn't make much sense to talk of the temperature of a system that is in a pure state. Let's assume that we're talking about a quantum system with disrete energy spectrum (with no accumuation points) in thermal equilibrium. Let $\beta = 1/kT$ be the ...

4

A maybe more mathematical awnser: You can define temperature as a scalar field (e.g. on earth). So given a certain position on the surface of the earth (or in three dimensions if you wish, it does not change anything) you have a scalar, the temperature on this position. Now you can take the gradient of this field, and now you have a vector. More directly on ...

3

Nothing - that is the correct definition. One little caveat is that small systems are usually in contact with a larger system with a temperature that's more easily controlled or measured (the "heat bath"), and $T$ usually stands for the temperature of the heat bath rather than the small system itself. However, this doesn't make a lot of difference in ...

2

I think the best physical references would be at the high end, such as the maximum volume of undistorted sound around 194 dB. There are several other examples of sound pressure level including one on Wikipedia. I don't know enough to know if a thermoacoustic device would meet your requirements, but that's another possibility, anyway. As an added note, I ...

2

Length scales are not accounted for properly in your question. When you have a system at local equilibrium where a temperature gradient can be defined then each "point" in this description contains say $10^{10}$ molecules and can be seen as a thermostatistical system at equilibrium. We call that "local" equilibrium because intensive quantities such as ...

1

Let's assume there is excellent thermal conductivity from the heater to the block, and from the block to the inner surface of the radiator. This is a convective radiator. The rate of transfer of heat energy depends linearly on the difference in temperature between the block and the air (since you're holding other things fixed). The whole system will reach ...

1

The maximum entropy probability distribution for a system of fixed expected total energy is the Boltzmann distribution. This means that Boltzmann distribution is appropriate for systems where we know the total energy but little else. The distribution is given by $p(E) = Z^{-1} e^{-E/kT}$, where $Z$ is the normalization constant (making sure probabilities add ...

1

A "point" in a macroscopic system is not a geometrical point. It is a volume element that is small on a macroscopic scale and yet has a large number of molecules for entropy and internal energy to be defined. Your temperature probe does not measure its value at a geometrical point but for a small volume of the system in whose contact it is put. The local ...

1

Generally speaking low and high pressure areas are associated with vertical movement of the air. Air rises in a low pressure area and falls in a high pressure area. In a low pressure area the rising air cools and this is likely to condense water vapour and form clouds, and consequently rain. The opposite is true in a high pressure area, which is why high ...

1

Yes, colouring the water could make the pool heat faster, though whether the colour you noticed has an significant effect is debatable. Swimming pools exchange heat with their environment by conduction through their walls, evaporation at the surface and absorption of sunlight. I have little direct experience of swimming pool thermodynamics, but some ...

1

This kind of exponential decay toward "equilibrium" can be derived when one looks at a Markov process. In this case, if we call $S_t$ the state of the system at time $t$ and $S_{t+1}$ the state at time $t+1$, one has for the evolution: $S_{t+1} = T S_t$ where $T$ is called the transition matrix. This implies that $S_t = T^t S_0$. The idea is then to ...

1

The valve will, if not to big, not let cold air in, but will let warm air out, which will be replenished by cold air from the bottom (I am assuming that the balloon is open at the bottom). If the valve would be very big, then cold air would also flow in through the valve, but I don't know how you could calculate this. For the flow rate you could use ...

1

Assuming the surface of the metal remains smooth, the reflection from it will be specular and the metal will look shiny regardless of the temperature. However the amount of light metals absorb, instead of reflecting, generally increases with increasing temperature because you get more scattering of the conduction electrons by lattice vibrations. So the metal ...

1

Actually, this is an assumption of the Landau theory: the simplest field model exhibiting a phase transition is analogous to a $\phi^4$ theory, which has the lagrangian density $${\mathcal L} = \partial_\mu \phi \partial^\mu \phi - \frac{m^2}{2} \phi^2 - \frac{\lambda}{4} \phi^4\,.$$ For $m^2 > 0$ the potential in the above lagrangian has a single ...

1

"Absolute Hot" is a cute phrase, but meaningful only in a limited context, where the concept of negative temperature applies. That is, there's a finite number of energy levels in a finite energy range, and whatever particles/quanta/excitations we're studying stay in that context. For example we may be interested in spins of nuclei in a crystal, and how ...

1

Temperature gradient is actually an object called a one-form. A temperature gradient does not have a direction. Instead you combine it with a vector to get a scalar (the temperature change). It's the vector that gives the direction. To take a simple 1-D example, suppose we have a temperature that varies along the $x$ axis as: $$T = 298 + x$$ so at \$x = ...

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As for the question of whether anything can be hotter than the sun. The Sun is composed of plasma, an energetic phase of matter in which electrons get ripped off of atoms, and electrons and ions coexist in something that might best be described as an ionized gas. According to this wiki page, the so-called Z machine has achieved temperatures on the order of ...

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I apologize "basics foundations of thermodynamics" still does not make a lot of sense to me. Steve B already provided some answer associated to one way to interprete the word "foundation" that is from statistical mechanics. I will kinda play here devil's advocate and assume that you are refering to axiomatic thermodynamics. As far as I am concerned, the ...

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There's a group I call "thermodynamic purists" who think that thermodynamics is a self-contained system based on semi-mathematical "axioms". I disagree! I think that thermodynamics is fundamentally a consequence of statistical mechanics, and that this is the best way to think about it and understand it. I acknowledge that reasonable people can differ on ...

1

Temperature is the measurement of kinetic energy per unit particle mass. Since you've added the same amount of heat energy to each object, the finite object will have a higher temperature because its heat energy is distributed across a smaller collection of mass. Taking something's temperature is indeed a meaningful measurement ;)

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