# Tag Info

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The notion of temperature is all about how the equilibrium an otherwise isolated system shifts when the system's internal energy changes. So you do not need to worry about whether this internal energy is kinetic, potential, whatever. Actually the temperature is not quite the ensemble average kinetic energy. Your statement is true for an ideal gas and also ...

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The mean free path can be meaningful quantity in quantum mechanics, although usually only in a semi-classical regime. It is particularly useful in the kinetic theory of quantum liquids at low temperature, where the excitations of the system can be described as quasiparticles that propagate approximately ballistically and interact only rarely. You can define ...

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What is most intuitive to me is to look at what we're asking. What is the probability that we find the system in a state with total energy $\hat{E}$? It is just the fraction of all possible states that have total energy $\hat{E}$, i.e. $$p(E) \equiv \frac{\int \delta\left(E(\Omega)- \hat{E}\right)d\Omega}{\int d\Omega}\sim N_\hat{E}$$ But $N_\hat{E}$ is ...

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You've answered your own question: The pitfall is obviously due to exponentiating a Taylor series with higher order terms dropped. To bring this into sharper focus, witness that the method, as you say, does indeed have some validity for any function analytic in the neighbourhood of the point in question. Unless we are at a stationary point, there is a ...

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The analytic continuation of $S^d$ is de Sitter space, often denoted as $dS_d$. Euclidean QFT on $S^d$ then corresponds to Lorentzian QFT on $dS_d$. This can be seen a number of ways, but the quickest is to simply note that the sphere is the maximally symmetric Euclidean signature space with positive curvature, and de Sitter is the maximally symmetric ...

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Ok. Let's put in this way: We know the average energy: $\langle E\rangle=\sum P_k E_k$ $P_K$ is a probability distribution: $\sum P_k=1$ We don't know nothing more about the system. The question is, what probability distribution $P_k$ capture the previous assertions in an unbiased way? The answer is, the distribution that maximize the Shannon's entropy ...

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I think the wrong step was the assumption that entropy increases, where in fact maintaining the temperature would require a an outflow of heat, which means the entropy of the gas is decreasing. To see how this relates to your formula, notice that this decrease in entropy would also increase the enthalpy by the same amount, however enthalpy will also ...

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The wave function for a pair of independent, indistinguishable bosons/fermions with no internal degrees of freedom can be written in the position representation as $$\Psi(r_1,r_2) = \frac{1}{\sqrt{2}} \left[\psi_1(r_1)\psi_2(r_2) \pm \psi_2(r_1)\psi_1(r_2) \right ].$$ You can calculate your desired probability distribution from \$P_{12}(r_1,r_2) = ...

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I don't know if this is standard, but consider a pendulum that can swing a full circle in a plane. Vibrate the point of suspension up and down at the appropriate frequency. The pendulum will gain energy and spin either clockwise or counterclockwise.

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