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The difference comes from the distinction between microstates and macrostates. It is the distribution over the microstates that is uniform. However, macrostates encompass many microstates with varying …

On a side note, you can avoid the Trotter formula by using instead the standard interaction picture (if you are already familiar with it from previous QM courses). I will write $\partial_{x_i} = \part …

You were actually using the density of states for the 1D quantum harmonic oscillator. Setting $\hbar \omega=1$ and $\beta = \frac1{k_BT}$, for $N$, $D$-dimension, oscillators, The full partition funct …

Your first assumption is wrong, you can have a first order transition with symmetry breaking. Perhaps your misleading intuition was guided by the Ising model and the usual quartic Landau theory, where …

I will add quantitative details to Bob D's answer. Joule expansion There is no work or heat exchange $W=Q=0$ so to internal energy variation $\Delta U=0$ and volume $V$ varies. For a gas, you only hav …

With no loss of generality, you can assume that $E_0=0$. You are assuming a discrete spectrum, but in most cases, even if you have a spectral gap, you often have a continuum right after the gap $E_g>0 …

It is just thermodynamics. If systems can exchange particles, at equilibrium, they need to have the same chemical potential. …

Neglecting pressure and volume, your energy is: $$ dU=TdS+HdM $$ You can therefore apply the same method by formally substituting $P\to-H$ and $V\to M$. The analogue for enthalpy is: $$ \mathcal H = U …

In general, the two distributions have little to do with each other. After all, $\rho_e$ is fixed but $\rho_{mc}$ depends on the variable energy $E$. I will therefore reformulate your question as: can …

More commonly, the mechanical/optical analogy is rather between two position/momentum pairs at different instants in time related by a canonical transformation related by the generating function: $$ d …

Your question is purely mathematical. I will set $\mu=1$ with no loss of generality in the following. If you think in terms of fugacity: $$ z = e^\beta $$ you can use polylogarithms: $$ \text{Li}_s(x) …

Start by the definition of entropy. By definition: $$ S(B)-S(A) = \int_{Rev,\,A\to B}\frac{\delta Q}{T_e} $$ for any reversible path from $A\to B$. The definition is well posed since Clausius' inequal …

Actually, your question has little to do with statistical mechanics, but more about classical thermodynamics. … It typically holds in the thermodynamic limit, which is why for most applications classical thermodynamics applies. …

While you can choose any two independent variables as your basis function to express any quantity, you need to be careful in the choice of your variables when defining new quantities as derivatives. D …

In your root problem, it is indeed a misconception that cooling down a fluid unconditionally reduces turbulence. Indeed, if you cool it at the top, the fluid will sink so you’ll have the similar Rayle …