Title: ETH_Sem3 / Physik_III / Woche5

Q: Laws of Thermodynamics
- Zeroth law: [...]
- First law: [...]
- Second law: [...]
A: Two systems in thermal equilibrium have the same temperature.; Heat is a form of energy. Conservation of energy means that the change in energy of a system \(\Delta U\) is the sum of the heat transferred to the system Q and the work done on the system \(W: \Delta U = Q + W\).; The entropy of an isolated system never decreases.

Q: What is an isolated system?
A: A system which can exchange neither particles nor energy with the outside.

Q: What is a reversible process?
How are the infinitesimal heat and the change in entropy related ?
A: A process where all the systems involved remain in thermal equilibrium.
This is an idealized process, as all the changes happen in infinitesimal amounts, leading to an infinite time needed.
Still a good approximation for some slow processes.
\(\text{đ} Q = TdS\)

Q: What is a bath (in a thermodynamics context)?
A: A system so large that no matter how much energy we take from or add to it, its temperature doesn't change.

Q: What is the Boltzmann factor?
A: \[e^{-\beta\varepsilon}\]with \(\beta=\frac1{k_BT}\)

Q: What is the partition function?
A: \[Z = \sum_s e^{-\beta\varepsilon_s}\]

Q: Stirling's theorem
\[[...] \approx [...]\]
A: \ln(x!); x\ln(x)-x

Q: Probability for the system to be in the microstate with energy \(\varepsilon_s\) if the temperature is fixed?
A: \[p(\varepsilon_s) = \frac{e^{-\beta\varepsilon_s}}{Z}\]

Q: Many-particle partition function for \(N\) distinguishable particles:
\[Z_D = [...]\]
A: Z_{SP}^N

Q: Many-particle partition function for \(N\) indistinguishable particles and its approximation in the classical regime:
\[Z_I = [...]\]\[Z_I \approx[...]\]What is the assumption in the classical regime?
A: \sum_s e^{-N\beta\varepsilon_s} + \text{a bunch of other terms we can neglect} + \frac1{N!} \sum_{s_1}\cdots\sum_{s_N}e^{-\beta(\varepsilon_{s_1} + \cdots + \varepsilon_{s_N}) }; \frac1{N!}Z_{SP}^N
Back Extra: In a gas, there is a very small probability for any single-particle state to be occupied by more than one particle.

Q: What is the average energy of a gas with \(N\) indistinguishable particles?
(total and per particle)
A: \[\begin{align}\langle E \rangle &= \frac32Nk_BT \\ \langle \varepsilon \rangle &= \frac32k_BT \end{align}\]

Q: What is the equipartition theorem?
A: Each term in the energy that is quadratic in some coordinate contributes \(k_BT / 2\) to the average energy per particle.
What we mean by coordinate here include external degrees of freedom like the position and momentum, and also internal degrees of freedom like the relative position between atoms in a molecule that can stretch, or the rotation angle in a molecule that can rotate.

Q: The heat capacity at constant volume:
\[C_V = [...]\]
A: \bigg(\frac{\partial U}{\partial T}\bigg)_V

Q: Ideal gas law
A: \[pV = Nk_BT = nRT\]

Q: Maxwell-Boltzmann distribution
(Probability distribution for the speed of particles)
\[f(v) = [...]\]
A: 4\pi v^2\bigg(\frac{\beta m}{2\pi}\bigg)^{\frac32}e^{-\beta mv^2/2}

Q: According to the Maxwell-Boltzmann distribution, the average and most probable speeds are given by:
\[\begin{align}\langle v \rangle &= [...] = [...]\\ v_{\text{most prob.} } &= [...]\end{align}\]
A: \int_0^\infty vf(v)\;dv; \sqrt{\frac{8k_BT}{\pi m} }; \sqrt{\frac{2k_BT}m}