List of top Physics Questions on Thermodynamic and Statistical Physics

A system of two atoms can be in three quantum states having energies 0, $\epsilon$ and $2\epsilon$. The system is in equilibrium at temperature \( T = (k_B\beta)^{-1} \). Match the following Statistics with the Partition function.

Consider a single one-dimensional harmonic oscillator of angular frequency \( \omega \), in equilibrium at temperature \( T = \left( k_B \beta \right)^{-1 }\). The states of the harmonic oscillator are all non-degenerate having energy \( E_n = \left( n + \frac{1}{2} \right) \hbar \omega \) with equal probability, where \( n \) is the quantum number. The Helmholtz free energy of the oscillator is
 

Consider a system of three distinguishable particles, each having spin \( S = \frac{1}{2} \) such that \( S_z = \pm \frac{1}{2} \) with corresponding magnetic moments \( \mu_z = \pm \mu \). When the system is placed in an external magnetic field \( H \) pointing along the z-axis, the total energy of the system is \( \mu H \). Let \( x \) be the state where the first spin has \( S_z = \frac{1}{2} \). The probability of having the state \( x \) and the mean magnetic moment (in the +z direction) of the system in state \( x \) are
For a finite system of Fermions where the density of states increases with energy, the chemical potential