Statistical physics

adiabatic process angular momentum atoms Boltzmann Bose Bravais lattice calculate chemical potential classical statistics closed system coefficient components condition coordinates corresponding crystal curve defined degrees of freedom denote density density matrix depends derivative determined differentials distribution function energy levels entropy equal equation equilibrium expansion external Fermi Fermi gas field fluctuations formula free energy frequency gases Gibbs distribution given gives Hamiltonian Hence ideal gas independent integral interaction kinetic energy liquid macroscopic body magnetic mean value molecule momenta motion normalisation number of particles obtain oscillator parameter partition function phase space phase transition phonon pressure probability distribution PROBLEM properties Quantum Mechanics radiation regarded relation respect result rotation solid SOLUTION specific heat spectrum spin statistical physics statistical weight Substituting subsystem suffix summation symmetry temperature thermal thermodynamic potential thermodynamic quantities tion total number transition point vapour variables velocity vibrations volume wave functions zero