Thermodynamics

Consider a three-dimensional system of non-interacting bosons with zero chemical potential. The energy of the system , where is the wave vector. The low temperature specific heat of the system at constant volume depends on the temperature as . The value of is ___ (in integer).

The Hamiltonian of a system of particles in volume at temperature is where and are positive constants. The ensemble average of the Hamiltonian is , where is the Boltzmann constant. The value of is _____ (in integer).

If a thermodynamical system is adiabatically isolated and experiences a change in volume under an externally applied constant pressure, then the thermodynamical potential minimized at equilibrium is the