Theory of the Robin quantum wall in a linear potential. II.   Thermodynamic properties
  A theoretical analysis of the thermodynamic properties of the Robin wall characterized by the extrapolation length $\Lambda$ in the electric field $\mathscr{E}$ that pushes the particle to the surface is presented both in the canonical and two grand canonical representations and in the whole range of the Robin distance with the emphasis on its negative values which for the voltage-free configuration support negative-energy bound state. For the canonical ensemble, the heat capacity at $\Lambda<0$ exhibits a nonmonotonic behavior as a function of the temperature $T$ with its pronounced maximum unrestrictedly increasing for the decreasing fields as $\ln^2\mathscr{E}$ and its location being proportional to $(-\ln\mathscr{E})^{-1}$. For the Fermi-Dirac distribution, the specific heat per particle $c_N$ is a nonmonotonic function of the temperature too with the conspicuous extremum being preceded on the $T$ axis by the plateau whose magnitude at the vanishing $\mathscr{E}$ is defined as $3(N-1)/(2N)k_B$, with $N$ being a number of the particles. The maximum of $c_N$ is the largest for $N=1$ and, similar to the canonical ensemble, grows to infinity as the field goes to zero. For the Bose-Einstein ensemble, a formation of the sharp asymmetric feature on the $c_N$-$T$ dependence with the increase of $N$ is shown to be more prominent at the lower voltages. This cusp-like dependence of the heat capacity on the temperature, which for the infinite number of bosons transforms into the discontinuity of $c_N(T)$, is an indication of the phase transition to the condensate state. Qualitative and quantitative explanation of these physical phenomena is based on the variation of the energy spectrum by the electric field. 
O. Olendski
quant-ph cond-mat.mes-hall math-ph math.MP