  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.
