Theory of the Robin quantum wall in a linear potential. II. Thermodynamic properties

Abstract

A theoretical analysis of the thermodynamic properties of the Robin wall characterized by the extrapolation length Λ in the electric field 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 Λ < 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 E and its location being proportional to (− ln E ) −1 . For the Fermi-Dirac distribution, the specific heat per particle cN 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 E is defined as 3(N − 1)/(2N )kB, with N being a number of the particles. The maximum of cN 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 cN -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 cN (T ), is an indication of the phase transition to the condensate state. Some other physical characteristics such as the critical temperature Tcr and ground-level population of the Bose-Einstein condensate are calculated and analyzed as a function of the field and extrapolation length. Qualitative and quantitative explanation of these physical phenomena is based on the variation of the energy spectrum by the electric field.

Theory of the Robin quantum wall in a linear potential. I. Energy spectrum, polarization and quantum‐information measures

“…The energy spectrum can be efficiently controlled by the applied voltage; for example, for the attractive Robin surface at the small fields it presents a quasi continuous part where the tiny energy difference between the levels decreases with the increase of the quantum number n and, additionally, downward-split single bound state separated from the nearest upper lying counterparts by the unity gap. Spectacular thermodynamic consequences of such energy arrangement are discussed in the following paper [89].…”

“…4. Similar to the quantum well with miscellaneous combinations of the Dirichlet and Neumann BCs at the opposite walls in the electric field [39,88], a companion paper [89] calculates statistical properties of the same structure.…”

Thermodynamic Properties of the 1D Robin Quantum Well

“…This is exemplified in panel (a) of of E means, according to Equation (4), larger values of the heat capacity whose expression in the same limit reads: This is exemplified in panel (a) of of E means, according to Equation (4), larger values of the heat capacity whose expression in the same limit reads:…”

“…Note that it turns to zero at the temperature β E =0 , which is found from equation: 1 π 2 2 e β/(π 2 2 ) = π 1/2 4β 3/2 (21) Its solution is expressed via the Lambert W function: [4, β E =0 = 3 2 π 2 2 W 16 2/3 24 1 π 1/3 2/3 (22) Asymptotics of the Lambert function [22] W(x) → ln…”

“…(7) Just this parameter was used for explaining a giant enhancement of the specific heat of the attractive Robin wall in vanishingly small electric fields [4] when it is practically the same for many quantum orbitals with n ≥ 1. .…”