Bose gas with generalized dispersion relation plus an energy gap

Abstract

Bose-Einstein condensation in a Bose gas is studied analytically, in any positive dimensionality (d > 0) for identical bosons with any energy-momentum positive-exponent (s > 0) plus an energy gap ∆ between the ground state energy ε0 and the first excited state, i.e., ε = ε0 for k = 0 and ε = ε0 + ∆ + csk s , for k > 0, wherehk is the particle momentum and cs a constant with dimensions of energy multiplied by a length to the power s > 0. Explicit formula with arbitrary d/s and ∆ are obtained and discussed for the critical temperature and the condensed fraction, as well as for the equation of state from where we deduce a generalized ∆ independent thermal de Broglie wavelength. Also the internal energy is calculated from where we obtain the isochoric specific heat and its jump at Tc. When ∆ > 0, for any d > 0 exists a Bose-Einstein critical temperature Tc = 0 where the internal energy shows a peak and the specific heat shows a jump. Both the critical temperature and the specific heat jump increase as functions of the gap but they decrease as functions of d/s. At sufficiently high temperatures ∆-independent classical results are recovered while for temperatures below the critical one the gap effects are predominant. For ∆ = 0 we recover previous reported results.PACS numbers:

Two-step BEC coming from a temperature dependent energy gap

“…which decays exponentially as given by expression (28) of Ref. [11] with s = 2 and d = 3, and experimentally observed in helium four near absolute zero temperature. When ∆(T ) → 0 Eq.…”

“…To begin with, we confirm that for a temperature independent constant gap: a) All the calculated thermodynamic properties are constant ground state energy ε 0 independent, except for the internal energy, which is measured from the reference energy, N ε 0 , instead of from the zero ground state energy of an IBG, b) A constant gap notoriously increases the magnitude of the BEC critical temperature, where the isochoric specific heat shows a jump, while for T near zero its temperature dependence is exponential rather than proportional to T 3/2 , as for a 3D ideal Bose gas, as already observed in Ref. [11] The decay of the undamped BCS gap with infinite slope at T B causes abrupt changes in the behavior of the thermodynamic properties of the boson gas as they go through the temperature T B . In order to explore the effect of Cooper pairs creation before they reach the BEC critical density we propose a smooth transition from a non-zero gap to an equal to zero one at T B , multiplying the undamped BCS gap by an exponential decreasing function of temperature.…”

Quantum thermochemical engines

“…A second generalisation is the presence of general trapping potentials and density of states [135][136][137][138][139]. In this cases the range of chemical potentials are not changed and the equations of state show the same qualitative behaviour of the ideal homogeneous gas, e.g., with upperbounded bosonic particle number in the continuum approximation and the emergence of BEC phases above critical densities and at vanishing chemical potentials.…”

“…The rest of the paper is dedicated to concrete cycles where the efficinecy η rev is maximised with quantum degenerate gases, while small efficiency and load are obtained in the classical regime. Ideal homogeneous gases are discussed, and similar behaviours extend to interacting models that do not limit the efficiency range (as those discussed above), and in the presence of general trapping potentials and density of states [135][136][137][138][139] that do not alter the qualitative behaviours of homogeneous gases (e.g., the monotonicity and the (un)boundedness of the average particle number discussed in Appendix B).…”

“…In particular, BECs above a critical density occur when the polylogarithm functions in the expressions of N are bounded from above, i.e., s > 1 (see equation (B4). BECs above a critical density also occur with different mathematical forms of N [138,139].…”