Thermodynamic properties of condensates with spin- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> atoms ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> <mml:mo>≥</mml:mo> <mml:m

Abstract When the Hamiltonian of a condensate containing spin- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="fraktur">f</mml:mi> </mml:math> bosons is dominated by attractive singlet pairing interaction with the strength − ∣ G 0 ∣ and when the temperature T is in the neighborhood of zero, the many-body Schrödinger equation can be exactly solved. Based on the complete and exact spectra the thermodynamic properties of these systems different in particle number N and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="fraktur">f</mml:mi> </mml:math> have been studied. The variation of the function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mfrac> <mml:mrow> <mml:mo>∂</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>∂</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> </mml:mfrac> </mml:math> against x is firstly studied, where C is the capacity, x ≡ lg( k B T /∣ G 0 ∣), and k B is the Boltzmann constant. It was found that the variation can be divided into four stages. In the very narrow first stage, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mfrac> <mml:mrow> <mml:mo>∂</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>∂</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> </mml:mfrac> </mml:math> appears as a sharp peak. The second stage is broad where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mfrac> <mml:mrow> <mml:mo>∂</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>∂</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> . The third stage is also very narrow where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mfrac> <mml:mrow> <mml:mo>∂</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>∂</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> </mml:mfrac> </mml:math> appears as a sharp anti-peak. In the fourth stage, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mfrac> <mml:mrow> <mml:mo>∂</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>∂</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> </mml:mfrac> </mml:math> becomes zero again. All the thermodynamic functions: the internal energy, the capacity, and the entropy vary accordingly. How these functions are affected by N , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="fraktur">f</mml:mi> </mml:math> , and ∣ G 0 ∣ in each stage has been clarified. The spin-thermodynamics would replace the traditional thermodynamics in the neighborhood of zero temperature, where the spin degrees of freedom play an essential role.