Statistical Physics

10.5 Properties of a Free-Boson Gas 175Thus, at T = T c , the pressure is reduced to about one-half of the classicalvalue given by PV = Nk B T . From this equation, it might be thought thatBoyle’s law, PV = constant at fixed temperature, is obeyed. However, this isnot correct. Since T c ∝ (N/V ) 2/3 , the pressure at T ≤ T c is independent ofthe volume and depends only on T :P (T,V,N)=k B T( ) 3/2 ( )mkB T 52π 2 ζ . (10.94)2From the internal energy, the heat capacity at constant volume can becalculated:( )∂U(T,V,N)C V =∂TV,N= 15ζ(5/2) ( ) 3/2 ( ) 3/2 T T4ζ(3/2) Nk B =1.926Nk B . (10.95)T c T cThat is, it decreases in proportion to T 3/2 .Thermodynamic Variables at T>T cAt T > T c , calculation becomes difficult. In order to proceed, we introducea function F σ (α), whereF σ (x) ≡ 1Γ (σ)∫ ∞0y σ−1 ∞dye x+y − 1 = ∑n −σ e −nx . (10.96)For σ>1, the value at x = 0 is given by the zeta function: F σ (0) = ζ(σ)(σ >1). The derivative with respect to x is related to F σ−1 , i.e.n=1ddx F σ(x) =−F σ−1 (x) . (10.97)In terms of this function, the thermodynamic functions can be written asN(T,V,µ)=V4π 2 3 (2mk BT ) 3/2 F 3/2 (−βµ) , (10.98)U(T,V,µ)=V4π 2 3 (2mk BT ) 3/2 k B TF 5/2 (−βµ)= 3 2 Nk BT F 5/2(−βµ)F 3/2 (−βµ) , (10.99)