已经导出,少体系统的热力学量的表达式为:

T=2ENkBp=2ELV=LS=12NkBln(EL2)+C\begin{aligned} T&=\dfrac{2E}{Nk_B} \\ p&=\dfrac{2E}L \\ V&=L \\ S&=\frac12Nk_B\ln(EL^2)+C \\ \end{aligned}

带入麦克斯韦关系式:

(TV)S=(pS)V(Tp)S=(VS)p(SV)T=(pT)V(Sp)T=(VT)p\begin{aligned} \left(\dfrac{\partial T}{\partial V}\right)_S&=-\left(\dfrac{\partial p}{\partial S}\right)_V \\ \left(\dfrac{\partial T}{\partial p}\right)_S&=\left(\dfrac{\partial V}{\partial S}\right)_p \\ \left(\dfrac{\partial S}{\partial V}\right)_T&=\left(\dfrac{\partial p}{\partial T}\right)_V \\ \left(\dfrac{\partial S}{\partial p}\right)_T&=-\left(\dfrac{\partial V}{\partial T}\right)_p \end{aligned}

  • 对于第一式,由

    S=f(EL2)T=2EL2NkB1L2V=LS=f(EL^2) \\ T=\dfrac{2EL^2}{Nk_B}\cdot\dfrac{1}{L^2} \\ V=L

    故:

    (TV)S=2L32EL2NkB=4ELNkB\left(\dfrac{\partial T}{\partial V}\right)_S=-\dfrac{2}{L^3}\cdot\dfrac{2EL^2}{Nk_B}=-\dfrac{4E}{LNk_B}

    V=f(L)p=2ELS=12NkBlnE+NkBlnL+CV=f(L) \\ p=\dfrac{2E}L \\ S=\frac12Nk_B\ln E+Nk_B\ln L+C

    故:

    (pS)V=2LdE12NkBdlnE=4ELNkB-\left(\dfrac{\partial p}{\partial S}\right)_V=-\dfrac{\dfrac{2}{L}\text dE}{\frac12Nk_B\text d\ln E}=\dfrac{4E}{LNk_B}

    故:

    (TV)S=(pS)V\left(\dfrac{\partial T}{\partial V}\right)_S=-\left(\dfrac{\partial p}{\partial S}\right)_V

  • 对于第二式,由

    S=f(EL2)T=2EL2NkB1L2p=2EL21L3S=f(EL^2) \\ T=\dfrac{2EL^2}{Nk_B}\cdot\dfrac{1}{L^2} \\ p=2EL^2\cdot\dfrac{1}{L^3}

    故:

    (Tp)S=2L33L41NkB=2L3NkB\left(\dfrac{\partial T}{\partial p}\right)_S=-\dfrac{\dfrac{-2}{L^3}}{\dfrac{-3}{L^4}}\cdot\dfrac{1}{Nk_B}=\dfrac{2L}{3Nk_B}

    p=f(EL)V=LEES=32NkBlnE+NkBlnLE+Cp=f(\dfrac EL) \\ V=\dfrac{L}{E}\cdot E \\ S=\frac32Nk_B\ln E+Nk_B\ln \dfrac LE+C

    故:

    (VS)p=LE32NkBdEdlnE=2L3NkB\left(\dfrac{\partial V}{\partial S}\right)_p=\dfrac{\dfrac LE}{\frac32Nk_B}\cdot\dfrac{\text dE}{\text d\ln E}=\dfrac{2L}{3Nk_B}

    故:

    (Tp)S=(VS)p\left(\dfrac{\partial T}{\partial p}\right)_S=\left(\dfrac{\partial V}{\partial S}\right)_p

  • 对于第三式,由

    T=f(E)S=NkBlnL+12NkBlnE+CV=LT=f(E) \\ S=Nk_B\ln L+\frac12Nk_B\ln E+C \\ V=L

    故:

    (SV)T=1LNkB=NkBL\left(\dfrac{\partial S}{\partial V}\right)_T=\dfrac{1}{L}\cdot Nk_B=\dfrac{Nk_B}{L}

    V=f(L)p=2ELT=2ENkBV=f(L) \\ p=\dfrac{2E}{L} \\ T=\dfrac{2E}{Nk_B}

    故:

    (pT)V=2L2NkB=NkBL\left(\dfrac{\partial p}{\partial T}\right)_V=\dfrac{\dfrac 2L}{\dfrac{2}{Nk_B}}=\dfrac{Nk_B}{L}

    故:

    (SV)T=(pT)V\left(\dfrac{\partial S}{\partial V}\right)_T=\left(\dfrac{\partial p}{\partial T}\right)_V

  • 对于第四式,由

    T=f(E)S=NkBlnL+12NkBlnE+Cp=2E1LT=f(E) \\ S=Nk_B\ln L+\frac12Nk_B\ln E+C \\ p=2E\cdot\dfrac 1L

    故:

    (Sp)T=dlnLd1LNkB2E=NLkB2E\left(\dfrac{\partial S}{\partial p}\right)_T=\dfrac{\text d\ln L}{\text d \dfrac1L}\cdot \dfrac{Nk_B}{2E}=-\dfrac{NLk_B}{2E}

    p=f(EL)V=LEET=2ENkBp=f(\dfrac EL) \\ V=\dfrac LE\cdot E \\ T=\dfrac{2E}{Nk_B}

    故:

    (VT)p=LE2NkB=NLkB2E-\left(\dfrac{\partial V}{\partial T}\right)_p=-\dfrac{\dfrac LE}{\dfrac{2}{Nk_B}}=-\dfrac{NLk_B}{2E}

    故:

    (Sp)T=(VT)p\left(\dfrac{\partial S}{\partial p}\right)_T=-\left(\dfrac{\partial V}{\partial T}\right)_p

可以看到,少体硬球系统仍然满足麦克斯韦关系式。