The Bethe Ansatz

The pressure is defined as the derivative of the free energy with respect to volume at fixed \(T\):

\begin{equation} P = - \left(\frac{\partial G (\mu, T)}{\partial L} \right)_T = \frac{T}{2\pi} \int_{-\infty}^\infty d\lambda \ln \left[ 1 + e^{-\epsilon(\lambda)/T}\right]. \tag{l.pr}\label{l.pr} \end{equation}

The isothermal and isoentropic (adiabatic) bulk moduli \(\kappa_T\) and \(\kappa_S\) are defined as

\begin{equation*} \kappa_{T,S} = -L \left( \frac{ \partial P}{\partial L}\right)_{T,S} = n \left( \frac{\partial P}{\partial n} \right)_{T,S}. \end{equation*}

The isothermal and isoentropic compressibilities \(\beta_{T,S}\) are the inverses of the bulk moduli,

\begin{equation*} \frac{1}{\beta_{T,S}} = \kappa_{T,S} = -L \left( \frac{\partial P}{\partial L} \right)_{T,S}. \end{equation*}

The speed of sound is defined macroscopically as (note: the factor of 2 comes from our conventions, in which we put the mass of the particles to \(1/2\))

\begin{equation} v_s^2 = 2 n \left( \frac{\partial P}{\partial n} \right)_{S} \hspace{10mm} \mbox{so} \hspace{10mm} v_s = \sqrt{\frac{2}{n \beta_S}} = \sqrt{\frac{2 \kappa_S}{n}}. \tag{l.vs}\label{l.vs} \end{equation}