The Bethe Ansatz

Equilibrium at finite $T$ : the Thermodynamic Bethe Ansatz
Lieb-Liniger
Thermodynamic properties

Thermodynamic propertiese.l.tp

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

\begin{matrix} (l.pr) & P = - {(\frac{\partial G (μ, T)}{\partial L})}_{T} = \frac{T}{2 π} \int_{- \infty}^{\infty} d λ \ln [1 + e^{- ϵ (λ) / T}] . \end{matrix}

The isothermal and isoentropic (adiabatic) bulk moduli $κ_{T}$ and $κ_{S}$ are defined as

κ_{T, S} = - L {(\frac{\partial P}{\partial L})}_{T, S} = n {(\frac{\partial P}{\partial n})}_{T, S} .

The isothermal and isoentropic compressibilities $β_{T, S}$ are the inverses of the bulk moduli,

\frac{1}{β_{T, S}} = κ_{T, S} = - L {(\frac{\partial P}{\partial L})}_{T, S} .

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{matrix} (l.vs) & v_{s}^{2} = 2 n {(\frac{\partial P}{\partial n})}_{S} so v_{s} = \sqrt{\frac{2}{n β_{S}}} = \sqrt{\frac{2 κ_{S}}{n}} . \end{matrix}

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Author: Jean-Sébastien Caux

Created: 2024-01-18 Thu 14:24

Thermodynamic properties e.l.tp

Thermodynamic propertiese.l.tp