User:Frobnitzem/AmsTex

$$ \begin{alignat}{3} \beta \mu^{ex} &= \beta \mu_{IS}^{ex} + \beta \mu_{HS}^{ex} + \beta \mu_{att}^{ex} && = \ln x_0 - \ln p_0 + \beta \mu_{att}^{ex} && \\

\beta \mu_{IS}^{ex} &\equiv \quad \ln \left( \operatorname{P}_{\Delta U} \left(r_{\text{min}} > \lambda \right) \right) && = \quad \ln {\left\langle e^{-\beta \Delta U_{HS}} \right\rangle}_{\Delta U} && \equiv \ln x_0 \\

\beta \mu_{HS}^{ex} &\equiv - \ln \left( \operatorname{P}_0 \left(r_{\text{min}} > \lambda \right) \right) && = - \ln {\left\langle e^{-\beta \Delta U_{HS}}\right\rangle}_0 && \equiv - \ln p_0 \\

\beta \mu_{att}^{ex} &= - \ln {\left\langle e^{-\beta \Delta U} | r_{\text{min}} > \lambda \right\rangle}_0 && = - \ln {\left\langle e^{-\beta \Delta U} \right\rangle}_{HS} && \\ \end{alignat} $$

$$ e^{-\beta \mu^{ex}} =

\left( \frac{\int{e^{-\beta \left( U + \Delta U + \Delta U_{HS} \right)} dx}} {\int{e^{-\beta \left( U + \Delta U \right)} dx}} \right)^{-1}

\left( \frac{\int{e^{-\beta \left( U + \Delta U_{HS} \right)} dx}} {\int {e^{-\beta U} dx}} \right)

\left( \frac{\int{e^{-\beta \left( U + \Delta U + \Delta U_{HS} \right)} dx}} {\int{e^{-\beta \left( U + \Delta U_{HS} \right)} dx}} \right) $$