Partition function (statistical mechanics): Difference between revisions

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Applying [[calculus of variations|variational calculus]] with constraints (analogous in some sense to the method of [[Lagrange multipliers]]), we write the Lagrangian (or Lagrange function) <math> \mathcal{L} </math> as

<math display="block">

\mathcal{L} = \left( -k_\text{B} \sum_i \rho_i \ln \rho_i \right) -+ \lambda_1 \left( 1 - \sum_i \rho_i \right) -+ \lambda_2 \left( U - \sum_i \rho_i E_i \right) .</math>

Varying and extremizing <math> \mathcal{L} </math> with respect to <math> \rho_i </math> leads to

<math display="block">\begin{align}

0 & \equiv \delta \mathcal{L} \\

&= \delta \left( - \sum_i k_\text{B} \rho_i \ln \rho_i \right) + \delta \left-( \lambda_1 - \sum_i \lambda_1 \rho_i \right) + \delta \left-( \lambda_2 U - \sum_i \lambda_2 \rho_i E_i \right) \\

&= \sum_i \bigg[ \delta \Big( - k_\text{B} \rho_i \ln \rho_i \Big) + \delta \Big( \lambda_1 \rho_i \Big) + \delta \Big( \lambda_2 E_i \rho_i \Big) \bigg] \\

&= \sum_i \left[ \frac{\partial}{\partial \rho_i } \Big( - k_\text{B} \rho_i \ln \rho_i \Big) \, \delta ( \rho_i ) + \frac{\partial}{\partial \rho_i } \Big( \lambda_1 \rho_i \Big) \, \delta ( \rho_i ) + \frac{\partial}{\partial \rho_i } \Big( \lambda_2 E_i \rho_i \Big) \, \delta ( \rho_i ) \right] \\