PX284 - F2a - systems with fixed T and V or canonical ensemble

in thermodynamics, the hemholtz free energy:

F = U - T_{0} S

where, $T = T_{0}$ is fixed

d F = d Q + d W - T_{0} d S

but, $d Q \leq T_{0} d S$

∴ d F \leq d W ⟹ d F \leq 0

for a mechanically isolated system, ie. $V$ is fixed

$F$ decreases until a minimum is reached at equilibrium
similarly, for fixed $S$ and $p$ , the enthalpy:

d H \leq 0

and, for fixed $T$ and $p :$

d G \leq 0

PX284 - F2b - aside on lagrange multipliers

minimizing $F = U - T S$ under the constraint $\sum_{i} P_{i} = 1$ , ie:

{\vec{\nabla}}_{P_{i}} | U - T S + λ \sum_{i} P_{i} | = 0

trying a particular state with the probability, $P_{j} :$

\begin{matrix} \frac{\partial}{\partial P_{j}} (\sum_{i} E_{i} P_{i} + k_{B} T \sum_{i} P_{i} \ln P_{i} + λ \sum_{i} P_{i}) = 0 \\ E_{j} + k_{B} T (\ln P_{j} + 1) + λ = 0 \\ \ln P_{j} = - \frac{E_{j}}{k_{B} T} - \frac{λ}{k_{B} T} - 1 \\ P_{j} = A \exp (- \frac{E_{j}}{k_{B} T}) \end{matrix}

where, $A = \frac{1}{Z}$

this is the boltzmann distribution
therefore, minimizing $F \equiv$ boltzmann distribution
this is the definition of equilibrium in a canonical ensemble
it shows that thermodynamics and statistical mechanics are consistent