PX284 - F1 - isolated system or microcanonical ensemble

from gibb's formula:

\begin{matrix} S = - k_{B} \sum_{i} P_{i} \ln P_{i} \\ U = \sum_{i} E_{i} P_{i} \end{matrix}

thermodynamics

the second law of thermodynamics states that, for all processes in an isolated system:

d S \geq 0

where, the equality is for reversible changes

a process is reversible if it:
- is performed quasi-statically (slow changes via equilibrium states)
- involves no dissipation (eg.: no turbulence, electrical resistance)
for a general process, $S$ increases until $d S = 0$ at equilibrium, ie. $S$ is maximized

statistical mechanics

in statistical mechanics: $S$ for an isolated two-state system:

S = - k_{B} \sum_{i} P_{i} \ln P_{i} = - k_{B} [P_{1} \ln P_{1} + P_{2} \ln P_{2}]

\frac{d S}{d P_{1}} = - k_{B} [\ln P_{1} + 1 + \frac{d P_{2}}{d P_{1}} (\ln P_{1} + 1)] = 0

at maximum (equilibrium)

but, $P_{1} + P_{2} = 1$
so, $\frac{d P_{2}}{d P_{1}} = 1$
at maximum $S : \ln P_{1} + 1 - \ln P_{2} - 1 = 0 ⟹ l n P_{1} = \ln P_{2} ⟹ P_{1} = P_{2}$
this shows that the microstates are equiprobable
it can be shown that this is true for an $N$ state system
therefore, maximizing $S \equiv$ equiprobable microstates
this is he definition of equilibrium in microcanonical ensemble