Entropy

Εντροπία

Isolated system - Microcanonical ensemble (δεν ανταλλάζει ενέργεια μάζα σωματίδια)

S = S (E, V, N) = k_{B} * l n (Ω (E, V, N))

where
Ω : Statistical weight
$k_{B}$ : Boltzmann constant ( $1.38 * 10^{- 23} \frac{J}{K}$ )

Ω(Α+Β)=Ω(Α)Ω(Β)
Ενώ S(A+B)=S(A)+S(B)

Derivation from general formula:

Isolated system -> (E,V,N)=const. with $Ω$ microstates (we cannot have a microstate not corresponding to energy E) each with $p = \frac{1}{Ω}$ be

S = - k \sum p_{r} \ln p_{r} = - k (\frac{1}{Ω} \ln \frac{1}{Ω} + \frac{1}{Ω} \ln \frac{1}{Ω} + \frac{1}{Ω} \ln \frac{1}{Ω} . . . + \frac{1}{Ω} \ln \frac{1}{Ω}) = - k Ω (\frac{1}{Ω} \ln \frac{1}{Ω}) $ $ $ $ S = k \ln Ω

General definition

S (T, V, N) = k \cdot \ln Ω (\bar{E}, V, N) = - k \sum p_{r} \ln p_{r}

In a Closed system - Heat Bath - Canonical ensemble we have v systems where total entropy is the sum of entropies: $S_{v} = v * S$ And $v_{r} = v * p$
Supposing they are weakly interacting

Ω_{v} = \frac{v!}{v 1! v 2! v 3! . . . v r!}

Where vi the amount of systems in a microstate

S = k * l n Ω = k * l n \frac{v!}{v 1! v 2! v 3! . . . v r!} = k [v \cdot l n v - \sum_{r} v_{r} \cdot l n v_{r}] = . . . = - k \sum p_{r} \ln p_{r}

S = S (T, V, N)

Independen of energy fluctuations in closed system.
from Stirling law for N>> :

\ln N! = N \ln N - N

... mandle σελ. 63

Entropy.png.png