Microstates and Macrostates

The microstates are a microscopic description of the possible states of the world.
- always well defined
- in an equilibrium, each microstate is equally likely
The macrostates are the states of the world that are still distinguishable on the basis of the information we retain.
- only well defined when the system is in the equilibrium
Phase space – a set of all possibly physical states of the system described by a given parametrization

Microstates map onto macrostates

Rolling a dice gives us 36 microstates, but only 11 macrostates, corresponding to sums of 2 through 12.

Using binomial distribution:

successes = list(range(n+1))
combinations = [comb(n,k) for k in successes]

Entropy: the universe is more likely to move towards more likely states.

There are too many atoms, so we think on logarithmic scales: $$S = k_B\ln \Omega$$ $k_{B} = 1.38 \times 10^{- 23} J / K, Ω - the number of microstates, S - entropy$

The equilibrium probability distribution describing how energy is spread between the two systems is given by:

P_{e q} (E_{A}) \propto Ω_{A} (E_{A}) Ω_{B} (E_{A B} - E_{A})

Or in terms of the entropy:

P_{e q} (E_{A}) \propto e^{S_{A} (E_{A}) / k_{B}} e^{S_{B} (E_{A B} - E_{B}) / k_{B}}