This question is based on example 8.3 in Molecular Driving Forces, Dill and Bromberg (https://books.google.ch/books?id=1gYPBAAAQBAJ&pg=PA136&lpg=PA136&dq=polymer+collapse+a+toy+model&source=bl&ots=lQrGS148oi&sig=ACfU3U1-3V0Vt8tHwba9cyWAD_cYSTCTyw&hl=en&sa=X&ved=2ahUKEwjMkrfI99XiAhXwx6YKHTHjCwQQ6AEwAXoECAkQAQ#v=onepage&q=polymer%20collapse%20a%20toy%20model&f=false)
In this problem Dill and Bromberg calculate the change in Helmholtz Free energy when a polymer collapses from an open configuration (higher energy, higher entropy) into a closed configuration (lower energy, lower entropy). They find that above a critical temperature the open configuration has a lower Helmholtz free energy and therefore conclude that this configuration is favoured at temperatures higher than this value. However, they also say that the exact composition needs to be calculated by means of considering a canonical ensemble.
My question now is: One can easily calculate the Boltzmann distribution corresponding to this system and see that at high temperatures one will get a closed configuration with a probability of 1/5 and an open configuration with a probability of 4/5 when evaluating as T goes to infinity. (They assume that there are four open configurations (energy: 0) and only one closed configuration (energy: -E)). But why do no ALL polymers collapse when such a process would clearly be favourable at high temperatures in terms of the Helmholtz free energy?
Of course I agree with the result obtained from the Boltzmann distribution! I just don't seem to grasp why not ALL of the polymers collapse.