I Fermi energy definition and Fermi-Dirac distribution

[chikchok]

TL;DR

Fermi energy definition and fermi-dirac distribution

1)In my book , there is a definition of fermi energy as topmost filled level in the ground state of an N electron system. This definition holds only for absolute zero,right? If it is not absolute zero,fermi energy is the energy at which the probability of a state being occupied is 50 percent. Please, tell me if I am understanding this correctly.
2)I was wondering why at low temperatures Fermi-Dirac function goes sharply from 1 to 0 and for higher temperature it goes down smoothly. Is it reasonable to assume that for low temperature, levels below the fermi level are filled and all above are empty? But why it does not happen with higher temperatures. Thank you in advance.

 

[hutchphd]

1) Stating "In the ground state" obviates mention of temperature.
2) Temperature is a measure of average internal energy. An isolated system at lowest energy will be in the ground state. If connected to a thermal reservoir it will have a finite probability of being in an excited state with a probability that is roughly negative exponential in energy e.g. the Boltzmann factor. Careful consideration of quantum effects gives rise to Fermi-Dirac statistics for half integer spin which gives a "step" function for T=0

 

[chikchok]

1) Stating "In the ground state" obviates mention of temperature.
2) Temperature is a measure of average internal energy. An isolated system at lowest energy will be in the ground state. If connected to a thermal reservoir it will have a finite probability of being in an excited state with a probability that is roughly negative exponential in energy e.g. the Boltzmann factor. Careful consideration of quantum effects gives rise to Fermi-Dirac statistics for half integer spin which gives a "step" function for T=0

so for non-absolute temperature fermi energy is not the energy of topmost filled level anymore?

 

[hutchphd]

The Fermi Energy is defined as the zero T result. It is "5" for your graphed system. With finite temperature occupation of levels is fluctuating.