# Is it possible to define a notion of temperature in a microcanonical ensemble?

I am thinking of a mircrocanonical ensemble as a finite system for which the number of particles, volume and the total energy is fixed. Is there a more refined view of this?

Can I think of temperature of this system as the average kinetic energy for all the particles?

To put the thing in a larger context - what is the implicit choice of ensemble when people define the entropy of a black-hole either as (1) horizon area or (2) via microstate counting or (3) macroscopically via quantum entropy function ?

Naively it feels that for a black-hole entropy one must think of a microcanonical ensemble because its not clear to me as to with what "bath" will it be able to exchange anything to maintain any chemical potentials for the conserved charges, volume or energy...may be I am being too naive...

• What about the standard definition of temperature for a microcanonical ensemble (appearing in every textbook on statistical mechanics) $\frac{1}{T} = \frac{\partial S}{\partial E}$? Where $S$ is the entropy of the system and $E$ its energy. Commented Apr 14, 2014 at 6:45
• @V.Moretti Well - the question is - whether this mathematical definition makes physical sense. A system in a microcanonical ensemble is an isolated system which can't exchange energy or particles or volume with anything else. Then how is the system equilibrating to a temperature? Can a system spontaneously determine its temperature without an infinite heat-bath? Isn't this the notion of temperature that leads to freaky things like negative temperature a finite system of spins? [...think of black-hole thermodynamics - isn't that a microcanonical ensemble?...] Commented Apr 14, 2014 at 19:14
• What I've read is that the microcanonical ensemble of a black hole in qft doesn't make sense, because infalling matter never reaches equilibrium. In string theory, it seems that strings would equilibrate with the black hole. Commented Jul 3, 2016 at 11:22
• Meaning of temperature in different thermostatistical ensembles (link to arXiv repository). Related: physics.stackexchange.com/q/623364/226902 Commented Aug 1, 2023 at 21:43