I Concept of Thermal Equilibrium in the Context of Canonical Ensemble

[Dario56]

Canonical ensemble can be used to derive probability distribution for the internal energy of the closed system at constant volume $V$ and number of particles $N$ in thermal contact with the reservoir.

Also, it is stated that the temperature of both system and reservoir is the same, i.e. they are in the thermal equilibrium.

In statistical thermodynamics, the thermal equilibrium arises naturally from following consideration:

If we consider closed system of interest in thermal contact with the reservoir such that these two form an isolated system than total energy $E$ is defined as: $E = E_S + E_R$

Total number of microstates of the isolated system with internal energy $E$ is given by: $$ \Omega(E) = \sum_{E_S}\Omega_S(E_S)\Omega_R(E - E_S) = \sum_{E_S}f(E_S) $$

If we wanted to find the internal energy of the system $E_S$ for which the number of microstates of the closed system $f(E_S)$ has a maximum value, we do the following: $$\frac {df(E_S)} {dE_S} = 0$$

This leads us to: $$ \Omega_R\frac {d\Omega_S} {dE_S} + \Omega_S \frac {d\Omega_R} {dE_S} = 0 $$

Because the total system is isolated: $$dE_S = - dE_R$$

Which leads us towards: $$ \frac {dln\Omega_S} {dE_S} = \frac {dln\Omega_R} {dE_R} $$

Taking into account Boltzmann definition of entropy, we know that the previous expression defines temperature equality of the closed system and the reservoir or thermal equilibrium condition.

Important thing to note is that thermal equilibrium is defined when the closed system of interest has the internal energy which corresponds to the maximum number of microstates of the isolated system.

Such state is the most probable and in the thermodynamic limit we are basically certain to find the system's internal energy very close to the value which maximizes the number of microstates.

However, canonical ensemble can be used, as it is mentioned earlier, to derive the probability distribution of the internal energy of the system in the more general case when the thermodynamic limit isn't supposed.

It is from the canonical ensemble that we know that the variance of internal energy for the closed system tends to zero when the number of particles tends to infinity.

This means that if the thermodynamic limit isn't assumed, thermal equilibrium condition losses its significance as system can be found in many different energy states with considerable probability not only in the energy state which maximizes the number of microstates as it is the case in the thermodynamic limit.

This is what doesn't really make sense because the canonical ensemble should be applicable in more general case than thermodynamic limit and yet it uses the concept of thermal equilibrium as its condition which only has sense in the thermodynamic limit.

I think solution to this question may be that the thermal equilibrium isn't the general requirement for the canonical ensemble and that it is only valid in the thermodynamic limit which covers almost all cases in reality.
Although, if you look on the Wikipedia: https://en.m.wikipedia.org/wiki/Canonical_ensemble
you'll find that thermal equilibrium is included in the definition of the canonical ensemble.

 

[anuttarasammyak]

I have no idea whether canonical ensemble should be applicable beyond thermal equilibrium. Thermal equilibrium with canonical ensemble in its foundation is kept during thermodynamical process to explain our daily experience. I am not sure whether canomical ensemble plays more role than that or not.

 

[Dario56]

I have no idea whether canonical ensemble should be applicable beyond thermal equilibrium. Thermal equilibrium with canonical ensemble in its foundation is kept during thermodynamical process to explain our daily experience. I am not sure whether canomical ensemble plays more role than that or not.

Yes, I get your point. I think that canonical ensemble is more general than the thermodynamic limit condition because we derive what happens in the thermodynamic limit from the canonical ensemble.

The fact that we know that variance of the closed system's internal energy tends to zero when number of particles tends to infinity comes from the probability distribution of closed system's internal energy. This distribution is derived exactly from the canonical ensemble (CE) and so CE should be more general than the thermodynamic limit.

Things aren't clear to me here. Question can be formulated as: What is the significance of thermal equilibrium if our isolated system isn't in the state in which the number of microstates is maximized?

While we will find our isolated system at this state always in the thermodynamic limit, what if the thermodynamic limit isn't assumed?

I think answer to my question may be that when it is said that temperature is fixed in the canonical ensemble, that doesn't neccesarily imply thermal equilibrium, but rather it just specifies temperature as a variable which defines the canonical ensemble along with the volume and number of particles.

 

[Demystifier]

This is what doesn't really make sense because the canonical ensemble should be applicable in more general case than thermodynamic limit

Not really. The canonical ensemble is much more than the statement that energy is well defined and conserved. In the canonical ensemble, all states (consistent with the given energy) have the same probability, which corresponds to the thermodynamic equilibrium. Thermodynamic equilibrium is not exactly the same as thermodynamic limit, but they are closely related.

 

[vanhees71]

That's the microcanonical ensemble. In the canonical ensemble you consider a system which can exchange energy with a "heat bath" but no particles. Thus not the energy is fixed but the temperature, while the particle number is fixed and constant.

You can do thermodynamics with finite systems. The thermodynamical limit is an idealization for "very large systems". In some cases, where you look at "mesoscopic systems" "finite-size effects" can be important.

 

[Demystifier]

That's the microcanonical ensemble.

Yes, that's what I meant. I was mislead by the initial post which talks of "canonical" ensemble but really presents equations of the microcanonical one.