I Questions about a System in Equilibrium

[Kashmir]

Pathria, Statistical mechanics, pg 93

"We consider the given system $A$ as immersed in a large reservoir $A'$, with which it can exchange both energy and particles. After some time has elapsed, the system and the reservoir are supposed to attain a state of mutual equilibrium. Then, according to Section 1.3, the system and the reservoir will have a common temperature $T$ and a common chemical potential $\mu$. The fraction of the total number of particles $\mathrm{N^0}$ and the fraction of the total energy $E^{\text {o }}$ that the system A can have at any time $t$ are, however variables..."

Why does author say "The fraction of the total number of particles $\mathrm{N^0}$ and the fraction of the total energy $E^{\text {o }}$ that the system A can have at any time $t$ are, however variables...". Since A is at equilibrium then total number of particles and energy of A shouldn't change but author says otherwise.

Why is it so?

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[Steve4Physics]

Since A is at equilibrium then total number of particles and energy of A shouldn't change but author says otherwise.

Why is it so?

Maybe this is a reference to random fluctuations.

For example, suppose the systems are gaseous. There will be continual fluctuations in the distribution of particles and energy between A and A' due to the random particle motion. (The time-averages of the various quantities will be constant though.)

Imagine A is a very small system. The fractional variations of the number of particles and energy in A could then be large.

Just a guess though.

 

[DrClaude]

Since A is at equilibrium then total number of particles and energy of A shouldn't change but author says otherwise.

$A$ represents an open system, so it can exchange both energy and particles with $A'$. Only the temperature and chemical potential are fixed at equilibrium. As @Steve4Physics wrote, the energy and the number of particles will fluctuate. This is characteristic of the grand canonical ensemble.

 

[Kashmir]

$A$ represents an open system, so it can exchange both energy and particles with $A'$. Only the temperature and chemical potential are fixed at equilibrium. As @Steve4Physics wrote, the energy and the number of particles will fluctuate. This is characteristic of the grand canonical ensemble.

if number of particles and energy of system A changes why won't it's temperature change?

 

[Steve4Physics]

if number of particles and energy of system A changes why won't it's temperature change?

It will!

For example, imagine A and A' are filled with an ideal gas. At equilibrium there will still be fluctuations in the number of particles in A, and their mean-square speed (hence their average kinetic energy and hence the temperature of A).

For a short time the temperature of A might rise/fall - and the temperature of A' would fall/rise (by a lesser amount due to A' having a larger size).