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**The first approach:**

Suppose we are dealing with a closed system where the system is composed of heat bath (R) and inside it there is a very small system (E), the two systems are in thermal equilibrium with temperature T.

This is called the canonical partition function. I will not write the derivation but the canonical partition function obtained is [itex]Z=\sum_{l=0}^{Ω_{s}} e^{-E_{l}/k_{B}T}[/itex] Where [itex]E_{l}[/itex] is the energy of the system (E) and [itex]Ω_{s}[/itex] is the total number of microstates for system (E).

[itex]p(E_l)=e^{-E_l/k_BT}/Z[/itex]

The system (E) +(R) is isolated.

To emphasis here the system is composed of a heat reservoir (R), which is a huge system and a very small system (E).

**The second approach:**

we have here a big isolated system (G), the number of microstates at thermal equilibrium is approximately equal to t{nj*}, where t{nj*} is the number of microstates corresponding to the distribution{nj*} which is the average distribution. The average distribution is the thermal distribution.

So [itex]Ω\approx t{n_j*}[/itex] ( when N the number of particles is very large). This means that the thermal distribution is the most probable distribution.

We have that t{nj}=N!/nj! then ln(t)=lnN!-∑ln(nj!)⇒lnt=(NlnN-N)-∑(njln(nj)-nj) Note : nj is the number of particles in the state j.

The most probable distribution correspond to the maximum t, which satisfies the equation d(lnt )=-∑dnjln(nj)=0.

As N and U are fixed then d(N)=∑d(nj)=0 and d(U)=∑εjdnj=0 where εj is the energy corresponding to the state j.

Using Lagrange multiplier method, we will get ∑(-ln(nj*)+α+βεj)dnj=0 for any values of α and β.

then -ln(nj*)+α+βεj=0 then"" nj*=exp(α+βεj)"" which is the boltzmann distribution.

the partition function is defined here by Z=∑exp(βεj) where β=-1/kT where k is the boltzmann constant.

Now lets divide the isolated system (G) into two systems "p" and "q".

where the system "p" has energy Up and number of particles Np,and "q" has energy Uq and number of particles Nq.

At thermal equilibrium, we have for system "p" nj=exp(αp+βεj) and for system "q" nj'=exp(αq+βεj') Where nj and nj' are the number of particles that have energies εj and εj' respectively at thermal equilibrium.

For system"p" Zp=Σexp(βεj) and for "q" Zq=∑exp(βεj').

My question: why the partition function is defined differently in the second case( where I divided the isolated system into two systems p and q),as εj here does not represent the energy of the system but the energy of specific particles in the system ? Is it because it is not considered an canonical ensemble as there is no heat bath?

How the partition function can be different from case to case and why?Thanks!