I Independence of Trace-Partition function

AI Thread Summary
The discussion revolves around calculating the partition function for two decoupled systems, confirming that the probability density function (PDF) is the product of each subsystem's PDF. The expression presented is validated, showing that the total Hamiltonian can be expressed as a product of the individual Hamiltonians. It is clarified that the partition function for the combined system is indeed the product of the partition functions of the individual subsystems. The Gibbs distribution for each subsystem further supports this conclusion, reinforcing that the total system's probability density is a product of the individual densities. Overall, the calculations and reasoning confirm the correctness of the approach to the partition function.
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Partition Function of a separable hamiltonian
I am trying to calculate the partition function of the system of two completely decoupled systems. Probability-wise, the decoupled nature means that the PDF is the product of the PDF of each subsystem. I just wanted to be sure that it would translate into:

$$
H = \sum_{k_i, s_i}e^{H_s(s_i)}e^{H_k(k_i)} = \sum_{k_i}(\sum_{s_i}e^{H_s(s_i)})e^{H_k(k_i)} = \sum_{s_i}e^{H_s(s_i)}\sum_{k_i}e^{H_k(k_i)} = Tr(e^{H_s(s_i)})*Tr(e^{H_k(k_i)})
$$

I know the question seems trivial, but I got a bit confused, and I would like to be 100% sure. Thank you for any help you can provide
 
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Your notation seems rather awkward (the "##H##" on the left-hand-side of your formula has nothing to do with the "##H##'s" in the exponents), but essentially yes - your expression is correct.

You can also see that you're right by considering, e.g., the Gibbs distribution for the subsystem ##n## (with the Hamiltonian ##H_n##) at the given temperature ##T## (##k_B## below is the Boltzmann constant):
$$ p_n(T) = \frac{e^{-H_n/k_B T}}{Z_n(T)} \rm{,}$$
where ##Z_n(T)## is the partition function for this subsystem,
$$ Z_n(T) = \text{Tr}\{e^{-H_n/k_B T}\} \rm{.}$$
Now, if you have two statistically independent subsystems ##s## and ##k##, the probability density of the total system ##s+k## is a product, as you've already noted:
$$p_{sk}(T) = p_s(T) \cdot p_k(T) \rm{.}$$
From this you immediately obtain that the partition function for the total ##s+k## system is ##Z_{sk}(T) = Z_s(T) \cdot Z_k(T)## - which is the result you got at the rightmost hand-side of your formula.
 
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