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This is a both physics and math question.

I have a Hamiltonian in the form 

H= q1^2 + q2^2 + q3^2 + q1 x q2 x q3

Therefore the probability distribution will have the form

p= exp ^ (-bH)

where b is beta/k_b*T

Now, I want to know if this probability function is separable, that is if I can write it as

p= exp ^(-b*f(q1)*f(q2)*f(q3))

where f functions are single variable functions of their arguments. I will need to check this condition for many other Hamiltonians so I want to know if there is a generally applicable test to check whether they are separable.

Cheers

user178882
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1 Answers1

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Well, in this case the distribution is obviously separable, it is exponential, thus using exponent laws the proof is trivial. A decisive test, where yo may not know in principle the exact functional form, can be done computing conditional probabilities. For instance you compute $P(q_1|q_2)$ (the probability that the event $q_1$ will happen, given that the event q_2 already happened). Thus, you proceed to show that it is equal to simply $P(q_1)$, proving statistical independence and thus separability of the distributions.

David Davalos
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