Distribution Functions Ideal Gas in a Force Field

Molecular Distribution Functions Ideal Gas in a Force Field 

Qg-ir/k.T d t ir, where dr represents the product of the ZN position differentials dxi dzu and d4 s represents the product of the ZN momentum differentials dpi dp-m- If we integrate this expression over all the momentum coordinates except those for particle 1 we obtain 

If we now integrate over all the possible coordinate positions (over drn) we obtain 

We see then that, since we have made no assumptions about the nature of the potential energy Uy the distribution of molecular velocities will be independent of the forces acting either between particles or through external fields.  

If now we select an ideal gas (no intcrmolecular forces) which is placed in some external force field, the potential energy is simply a sum of the individual potential energies of each molecule and the canonical distribution can be expressed as a product  

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