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The Cross Differences of the Interaction Energies
with Respect to Two Types of Particles Is Equal
to the Interaction Energy of the Last Particles
of Each Type with Each Other

Consider systems of two types of particles for which the potential energy is known for various arrangements. This potential energy is the sum of the potential energies of the pairs of individual particles. Let the two types of particles be designated as μ and ν particles and their numbers by m and n, respectively. There are the interactions of the μ particles with each other, the ν particles with each other and the μ particles with the ν particles. These are depicted visually below.

  

The black squares indicate there are not any interactions of a particle with itself.

What is being considered is the increment due to a change in one variable in the increment due to a change in the other variable. This difference is called the cross difference. It corresponds to the cross derivative in calculus. If the energy E is a functions of the number of μparticles m and the number of ν particles n, and m and n were continuous variables then the cross derivative would be ∂²E/∂n∂m, which is the same as ∂²E/∂m∂n. These cross differences have some remarkable properties. For example, the same values would arise if first the incremental differences were calculated for an increase in the number of ν particles and the differences in these incremental values were computed for an increase in the number of μ particles. A much more important property is:

The increase with respect to the number of ν particles in the incremental increase in energy due to an increase in the number of μ particles is equal to the interaction energy of the last μ particle with the last ν particle.

Proof:
Consider once again the depiction of the interactions of the m μ particles and the n ν particles.

  

Consider now the incremental energy with respect to a unit increase in the number of μ particles.

  

The interactions for m-1 μ particles are shown in color. The difference, the incremental energy, is then the sum of all the squares shown in white. These are shown below.

 

Below the increment with respect to the number of μ particles for n ν particles is shown with the the increment for n-1 ν particles superimposed over it in color.

 

The subtraction eliminates the interactions of the m-th μ particle with the other (m-1) μ particles. It also eliminates the interactions of the m-th μ particle with the (n-1) ν particles. What is left is the interaction of the m-th μ particle with the n-th ν particle.


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