San José State University

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Thayer Watkins
Silicon Valley
USA

 A Generalization of the Proposition That 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 or more 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. For the moment let the number of types of particles be three. Let the three types of particles be designated as μ, ν and π particles and their numbers by m, n and p, 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 and likewise the μ particles with the π particles and the ν particles with the π 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 μ particles 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 the depiction of the interactions of the m μ particles, the n ν particles and the p π particles shown previously.

Consider now the incremental energy with respect to a unit increase in the number of μ particles.      The interactions for the m-1 μ particles with each other and with the n ν and p π particles are shown in color. The difference between the white squares and the colored squares, the incremental energy, is then the sum of all the squares shown in white left after the elimination of the colored squares. These are shown below.   Below the increment with respect to the number of μ particles for n ν particles is shown with 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 and the interactions of the n ν particles with the p π particles. What is left is the interaction of the m-th μ particle with the n-th ν particle. Clearly if there had been more than three types of particles the interaction of all the types whose numbers did not change would be eliminated in the subtractions involved in computing the cross differences.