Partial Difference Equations

Let two atoms called $\mathbf{A}$ and $\mathbf{B}$ have an interaction with each other by exchanging another particle called $\sf{X}$. The interaction is caused when $\, \mathbf{A} \,$ emits $\, \sf{X} \,$ at event $\, \mathbf{A}_{\it{i}} \,$ which is called the initial event of the interaction

$\mathbf{A}_{ \it{i}\mathrm{ - 1}} \to \mathbf{A}_{\it{i}} + \sf{X}_{\it{i}}$

Particle $\sf{X}$ then has an effect on $\mathbf{B}$ by being absorbed at event $\mathbf{B}_{\it{f}}$ which is called the final event of the interaction

$\mathbf{B}_{\it{f}} + \sf{X}_{\it{f}} \to \mathbf{B}_{ \it{f} \mathrm{ + 1}}$

Recall that $n$ notes a particle's quark coefficients for any sort of quark Z. The radius vector is marked by $\ \overline{\rho}$. And $W$ is the work required to form a particle. Some difference equations are defined to describe the interaction

$\partial n \equiv n _{\it{f}} - n _{\it{i}}$ $\partial \Delta n \equiv \Delta n_{\it{f}} - \Delta n_{\it{i}}$ $\partial \overline{\rho} \equiv \overline{\rho} _{\it{f}} - \overline{\rho} _{\it{i}}$ $\partial W \equiv W_{\it{f}} - W_{\it{i}}$

And the following relationships obtain

$\partial n^{\mathbf{B}} = n^{\sf{X}}$

$\partial \Delta n^{\mathbf{B}} = \Delta n^{\sf{X}}$

$\partial \overline{\rho}^{\mathbf{B}} = \overline{\rho}^{\sf{X}}$

$\partial W^{\mathbf{B}} = F_{m}^{\mathbf{B}} \, \rho_{m}^{\sf{X}} + F_{e}^{\mathbf{B}} \, \rho_{e}^{\sf{X}} - F_{c}^{\mathbf{B}} \, \rho_{z}^{\sf{X}}$

If X is composed of just muonic quarks then $\partial \rho_{e} = \partial \rho_{z} = 0$ and $F_{m} = \partial W / \partial \rho_{m}$. That is, the magnetic force is just a ratio of the work done, to the change in magnetic radius. Similarly, an exchange consisting exclusively of electronic quarks is related to the just the electric-force, and only the electric-radius. This arrangement allows us to model specific forces with specific exchange particles that are logically distinct and independently variable. Different causes and effects are not entangled, and we can use a mathematical technique called separation of variablesXlink.png to solve these equations. Changes assessed in this discrete way are called partial difference equations to emphasize that they can be considered in parts. If an interaction produces effects that are not too big, then we may use differential calculus to work with these equations. Such an assumption is often implicit when employing the symbol $\partial$ rather than $\Delta$ to mark an exchange.


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