Internal Energy

Consider a generic particle P characterized by some repetitive chain of events noted as

$\Psi ^{\sf{P}} = \left( \sf{\Omega}_{1}, \sf{\Omega}_{2}, \sf{\Omega}_{3} \; \ldots \; \right)$

where each orbital cycle is a bundle of $N$ seeds

$\sf{\Omega} = \left\{ \sf{Z}_{1}, \sf{Z}_{2} \; \ldots \; \sf{Z}_{\it{i}} \; \ldots \; \sf{Z}_{\it{N}} \right\}$

Let each seed be described by its audibility $\varepsilon$ and its specific energy $\hat{E}$. We characterize $\sf{P}$ using a sum over all of these component seeds

\begin{align} U \equiv \sum_{i \, \sf{=1}}^{N} \varepsilon_{\it{i}} \hat{E}_{\it{i}} \end{align}

Definition: the number U is called the internal energy of P. The internal energy may be positive, negative or zero depending on a particle's composition and some choice for the calorimetric reference sensation.

To establish numerical values for the internal energy consider a down-quark defined by the pair of seeds

$\sf{d} \equiv \{ \sf{D}, \sf{O} \}$

Applying the foregoing definition of internal energy gives

$U^{\sf{d}} = \hat{E} \left( \sf{D} \right) - \hat{E} \left( \sf{O} \right)$

If a down-seed has just about the same specific energy as an ordinary conjugate-seed, then

$\hat{E} \left( \sf{D} \right) \simeq \hat{E} \left( \sf{O} \right)$ and $U^{\sf{d}} \simeq 0$

Let us require experimental practice to obtain this this consistently; for example, by using the down quark as a reference particle to set the null value when measuring internal energy. Down quarks are objectified from black sensations, so this requirement could be interpreted as closing any shutters and using insulation so that a measuring instrument is completely isolated and in the dark when indicating zero. The other numbers shown in the accompanying table are obtained by juggling quark coefficients and laboratory observations1 of nuclear particles. The conventional unit used for reporting these measurements is the electronvolt abbreviated as (eV). Theorem: an ordinary quark and its associated anti-quark have the same internal energy. Consider the generic quarks

$\sf{z} = \{ \sf{Z}, \sf{O} \}$ and $\bar{\sf{z}} = \{ \sf{Z}, \overline{\sf{O}} \}$

By the foregoing definition, the internal energies for these particles are given by

$U^{\sf{z}} = \hat{E} \left( \sf{Z} \right) - \hat{E} \left( \sf{O} \right)$ and $U^{ \sf{ \bar{z}}} = \hat{E} \left( \sf{Z} \right) - \hat{E} \left( \sf{\overline{O}} \right)$

But the assumption of conjugate symmetry asserts that $\hat{E} ( {\sf{O}} ) = \hat{E} ( \overline{\sf{O}} )$. So both quarks have the same internal energy and we can unambiguously use the quark index $\zeta$ to refer to either quark

$U^{\sf{z}} = U^{\sf{\bar{z}}} = U^{\zeta}$

## Conservation of Internal Energy

Consider that each each orbital cycle of P may also be described as a bundle of N quarks

$\sf{\Omega} = \left\{ \sf{q}_{1}, \sf{q}_{2} \; \ldots \; \sf{q}_{\it{i}} \; \ldots \; \sf{q}_{\it{N}} \right\}$

By definition each quark is composed from a pair of seeds $\sf{q} = \left\{ \sf{Z} , \sf{Z}^{\prime} \right\}$ and characterized by its internal energy

$U ^{ \sf{q}} = \varepsilon \hat{E} + \varepsilon^{\prime} \hat{E}^{\prime}$

where $\hat{E}$ is the specific energy and $\varepsilon$ is the audibility of each seed. Then by the definition of internal energy as a sum over all seeds

\begin{align} U ^{ \sf{P}} = \sum_{i\sf{=1}}^{N} \varepsilon _{\it{i}} \hat{E}_{\it{i}} + \varepsilon^{\prime} _{\it{i}} \hat{E}^{\prime}_{\it{i} } = \sum_{i\sf{=1}} ^{N} U_{\it{i}}^{\sf{q}} \end{align}

and so the internal energy of a compound quark is just a sum over the internal energies of its component quarks. Quarks are indestructible and the internal energy of each quark has a specific fixed value, so whenever some generic compound quarks $\mathbb{X}$, $\mathbb{Y}$ and $\mathbb{Z}$ interact, if

$\mathbb{ X} + \mathbb{ Y} \leftrightarrow \mathbb{ Z}$ then $U ^{ \mathbb{X} } + U ^{ \mathbb{Y} } = U ^{ \mathbb{Z} }$

We say that internal energy is conserved when particles are combined or decomposed. Also by the hypothesis of conjugate symmetry an ordinary quark and its anti-quark have the same internal energy. Swapping ordinary quarks with anti-quarks does not change the total number of quarks of a given type. So particles have the same internal energy as their associated anti-particles

$U \left( \sf{P} \right) = U \left( \overline{\sf{P}} \right)$

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 Summary
 Adjective Definition Internal Energy \begin{align} U \equiv \sum_{i \, \sf{=1}}^{N} \varepsilon_{\it{i}} \hat{E}_{\it{i}} \end{align} 4-7
page revision: 911, last edited: 01 Aug 2022 18:55