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Internal Energy

Internal energy extends the notion of specific energy so that the size of a particle can be established from calibrated laboratory experiments. Consider a generic particle P characterized by some repetitive chain of events noted as

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

where each orbital cycle is a bundle of N seeds

\mathsf{\Omega} = \left(    \mathsf{Z}_{1},   \mathsf{Z}_{2}   \;  \ldots \;      \mathsf{Z}_{\it{i}}        \;  \ldots \;   \mathsf{Z}_{\it{N}}     \right)

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

    \[ U \equiv \sum_{i \, {\mathsf{=1}}  } ^{N} \varepsilon_{\mathit{i}} \widehat{E}_{\mathit{i}} \]

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.

The internal energy of down quarks is represented by the black background in this icon for achromatic visual sensation.

To establish numerical values for the internal energy consider a down-quark defined by the pair of seeds \mathsf{d} \equiv \{ \mathsf{D}, \mathsf{O} \}. Applying the foregoing definition of internal energy gives U^{\mathsf{d}} = \widehat{E} \left( \mathsf{D} \right) - \widehat{E} \left( \mathsf{O} \right). If a down-seed has just about the same specific energy as an ordinary conjugate-seed, then

  \widehat{E} \left( \mathsf{D} \right)  \simeq  \widehat{E} \left( \mathsf{O} \right)

and

 U^{\mathsf{d}} \simeq 0

Internal Energy
 \zeta \mathsf{Z} \widetilde{U}^{\mathsf{Z}}(eV)
1U242 926 032 _{\bullet}
2D-0 _{\bullet}000 027 2
3E-31 966 250 _{\bullet}
4G298 359 162 _{\bullet}
5M1 185 795 604 _{\bullet}
6A3 122 059 _{\bullet}
7T149 556 239 _{\bullet}
8B-85 011 771 _{\bullet}
9S50 119 218 _{\bullet}
10C-53 062 870 _{\bullet}
11-2 _{\bullet}22
12-1 _{\bullet}80
13-2 _{\bullet}11
14-2 _{\bullet}55
15-0 _{\bullet}028 8
16-0 _{\bullet}049 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 with observations of molecular bond strength and nuclear particle mass. The conventional unit used for reporting these measurements is the electronvolt abbreviated as (eV). Results are presented without the use of scientific notation to graphically emphasize how quark energies range over about fifteen orders of magnitude. The structure of this huge variation governs the subsequent division of analysis into nuclear, atomic and ‘dark’ regimes.

Anti-Quark Energies

An ordinary quark and its associated anti-quark have the same internal energy if conjugate symmetry can be assumed. To see this, consider the generic quarks

\mathsf{z} = \{  \mathsf{Z}, \mathsf{O} \}

and

\overline{\mathsf{z}} = \{  \mathsf{Z}, \overline{\sf{O}} \}

By the foregoing definition, the internal energy for these particles is given by

U^{\mathsf{z}} = \widehat{E} \left( \mathsf{Z} \right)  - \widehat{E}  \left( \mathsf{O} \right)

and

U^{ \mathsf{ \overline{z}}} = \widehat{E} \left( \mathsf{Z} \right)   - \widehat{E}  \left( \mathsf{\overline{O}} \right)

The hypothesis of conjugate symmetry asserts that  \widehat{E} ( {\mathsf{O}} ) = \widehat{E} ( \overline{\mathsf{O}} ). Then both quarks have the same internal energy and we can use  \zeta the quark index to refer to either quark

 U^{\mathsf{z}} = U^{\mathsf{\overline{z}}}    =     U^{\zeta}

However, we cannot always assume conjugate symmetry. Then we use a conjugate difference  \Delta U ^{\mathsf{Z}} and a conjugate mean  \widetilde{U} ^{\mathsf{Z}} to describe the relationship between quarks and anti-quarks.

 \Delta U^{\sf{Z}} \equiv \dfrac{U^{\sf{\overline{z}}} - \, U^{\sf{z}}}{2}

and

 \widetilde{U}^{\sf{Z}} \equiv \dfrac{U^{\sf{\overline{z}}} + \, U^{\sf{z}}}{2}

Internal Energy is Conserved

Consider that each each orbital cycle of P may also be described as a bundle of N_{\mathsf{q}} quarks

\mathsf{\Omega^{P}} = \left\{    \mathsf{q}_{1},       \mathsf{q}_{2}    \;  \ldots \;      \mathsf{q}_{j}        \;  \ldots \;   \mathsf{q}_{N_{\mathsf{q}}}     \right\}

Each quark is composed from a pair of seeds \mathsf{q} = \left\{ \mathsf{Z} , \mathsf{Z}^{\prime} \right\}. And from the foregoing definition of internal energy

U ^{  \mathsf{q}}   =   \varepsilon  \widehat{E} +  \varepsilon^{\prime}   \widehat{E}^{\prime}

Then changing the sum over seeds, to a sum over quarks, gives

    \[     U ^{ \mathsf{P}} \equiv \sum_{i \, {\mathsf{=1}}  } ^{N} \varepsilon_{i} \widehat{E}_{i} = \sum_{j\mathsf{=1}}^{N_{\mathsf{q}}}   \varepsilon _{j} \widehat{E}_{j} +  \varepsilon^{\prime} _{j} \widehat{E}^{\prime}_{j}      =   \sum_{j\mathsf{=1}} ^{N_{\mathsf{q}}}    U_{j}^{\mathsf{q}} \]

The internal energy of a compound quark is just the sum its parts. But quarks are conserved. And the internal energy of each quark has a 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} }

And so internal energy is conserved. This also implies that any particle has the same internal energy as its anti-particle. Because If there is conjugate symmetry, then swapping quarks doesn’t change the total energy, so

  U \left( \mathsf{P} \right) = U \left( \overline{\mathsf{P}} \right)

Internal energy is understood as a balance between different sensations, somewhat like the ratios shown in this Sumatran tampan.
Tampan, Paminggir people. Lampung region of Sumatra, Kalianda district, 19th century, 77 x 67 cm. Photograph by D Dunlop.