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Temperature extends the notion of vis viva so that the urgency of objectified feelings 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}^{\mathsf{P}} = \left(    \mathsf{Z}_{1},   \mathsf{Z}_{2}   \;  \ldots \;      \mathsf{Z}_{i}        \;  \ldots \;   \mathsf{Z}_{N}     \right)

Let each seed be described by  \varepsilon, its audibility, and  \widehat{K}, its vis viva. We characterize P using a sum over all of these component seeds

    \[ T^{\mathsf{P}}  \equiv  \frac{2}{Nk_{B}} \sum_{i\mathsf{=1}}^{N} \varepsilon _{i} \widehat{K}_{i}   \]

where k_{B} is Boltzmann’s constant. The number  T is called the temperature of P. This temperature may be positive, negative or zero depending on the particle’s composition and the choice of a thermometric reference sensation.

The temperature of a bottom quark is represented by this snowflace icon.

To establish numerical values start with the bottom-quark defined by the pair of seeds \mathsf{b} \equiv \{ \mathsf{B}, \mathsf{O} \}. Applying the foregoing definition of temperature gives   T^{ \mathsf{b}} \, k_{B} = \widehat{K} \left( \mathsf{B} \right) - \widehat{K} \left( \mathsf{O} \right). If a bottom-seed has the same vis viva as a conjugate-seed, then

  \widehat{K}    \left( \mathsf{B} \right)  =  \widehat{K}   \left( \mathsf{O} \right)


T^{\mathsf{b}} =0  \hspace{0.25cm}  ( ^{\circ} \mathsf{C} )

 \zeta \mathsf{Z} T (℃)

Consider experimental practice to obtain this consistently; for example, by using bottom quarks as a reference to calibrate the measurement of temperature. This would depend on what we mean by touching ice because this feeling was used to specify freezing reference sensations and objectified to define bottom seeds. But there are many different kinds of ice and to make reliable measurements we therefore need to specify the reference sensation more precisely. So, by “touching ice” we mean touching a slushy mix of frozen solid water and clean pure liquid water in an open container near sea level on Earth. This is an utterly conventional way of specifying zero on the Celsius temperature scale. So we note such a convention by writing T ( ^{\circ} \mathsf{C} ) as the Celsius temperature.

The temperature of a charmed quark is suggested by this icon of a steaming pot.

We have also defined the charmed quarks using the reference sensation of touching steam. And since there are different kinds of steam we also need to specify this sensation more carefully. So, by “touching steam” we mean touching the vapors rising from an open container of pure boiling water near sea level on Earth. This is a very traditional way of defining 100 ( ^{\circ} \mathsf{C} ). Charmed quarks are objectified from this sensation, so we require that their temperature is 100 ( ^{\circ} \mathsf{C} ). The other temperatures listed in the accompanying table are obtained by juggling quark coefficients and laboratory observations of nuclear particles. The large negative temperatures are later interpreted to mean robust stability.

Anti-Quark Temperatures

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

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


\overline{\mathsf{z}} = \left\{  \mathsf{Z}, \overline{\mathsf{O}} \right\}

By the foregoing definition, the temperature of these particles is given by

  T^{\mathsf{z}}      = \dfrac {\widehat{K} \left( \mathsf{Z} \right) -  \widehat{K}  \left( \mathsf{O} \right)}{k_{B}}


  T^{ \mathsf{\overline{z}}}   = \dfrac{ \widehat{K} \left( \mathsf{Z} \right) -  \widehat{K}   \mathsf{(} \mathsf{\overline{O}} \mathsf{)}  }{k_{B} }

But the hypothesis of conjugate symmetry asserts that  \widehat{K} ( {\mathsf{O}} ) = \widehat{K} ( \overline{\mathsf{O}} ). So both quarks have the same temperature and we can use  \zeta the quark index to refer to either quark

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

The Temperature of Compound Quarks

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

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

And let seeds be paired in sequence such that \mathsf{q}_{j} = \left\{ \mathsf{Z}_{j} , \mathsf{Z}_{j+1} \right\}. Then the temperature, as defined above, can be changed from a sum over seeds to a sum over quarks

    \[ T ^{\mathsf{P}} =  \frac{2}{Nk_{B}} \sum_{j\mathsf{=1}}^{  N_{\mathsf{q}} } \varepsilon _{j} \widehat{K}_{j}  +  \varepsilon _{j+1} \widehat{K}_{j+1}    \]

If P happens to be a solitary quark  \mathsf{q}_{j} then the number seeds is just  N=2 and so

    \[ T ( \mathsf{q}_{j} )  =  \frac{1}{k_{B}} \left( \varepsilon _{j} \widehat{K}_{j}  +  \varepsilon _{j+1} \widehat{K}_{j+1} \right)   \]

Substituting this back into the general expression for P’s temperature gives

    \[ T ^{\mathsf{P}} =  \frac{2}{N} \sum_{j\mathsf{=1}}^{  N_{\mathsf{q}} } \; T ( \mathsf{q}_{j} )    \]

But  N=2N_{\mathsf{q}} because there are two seeds in every quark. So finally

    \[ T ^{\mathsf{P}} =  \frac{1}{N_{\mathsf{q}}} \sum_{j\mathsf{=1}}^{  N_{\mathsf{q}} } \; T ( \mathsf{q}_{j} )  =\widetilde{T}(\mathsf{q}) \]

The temperature of a compound quark is just the average temperature of its component quarks. Swapping ordinary quarks with anti-quarks cannot change this average because  T^{\mathsf{z}} = T^{\mathsf{\overline{z}}}. Thus any particle has the same temperature as its associated anti-particle

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

The temperature represents feelings of urgency and movement, somwhat like those presented in this Sumatran Tatibin.
Tatibin, Paminggir people. Lampung region of Sumatra, Kota Agung district, 19th century, 59 x 41 cm. Ship motif. Photograph by D Dunlop.