Dynamic Equilibrium

We characterize Newtonian particles as being in some kind of steady balance with their environment. They are presumably interacting with countless photons, bouncing around a lot, and colliding with other particles. But despite much agitation, there is still a central tendency that might be called realistic motion, or perhaps naturalistic movement. Particles that depart too far from the mean may be called non-Newtonian, or even unphysical. To be more exact about this we define the kinetic and potential energy.

Kinetic Energy

//Tampan//, Paminggir people. Lampung region of Sumatra, 19th century, 77 x 70 cm. Photograph by D Dunlop.
Tampan, Paminggir people. Lampung region of Sumatra, 19th century, 77 x 70 cm. Photograph by D Dunlop.

Consider a material particle P, described by its rest mass $m$ and momentum $p$. The kinetic energy of P is

$\begin{align} K \equiv \frac{\, p^{ 2}}{2m} \end{align}$

Since $m > 0$ for material particles, $K$ is never negative. And in an inertial frame, momentum is proportional to the wavenumber ${\kappa}$. So $K$ is proportional to $\kappa ^{2}$. Then recall that the wavenumber depends only on the coefficients of dynamic quarks. So the kinetic energy depends strongly on P's dynamic quark content. Sensory interpretation: Dynamic quarks and the wavenumber have already been interpreted as representations of somatic and visual sensations. So the kinetic energy depends mostly on audio-visual sensations.

Potential Energy

Let us also include the mechanical energy $\, E$ in the description of P. The difference between $E$ and the kinetic energy defines another number $\mathcal{U}$ called the potential energy

$\mathcal{U} \equiv E - K$

And recall that if the motion of a material particle is not relativistic, then

$\begin{align} E \simeq m c^{2} \left( 1 + \frac{p^{2}}{2m^{2}c^{2}} \right) = m c^{2} \left( 1 + \frac{K}{mc^{2}} \right) = m c^{2} +K \end{align}$

So the potential energy may be approximated as $\mathcal{U} \simeq m c^{2}$. For slowly moving Newtonian particles, the potential energy depends strongly on the mass. Then remember that for heavy particles, a sensory interpretation of the mass relates mainly to thermal sensations. And so for Newtonian particles, the potential energy is mostly associated with thermal sensations too.

Dynamic Equilibrium

We say that P is in dynamic equilibrium when $K = \mathcal{U}$ so that kinetic and potential energies are equal to each other. There is an equal sharing, or equipartition, of mechanical energy between kinetic and potential types. But the potential energy is defined by $E$ less $K \,$. So for a particle in dynamic equilibrium

$E=K+ \mathcal{U} =2K =2 \mathcal{U}$

This account of dynamic equilibrium is succinct. And it eventually links the equipartition of energy to our traditional understanding of momentum as the product of mass and velocity. But it is not very meaningful in practice. Difficulties arise from using a theoretical state of perfect isolation to set the zero-value for energy measurements. Recall that initially our discussion of energy adopted the reference sensation of not seeing the Sun to grasp the notion of having no internal energy. So, even in principle, we do not have a tangible laboratory reference standard for absolute-zero on the energy scale. This is noticeable now that energy measurements can be made to about a part in 1012. Moreover, there are conflicts with theories of dispersion and gravitation which may deny even the possibility of perfect isolation. Anyway, these difficulties are manageable because physical phenomena often occur within distinct energy regimes. So for example, nuclear energy can usually be ignored when doing benchtop chemistry because nuclei are so stable. Practically, we measure energy changes that are related to each other by $\Delta \mathcal{U} = \Delta E - \Delta K$. A null value for energy-difference measurements can be selected for experimental convenience. Then results are reported using a slightly different version of the potential energy

$\mathcal{U}^{\prime} = \mathcal{U} + \sf{\text{an arbitrary constant}}$

such that $\Delta \mathcal{U}^{\prime} = \Delta \mathcal{U}$. Energy differences are more susceptible of precise laboratory observation. But then, sweeping generic relationships between potential, kinetic and mechanical energy are no longer true

$K ≠ \mathcal{U}^{\prime}$ and $E ≠ K+ \mathcal{U}^{\prime}$

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Sensory interpretation: As noted above, in an ideal reference frame the kinetic energy characterizes the sensory magnitude of visual stimuli, whereas the potential energy depends more on thermal perception. So there must be a balanced experience of both thermal and visual sensation for events to be objectified as particles in dynamic equilibrium. This requirement for eyes-open visual awareness means that, for example, a dream about flying while asleep cannot meet equilibrium conditions. And neither can watching cartoons on TV, because television only represents audio-visual sensations, not thermal sensations. So dynamic equilibrium is more like experiencing ordinary circumstances in classrooms and laboratories on Earth. Unlike many movies, dreams and hallucinations.
Right.png Next step: Conservation of energy and mass for Newtonian particles.

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