28  Summary

• Thermodynamics inherits many results from nonthermal mechanics. Energy, momentum, and electrical charge are always well defined. Each obeys a strict local conservation law.
• Entropy is defined in terms of probability. It is always well defined. It obeys a strict local paraconservation law. Entropy is what sets thermodynamics apart from nonthermal mechanics.
• Entropy is not defined in terms of energy, nor vice versa. Energy and entropy are well defined even in situations where the temperature is unknown, undefinable, irrelevant, or zero.
• Entropy is not defined in terms of position. It involves probability spread out in state-space, not necessarily particles spread out in position-space.
• Entropy is not defined in terms of multiplicity. It is equal to the log of the multiplicity in the special case where all accessible states are equiprobable … but not in the general case.
• Work suffers from two inconsistent definitions. Heat suffers from at least three inconsistent definitions. Adiabatic suffers from two inconsistent definitions. At the very least, we need to coin new words or phrases, so we can talk about the underlying reality with some semblance of clarity. (This is loosely analogous to the way phlogiston was replaced by two more-modern, more-precise concepts, namely energy and oxygen.)
• Heat and work are at best merely means for keeping track of certain contributions to the energy budget and entropy budget. In some situations, your best strategy is to forget about heat and work and account for energy and entropy directly.
• When properly stated, the first law of thermodynamics expresses conservation of energy … nothing more, nothing less. There are several equally-correct ways to state this. There are also innumerably many ways of misstating it, some of which are appallingly widespread.
• When properly stated, the second law of thermodynamics expresses paraconservation of entropy … nothing more, nothing less. There are several equally-correct ways to state this. There are also innumerably many ways of misstating it, some of which are appallingly widespread.
• Some systems (not all) are in internal equilibrium. They are described by a thermal distribution. They have a temperature.
• Even more importantly, some systems (not all) are in internal equilibrium with exceptions. They are described by a thermal distribution with exceptions. They have a temperature.
• Two systems that are each in internal equilibrium may or may not be in equilibrium with each other. Any attempted theory of thermodynamics based on the assumption that everything is in equilibrium would be trivial and worthless.
• The idea of distinguishing thermal versus nonthermal energy transfer across a boundary makes sense in selected situations, but has serious limitations.
  • Heat exchangers exist, and provide 100% thermal energy transfer.
  • Thermally-insulated pushrods exist, and (if properly used) provide nearly 100% nonthermal energy transfer.
  • The idea of distinguishing thermal from nonthermal on the basis of transfers across a boundary goes to pot in dissipative situations such as friction in an oil bearing.
• There is a well-founded way to split the energy-change dE into a thermal part T dS and a mechanical part P dV (subject to mild restrictions).
• There is a well-founded way to split the overall energy E into a Lochrian (thermal) part and a non-Locrian (nonthermal) part (subject to mild restrictions).
• Not all kinetic energy contributes to the heat capacity. Not all of the heat capacity comes from kinetic energy. Not even close.

  More generally, none of the following splits is the same as another:

  • T dS versus P dV.
  • Locrian versus non-Locrian.
  • Cramped versus uncramped.
  • Kinetic versus potential energy.

  • Overall motion of the center-of-mass versus internal motion relative to the center-of-mass.
• There is a simple relationship between force and momentum, for any system, macroscopic or microscopic.
• For pointlike systems (no internal degrees of freedom), there is a simple relationship between overall force and total kinetic energy … but for more complex systems, the relationship is much more complicated. There are multiple inequivalent work-like quantities, depending on what length scale λ you look at.