Elementary Kinetic Theory of Gases

Abstract

It is convenient to introduce the notion of collisional invariants (or summational invariants) (e.g., [55], p. 460; [90], p. 150). The validity of (2.211) is commonly justified by the following arguments: Due to the symmetry properties of the collision term expression, interchanging variables gives the following equalities:These symmetry properties can be combined to give the following relationship:The linearity of the J operator allows the latter result to be manipulated as:In conclusion, a function ψ(c) is a summation invariant if:

Scaling in Biological Systems: A Molecular-Ensemble Duality

“…Similarly, it demonstrates that obscurus structural mechanisms often described in molecular biology cannot describe the omnia agency of biological function. We have shown how this solves the Gibbs paradox (19), and it appears to solve the paradox of Schrodinger’s cat. That is, if an omnia system is defined at a given instant of time, the obscurus states of its components can only be defined probabilistically at that same time.…”

“…This analysis inverts conventional understandings of entropy, scale, mechanistic agency, and the arrow of time. It solves many of entropy's paradoxes including Loschmidt's paradox (9), Gibbs' paradox (19)(20)(21), Maxwell's demon (22), and Schrodinger's cat (an entropic paradox), and it disproves all molecular mechanisms of irreversible ensemble processes.…”

Scaling in Biological Systems: A Molecular-Ensemble Duality

“…Similarly, it demonstrates that obscurus structural mechanisms often described in molecular biology cannot describe the omnia agency of biological function. We have shown how this solves the Gibbs paradox (19), and it appears to solve the paradox of Schrodinger’s cat. That is, if an omnia system is defined at a given instant of time, the obscurus states of its components can only be defined probabilistically at that same time.…”

“…This analysis inverts conventional understandings of entropy, scale, mechanistic agency, and the arrow of time. It solves many of entropy's paradoxes including Loschmidt's paradox (9), Gibbs' paradox (19)(20)(21), Maxwell's demon (22), and Schrodinger's cat (an entropic paradox), and it disproves all molecular mechanisms of irreversible ensemble processes.…”

Thermodynamic and kinetic investigation of the influence of external centrifugal field and the heat transfer on a confined neutral gas

“…In a frame co-moving with the fluid [9,[21][22][23], the system consisting of four nonlinear partial differential equations is solved analytically. The moment method [28][29][30] with suitable boundary conditions is applied. The manner of the macroscopic properties of the neutral gas such as temperature and concentration is investigated.…”

“…2. For the steady motion, a centrifugal force F C can be expressed in terms of the Boltzmann equation [28][29][30] in the BGK model as:…”

Scaling in Biological Systems: A Molecular-Ensemble Duality

“…Similarly, it demonstrates that obscurus structural mechanisms often described in molecular biology cannot describe the omnia agency of biological function. We have shown how this solves the Gibbs paradox (19), and it appears to solve the paradox of Schrodinger’s cat. That is, if an omnia system is defined at a given instant of time, the obscurus states of its components can only be defined probabilistically at that same time.…”

“…This analysis inverts conventional understandings of entropy, scale, mechanistic agency, and the arrow of time. It solves many of entropy's paradoxes including Loschmidt's paradox (9), Gibbs' paradox (19)(20)(21), Maxwell's demon (22), and Schrodinger's cat (an entropic paradox), and it disproves all molecular mechanisms of irreversible ensemble processes.…”

Thermodynamic and kinetic investigation of the influence of external centrifugal field and the heat transfer on a confined neutral gas

“…In a frame co-moving with the fluid [9,[21][22][23], the system consisting of four nonlinear partial differential equations is solved analytically. The moment method [28][29][30] with suitable boundary conditions is applied. The manner of the macroscopic properties of the neutral gas such as temperature and concentration is investigated.…”

“…2. For the steady motion, a centrifugal force F C can be expressed in terms of the Boltzmann equation [28][29][30] in the BGK model as:…”

Scaling in Biological Systems: A Molecular-Ensemble Duality

“…Similarly, it demonstrates that obscurus structural mechanisms often described in molecular biology cannot describe the omnia agency of biological function. We have shown how this solves the Gibbs paradox (19), and it appears to solve the paradox of Schrodinger’s cat. That is, if an omnia system is defined at a given instant of time, the obscurus states of its components can only be defined probabilistically at that same time.…”

“…This analysis inverts conventional understandings of entropy, scale, mechanistic agency, and the arrow of time. It solves many of entropy's paradoxes including Loschmidt's paradox (9), Gibbs' paradox (19)(20)(21), Maxwell's demon (22), and Schrodinger's cat (an entropic paradox), and it disproves all molecular mechanisms of irreversible ensemble processes.…”

Thermodynamic and kinetic investigation of the influence of external centrifugal field and the heat transfer on a confined neutral gas

“…In a frame co-moving with the fluid [9,[21][22][23], the system consisting of four nonlinear partial differential equations is solved analytically. The moment method [28][29][30] with suitable boundary conditions is applied. The manner of the macroscopic properties of the neutral gas such as temperature and concentration is investigated.…”

“…2. For the steady motion, a centrifugal force F C can be expressed in terms of the Boltzmann equation [28][29][30] in the BGK model as:…”