Boltzmann equation in plasma physics is a very fundamental equation in kinetic theory. What's the relation of the one in plasma physics and that in thermodynamics.
The Boltzmann equation is describe the evolution in space and time of the velocity distribution of particles subject to the an applied field. The equation is written as
d f(x,v,t)/dt+ v*grad(x) f(x,v,t) +a*grad(v) f(x,v,t)=Coll
whee the gradients refers firstly to the position (x) and the second to the velocity (v). Coll is the collisional operator. You can look at the chapter
Boltzmann and Vlasov equations in plasma physics
that you can download for free from the book
Plasma Modeling: Methods and Application (https://iopscience.iop.org/book/978-0-7503-1200-4).
In the case of plasma usually the Coll operatore is considered null and the acceleration is calculated coupling the equation with the Poisson equation.
This set of equations is known as Vlasov equations.
In thermodynamics you have the Boltzmann distribution f(E)=K exp(-E/kT), that is a particular case of distribution. In the case of translational energy it becomes the Maxwell distribution. The peculiarity of this distribution is that it makes null the collisional operator in the Boltzmann equation.
However the Boltzmann distribution is general and it is considered the distribution of the internal levels for atoms and molecules.
The Boltzmann equation is describe the evolution in space and time of the velocity distribution of particles subject to the an applied field. The equation is written as
d f(x,v,t)/dt+ v*grad(x) f(x,v,t) +a*grad(v) f(x,v,t)=Coll
whee the gradients refers firstly to the position (x) and the second to the velocity (v). Coll is the collisional operator. You can look at the chapter
Boltzmann and Vlasov equations in plasma physics
that you can download for free from the book
Plasma Modeling: Methods and Application (https://iopscience.iop.org/book/978-0-7503-1200-4).
In the case of plasma usually the Coll operatore is considered null and the acceleration is calculated coupling the equation with the Poisson equation.
This set of equations is known as Vlasov equations.
In thermodynamics you have the Boltzmann distribution f(E)=K exp(-E/kT), that is a particular case of distribution. In the case of translational energy it becomes the Maxwell distribution. The peculiarity of this distribution is that it makes null the collisional operator in the Boltzmann equation.
However the Boltzmann distribution is general and it is considered the distribution of the internal levels for atoms and molecules.
The starting point of thermodynamic logic is: 1) opposing perpetual motion. 2) The irreversibility of dynamics. Let's look at how to get Carnot efficiency = 1-T1/T2.
1.1 There are many kinds of type 2 perpetual motors, each of which is a natural phenomenon.
A) Machine A: against the irreversibility of thermodynamics (diffusion, heat conduction, friction, etc) - dynamics;
B) Machine B: Utilizing the Difference of Carnot Efficiency (Reversible Thermodynamics) - Thermodynamics
1.2 A and B belong to different disciplines. There is a parallel relationship between them and there is no logical mutual inevitability.
1.3. Logic of the Second Law of Thermodynamics:
Experience induction, deny that A machine==> B machine can not be manufactured==> All material Kano efficiency: 1-T1/T2.
1.4 The logic of the second law of thermodynamics violates the physical logic that A and B cannot be inferred from each other.
1.5 The second law of thermodynamics elevates the "irreversibility", which is in fact only a kinetic experience.
📷 Detailed discussion can be found in the following figure.
Comparison of New and Old Thermodynamics
1. Logic of the Second Law of Thermodynamics: Subjectivism, Logical Jump, Interdisciplinary Argumentation.
2. New thermodynamics pursues universality, two theoretical cornerstones:
2.1 Boltzmann formula: ro=A*exp(-Mgh/RT) - Isotope centrifugal separation experiments show that it is suitable for gases and liquids.
2.2. Hydrostatic equilibrium: applicable to gases and liquids.
3. The second and third sonic virial coefficients of R143a derived from the new thermodynamics are in agreement with the experimental results.
3.1. The third velocity Virial coefficient derived is in agreement with the experimental data, which shows that the theory is still correct when the critical density is reached.
4. See Appendix Pictures and Documents for details.
Bo Miao There is some fundamental points that need to be specified in your theory,
1) reversible and irreversible thermodynamic laws are obtained from observation of the macroscopic world and assume that the energy distribution follow the Boltzmann trend. Thermodynamics represents the asymptotic solution of the kinetic theory, when the time of observation is much longer than the characteristic relaxation time at molecular levels.
2) The irreversible thermodynamics assumes that some relaxation times are slower than the observation time, but these terms are usually those related to feeding energy to the system, while molecular relaxation times are still considered much faster than the slow varying variables.
3) In open systems, if mass or energy is supplied with a characteristic time comparable to the relaxation time, is it still possible to apply thermodynamics?
When the macroscopic characteristic time is comparable or much larger than the molecular relaxation time, the energy distributions does not follow the Boltzmann expression.
Therefore the previous question can be posed in different form: In open systems, molecular level non-equilibrium can manifest at macroscopic level?
The answer is yes. The most known aspect arise in the calculation of transport properties, as demonstrated by the Chapmann-Enskpg theory, where viscosity, diffusion etc are generated by non-Boltzmann distributions. Less known field is the non-Boltzmann distribution arising in very high enthalpy (hypersonic) flows .
Another question to be answered is: can an open system in stationary conditions be described by thermodynamics, or molecular distributions can be described by the Boltzmann law?
The answer is no, as observed in many different systems, like gas discharges.
In the case the Boltzmann law is not valid, the distribution is not dependent on a single parameter (temperature), and can be determined only by solving a kinetic equation.
In conclusion I should say that thermodynamics (reversible and irreversible) is very useful, but should be used with some criticism, because, as classical mechanics is an approximation of the relativity and quantum mechanics, thermodynamics must be considered an approximation of the kinetic theory for t->infinity.
It should be also pointed out, that the kinetic theory is also very complex and can be hardly used in many practical applications, and sometime thermodynamics is the only possible approach to the description of complex phenomena. However one should keep in mind its limits.
The problem lies in the pursuit of universality of the theory, and then there are some cases which are in accordance with the experiment.
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2. New thermodynamics pursues universality, two theoretical cornerstones:
2.1 Boltzmann formula: ro=A*exp(-Mgh/RT) - Isotope centrifugal separation experiments show that it is suitable for gases and liquids.
2.2. Hydrostatic equilibrium: applicable to gases and liquids.
Proving the Existence of Perpetual motion machine with COMSOL
Miao Bo
ABSTRACT: In this paper, we study charged conductors. The charge density at the tip is high, and the charge concentration potential is high. In the depression, the charge density tends to be zero, and the concentration potential tends to be negative infinite. Using the difference of the concentration potential between the two, the diffusion channel is established at the tip and the depression. During the diffusion process, the heat energy of the charge is converted into electric energy. The charge enters the depression and returns to the tip by conduction.
Key words: Law 2 of thermodynamics, electrostatic equilibrium, COSMOL of long-range interaction
PACC: 4460, 5130
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