H theorem

The Boltzmann H theorem allows the kinetic theory of gases , the Maxwell-Boltzmann distribution to find and the entropy to define. It is a central statement in the kinetic gas theory. The H-theorem can be used to describe the process of equilibration of a system, which occurs particularly in non-equilibrium .

The H-theorem is also called the Eta-theorem because the symbol H instead of the Latin letter  H , which does not stand for enthalpy , could also mean the Greek letter Eta, which often looks the same  . How the symbol is to be understood has been discussed for a long time and remains unclear due to the lack of written evidence from the time the theorem was created. However, some indications speak in favor of the interpretation as Eta.

statement

The contents of the H-theorem consists in a statement about the size , ${\displaystyle H}$

${\displaystyle H(t):=-\int \mathrm {d} ^{3}v\cdot f({\vec {v}},t)\ln f({\vec {v}},t)}$,

where is the Boltzmann distribution function , which gives the number of particles in a volume element of the phase space at . As a consequence of the thermodynamic limit, effects on the surface of the volume under consideration are neglected and freedom from external forces is assumed, thus establishing an independence of . ${\displaystyle f}$ ${\displaystyle \mathrm {d} ^{3}v}$ ${\displaystyle ({\vec {x}},{\vec {v}})}$ ${\displaystyle {\vec {x}}}$ ${\displaystyle f}$

The approach for can be varied depending on the problem; for a mixture of two gases and is roughly the approach ${\displaystyle H}$ ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle H=H_{A}+H_{B}}$

makes sense where and what is defined above with the distribution functions for and . ${\displaystyle H_{A}}$ ${\displaystyle H_{B}}$ ${\displaystyle H}$ ${\displaystyle A}$ ${\displaystyle B}$

With the help of the Boltzmann equation and the assumption of vanishing external forces, the time derivative of as is calculated ${\displaystyle H}$

${\displaystyle \partial _{t}H=-\int \mathrm {d} ^{3}v_{1}\cdot \mathrm {d} ^{3}v_{2}\cdot \mathrm {d} \Omega \ {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}\ |{\vec {v}}_{1}-{\vec {v}}_{2}|\ (f_{1}f_{2}-f'_{1}f'_{2})\left[\ln(f'_{2}f'_{1})-\ln(f_{2}f_{1})\right]}$.

With

• ${\displaystyle f_{i}':=f({\vec {v}}_{i}')}$
• ${\displaystyle {\vec {v}}_{1}}$and denote the velocities of two collision particles before the collision , ${\displaystyle {\vec {v}}_{2}}$

the crossed out variants their speeds after a shock

• ${\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}}$is the differential cross section of the collision particles.

From the form of we see the statement of the H-theorem: ${\displaystyle \partial _{t}H}$

${\displaystyle {\frac {\mathrm {d} H}{\mathrm {d} t}}\geq 0}$,

from which it follows that is a monotonic function . ${\displaystyle H(t)}$

Inferences

Equilibrium distribution

In the case of equilibrium must obviously apply. From the shape of , one recognizes that there must then be a conservation quantity in the impacts that occur. Assume that this is a linear combination of the following known conservation quantities of the shock: ${\displaystyle \partial _{t}H=0}$ ${\displaystyle \partial _{t}H}$ ${\displaystyle \ln f}$

• Mass of the collision particles ${\displaystyle m}$
• Total impulse and ${\displaystyle m{\vec {v}}}$
• Total energy , ${\displaystyle m{\vec {v}}\,^{2}/2}$

this gives the Maxwell-Boltzmann distribution

${\displaystyle f=C\cdot \mathrm {\exp(} -A({\vec {v}}-{\vec {v}}_{0})^{2})}$

with the constants , and . ${\displaystyle C}$ ${\displaystyle A}$ ${\displaystyle {\vec {v}}_{0}}$

entropy

From the H-theorem it follows that H is a monotonically increasing quantity, as is necessary for entropy. One defines

${\displaystyle S:=k\cdot H_{0}\cdot V}$

With

• ${\displaystyle k}$the Boltzmann constant
• ${\displaystyle H_{0}}$the size for the equilibrium distribution and ${\displaystyle H}$
• ${\displaystyle V}$ the volume of the gas,

in this way one obtains an extensive state variable that grows monotonically over time: an entropy.

Generalizations

There are generalizations for the H-theorem, including the relaxation theorem .

literature

• Hannelore Bernhardt : The recurrence objection to Boltzmann's H-theorem and the concept of irreversibility. NTM 6 (1969) 2, pp. 27-36.
• Kerson Huang : Statistical Mechanics . John Wiley & Sons 1987, ISBN 0-471-81518-7 , chapter 4.

Individual evidence

1. ↑ James C. Reid, Denis J. Evans, Debra J. Searles: Communication: Beyond Boltzmann's H-theorem: Demonstration of the relaxation theorem for a non-monotonic approach to equilibrium . In: The Journal of Chemical Physics . tape 136 , no. 2 , January 11, 2012, ISSN  0021-9606 , p. 021101 , doi : 10.1063 / 1.3675847 ( scitation.org [accessed June 25, 2019]).
2. ^ S. Chapman: Boltzmann's H-Theorem. In: nature , 139 (1937), p. 931, doi : 10.1038 / 139931a0 .
3. ↑ SG Brush: Boltzmann's “Eta Theorem”: Where's the Evidence? In: American Journal of Physics , 35 (1967), p. 892, doi : 10.1119 / 1.1974281 .
4. ^ S. Hjalmars: Evidence for Boltzmann's H as a capital eta. In: American Journal of Physics , 45 (1977), pp. 214-215, doi : 10.1119 / 1.10664 .
5. ↑ James C. Reid, Denis J. Evans, Debra J. Searles: Communication: Beyond Boltzmann's H-theorem: Demonstration of the relaxation theorem for a non-monotonic approach to equilibrium . In: The Journal of Chemical Physics . tape 136 , no. 2 , January 11, 2012, ISSN  0021-9606 , p. 021101 , doi : 10.1063 / 1.3675847 ( scitation.org [accessed June 25, 2019]).