H-Theorem

The Boltzmann's H-theorem makes it possible to find in the kinetic theory of gases, the Maxwell - Boltzmann distribution and to define the entropy. It thus is a central statement in the kinetic theory of gases.

The H-theorem is also Eta theorem called because with the symbol H instead of the Latin letter H, which is not available for the enthalpy here in any case, even the often look the same, Greek letter Eta days could be meant. As the symbol is to be understood, has long been debated and remains unresolved for lack of written evidence from the period of the theorem. However, some evidence favors the interpretation as Eta.

Statement

The contents of the H- theorem is a statement about the size,

Where Boltzmann distribution function that indicates the number of particles in a phase space volume in infinitesimal. Be neglected as a consequence of the thermodynamic limit effects on the surface of the considered volume and assumed freedom from external forces and thus establishes a - independence of.

The approach can be varied depending on the problem; for a mixture of two gases A and B is about the batch

Useful where and the defined above with the distribution functions for A and B is.

With the help of the Boltzmann equation and the assumption of zero external forces is calculated as the time derivative of

With

  • And denote the velocities of two Stoßteilchen before the collision,
  • Is the differential cross section for the Stoßteilchen.

From the form of the statement we see the H- theorem:

Conclusions

Equilibrium distribution

In the case of equilibrium must apply obviously. From the form of one realizes that then must be conserved in the collisions occur. Assuming that this is a linear combination of the following well-known conserved quantities of shock:

  • Mass of Stoßteilchen
  • Total momentum and
  • Total energy,

Is obtained from the Maxwell - Boltzmann distribution

Entropy

Follows from the H-theorem that H is a monotonically increasing size as is required for an entropy. If we define

With

  • Is Boltzmann's constant
  • The size of the equilibrium distribution and
  • The volume of the gas,

We obtain an extensive state variable that increases monotonically with time: an entropy.

Ludwig Boltzmann Kerson Huang
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