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Boltzmann's H-theorem[edit]

Boltzmann's H-theorem states that the entropy of a closed system can only increase in the course of time, and must approach a limit as time tends to infinity.

\sigma \geq 0

where \sigma is the entropy source strength, given by (Eq 36 Chap IX Ref. 2)

\sigma = -k \sum_{i,j} \int C(f_i,f_j) \ln f_i d {\mathbf u}_i

where the function C() represents binary collisions. At equilibrium, \sigma = 0.

Boltzmann's H-function[edit]

Boltzmann's H-function is defined by (Eq. 5.66 Ref. 3):

H=\iint f({\mathbf V}, {\mathbf r}, t) \ln f({\mathbf V}, {\mathbf r}, t) ~ d {\mathbf r} d{\mathbf V}

where {\mathbf V} is the molecular velocity. A restatement of the H-theorem is

\frac{dH}{dt} \leq 0

Gibbs's H-function[edit]

See also[edit]


  1. L. Boltzmann "", Wiener Ber. 63 pp. 275- (1872)
  2. Sybren R. De Groot and Peter Mazur "Non-Equilibrium Thermodynamics", Dover Publications
  3. Robert Zwanzig "Nonequilibrium Statistical Mechanics", Oxford University Press (2001)

Related reading