Thus, the one-dimensional

Boltzmann distribution obeys an interesting phenomenon that we have identified in a wide range of skewed-right complex systems, which (as we briefly discussed in the Introduction) we call restricted diversity and, more technically, the 60/40 rule [6].

Deriving the truncated Wigner function for [N.sub.2] in an equilibrium

Boltzmann distribution, the expected Gaussianlike features were found.

It is proven (Stefan, 2003) that the

Boltzmann distribution converges to uniform distribution as T goes to infinity (this is the exploration) and to the greedy distribution as T goes to 0 (this is the exploitation).

Here, assignments of initial positions are somewhat arbitrary while initial velocities are done using the Maxwell

Boltzmann distribution at the temperature of interest.

This means that the occupation probability f(v) of the responsible excited states does not follow a

Boltzmann distribution f(v) = exp (-hv/kT) but the rule f(v) = constant (Fig.3).

It allows introduction of concepts such as atomic transitions, black body radiation, the

Boltzmann distribution, molecular orbitals, the Fermi distribution, band theory, interference, soaps and soap films.

(At any given temperature, the simulation must proceed long enough for the system to reach a steady state.) The acceptance criterion is based on the

Boltzmann distribution. Thus the probability that the configuration will be accepted when [DELTA]E > 0 is P([DELTA]E)=exp {-[DELTA]E/[K.sub.B]T}.

As for the methods to obtain the real temperature by disengaging the contributions of thermal motion and flow effect, we can use the blast-wave model based on the

Boltzmann distribution [28-30], the blast-wave model based on the Tsallis distribution [31], the improved Tsallis distribution [32,33], some alternative methods [21,29,34-36], and others [37-40].

From a theoretical point of view, thermal vibrations are often quantized as phonons, obeying the

Boltzmann distribution. The phonon energy can be obtained from the spectra of 3C-SiC, 4H-SiC, and 6H-SiC, shown in Figure 4.

An MCMC sampler makes local moves about the energy surface, its sampling distribution constructed to form a Markov chain with the

Boltzmann distribution as its stationary distribution.

Ions in the plasma are assumed to be nonthermal, so their density can be described by the known Maxwell

Boltzmann distribution function.

Physical models demonstrate gravitational force near the earth's surface, potential energy, vibrational energy, collisions, entropy, and the

Boltzmann distribution.

It can easily be checked that the sampler above generates cliques according to the

Boltzmann distribution (1), if it does not early interrupt (returning [perpendicular]).