Figure 2 shows the [C.sub.c] curve thus calculated for the Boltzmann distribution in three dimensions.
In these figures, it appears that, by 6000 K, the Bose-Einstein distribution for helium-4 is indistinguishable from the 3D Boltzmann distribution. Also, while the Fermi-Dirac distribution has clearly not reduced to the Boltzmann distribution even at 15 x [10.sup.6] K, it appears to be trending towards it.
If we compare the GA to the combined used of reinforcement learning and simulated annealing with the help of Boltzmann distribution
in terms of idle times (in the case of 30 different 20-machine flow-shop problems increasing the number of jobs from one to thirty), we will experience that the genetic algorithm approaches the optimum with equally well than the RL at first (up to ten-job problems), but an increase in the number of jobs issues in more and more bigger difference for GA against reinforcement learning (Figure 10).
Instead of the Boltzmann distribution
f(v) = exp(-hv /kT), this open system becomes the subject of f(v) = 1/N, where N is the number of degrees of freedom.
Each class is asymptotically distributed as a Boltzmann distribution
, and the process of determining this distribution for any [THETA] is referred to as "equilibration."
They cover the fundamental concepts: heat, work, internal energy, enthalpy, and the First Law of Thermodynamics; thermochemistry; entropy and the Second Law of Thermodynamics; chemical equilibrium; the properties of real gases; phase diagrams and the relative stability of solids, liquids and gases; ideal and real solutions; electrochemical cells, batteries and fuel cells; probability; the Boltzmann distribution
; ensemble and molecular partition functions; statistical thermodynamics; the kinetic theory of gases; transport phenomena; elementary chemical kinetics; and complex reaction mechanisms.
For the description of the ion concentration distribution in diluted solutions, the Boltzmann distribution
can be applied in the form
Now, different models have been suggested to describe the [p.sub.T] distributions of the final-state particles in high energy collisions [10-13], such as Boltzmann distribution
, Rayleigh distribution, Erlang distribution, the multisource thermal model, and Tsallis statistics.
For example, in the framework of a multisource thermal model which was proposed by us some years ago , one can use the Rayleigh distribution [2-6], Boltzmann distribution
[7, 8], Fermi-Dirac distribution , and Erlang distribution [10, 11] to describe the transverse momentum spectrum contributed by a given isotropic emission source.
Each local equilibrium state can be described by the relativistic ideal gas model (Boltzmann distribution
) [22, 23] with a given temperature, and the emission of particles in the rest frame of the considered source is isotropic.
To give a further test to the multisource thermal model, in this paper, we will describe the transverse momentum distributions of charged particles produced in nucleus-nucleus collisions at RHIC energies by using the model in which the Boltzmann distribution
is revised to fit the data at high transverse momentum.