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Boltzmann's distribution is an exponential distribution. Boltzmann factor (vertical axis) as a function of temperature T for several energy differences ε i − ε j.. In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution [1]) is a probability distribution or probability measure that gives the probability that a system will be in a certain ...
An analytical solution to the Poisson–Boltzmann equation can be used to describe an electron-electron interaction in a metal-insulator semiconductor (MIS). [16] This can be used to describe both time and position dependence of dissipative systems such as a mesoscopic system. This is done by solving the Poisson–Boltzmann equation ...
Exact solutions to the Boltzmann equations have been proven to exist in some cases; [15] this analytical approach provides insight, but is not generally usable in practical problems. Instead, numerical methods (including finite elements and lattice Boltzmann methods ) are generally used to find approximate solutions to the various forms of the ...
The typical problem begins with a system for which the Hamiltonian is known, it is at a given temperature and it follows the Boltzmann statistics. To obtain the mean value of some macroscopic variable, say A, the general approach is to compute, over all the phase space, PS for simplicity, the mean value of A using the Boltzmann distribution:
This is often the case when dealing with atmospheres. If a medium is in Local Thermodynamic Equilibrium (LTE), then Schwarzschild's equation can be used to calculate how radiation changes as it travels through the medium. A medium is in LTE when the fraction of molecules in an excited state is determined by the Boltzmann distribution.
The equation predicts that for short range interactions, the equilibrium velocity distribution will follow a Maxwell–Boltzmann distribution. To the right is a molecular dynamics (MD) simulation in which 900 hard sphere particles are constrained to move in a rectangle.
Boltzmann distribution (separable system) [ edit ] If a system described by a canonical ensemble can be separated into independent parts (this happens if the different parts do not interact), and each of those parts has a fixed material composition, then each part can be seen as a system unto itself and is described by a canonical ensemble ...
Maxwell–Boltzmann statistics is used to derive the Maxwell–Boltzmann distribution of an ideal gas. However, it can also be used to extend that distribution to particles with a different energy–momentum relation, such as relativistic particles (resulting in Maxwell–Jüttner distribution), and to other than three-dimensional spaces.