Maxwell-Boltzmann velocity distribution mean energy

Here v is the (scalar) velocity, f v) is the normalized three-dimensional velocity distribution as determined by the molecular dynamics simulation at a given point in time. f u) is the three dimensional Maxwell-Boltzmann velocity distribution with a temperature determined by the condition that / (i/) has the same mean energy as the velocity distribution obtained in the simulation after a long propagation time. At thermal equilibrium DS = 0 and otherwise it is positive. The larger is DS, the more extreme is the deviation from equilibrium. The results for the entropy deficiency are shown in... 

Sometimes the thermal averaging (vaR(v)) required to compute k(T) is easy to implement. For example, for ion-molecule reactions for which, cf Eq. (3.6), vctr a constant, k(T) is independent of temperature. At other times, the averaging needs to be carried out. Explicitly, it means evaluating an integral over a Maxwell-Boltzmann velocity distribution /(v) of the (collision-energy-dependent) reaction cross-section... 

The zeroth moment of a distribution is 1, the first moment is < i>, the second moment is < P>, etc. The higher moments of a distribution hence compute successively higher averages of the distributions of the independent variable for example, in classical statistical thermodynamics the mean square velocity is the second moment of the Maxwell-Boltzmann speed distribution for an ideal gas, and is directly related to average kinetic energy < KE > = m < v >/2, and hence to temperature [= 3k TI2 for a monatomic gas]. 

Electrons which have energies greater than the thermal energy keT of the medium they are moving in are called warm or hot electrons. This definition implies that the electron velocity distribution is of Maxwell-Boltzmann type and that an electron mean energy, , can be defined as... 

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