Ensemble Maxwell-Boltzmann distribution

Consider, as an example, the calculation of the mean-square speed of an ensemble of molecules which obey the Maxwell-Boltzmann distribution law. This quantity is given by... 

In the general approach to classical statistical mechanics, each particle is considered to occupy a point in phase space, i.e., to have a definite position and momentum, at a given instant. The probability that the point corresponding to a particle will fall in any small volume of the phase space is taken proportional to die volume. The probability of a specific arrangement of points is proportional to the number of ways that the total ensemble of molecules could be permuted to achieve the arrangement. When this is done, and it is further required that the number of molecules and their total energy remain constant, one can obtain a description of the most probable distribution of the molecules in phase space. Tlie Maxwell-Boltzmann distribution law results. 

If we now supplement Gibbs s discussion with the investigations of Boltzmann as presented in Section 13(1), we come to the following conclusion In a canonically distributed ensemble of gas models the overwhelming majority of the individual members are in a state described by the Maxwell-Boltzmann distribution given in Eq. (46) with the parameters n, , rm, and with the energy E—E. 

Mb is interpreted as the mass of the heat bath . For appropriate choices of Mb, the kinetic energy of the particles does indeed follow the Maxwell-Boltzmann distribution, and other variables follow the canonical distribution, as it should be for the AfVT ensemble. Note, however, that for some conditions the dynamic correlations of observables clearly must be disturbed somewhat, due to the additional terms in the equation of motion [(38) and (39)] in comparison with (35). The same problem (that the dynamics is disturbed) occurs for the Langevin thermostat, where one adds both a friction term and a random noise term (coupled by a fluctuation-dissipation relation) [75, 78] ... 

The activation free energy AA can be used to compute the TST approximation of the rate constant = Ce, where C is the preexponential factor. Because not every trajectory that reaches the transition state ends up as products, the actual rate is reduced by a factor k (the transmission coefficient) as described earlier. The transmission coefficient can be calculated using the reactive flux correlation function method. " " " Starting from an equilibrated ensemble of the solute molecules constrained to the transition state ( = 0), random velocities in the direction of the reaction coordinate are assigned from a flux-weighted Maxwell-Boltzmann distribution, and the constraint is released. The value of the reaction coordinate is followed dynamically until the solvent-induced recrossings of the transition state cease (in less than 0.1 ps). The normalized flux correlation function can be calculated using " ... 

The initial configuration is taken from the equilibrium ensemble and the initial momenta are chosen to follow the Maxwell-Boltzmann distribution corresponding to the temperature. In this study, we repeat simulations 200 times for each initial structure. The time interval t is chosen to be 1.5t at each temperature, where t s have been determined in Sec. 3. The averages over the IC ensemble exclude the influence of the direction and magnitude of initial momenta on the resultant dynamics. Thus, p would reveal how the ion i tends to be mobile, resulted by, if any, the initial configuration. Previous studies of supercooled liquids... 

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