Statistical mechanics Maxwell velocity distribution

Thirdly it is easy to see that the condition that the X are independent is important. If one takes for all r variables one and the same X the result cannot be true. On the other hand, a sufficiently weak dependence does not harm. This is apparent from the calculation of the Maxwell velocity distribution from the microcanonical ensemble for an ideal gas, see the Exercise in 3. The microcanonical distribution in phase space is a joint distribution that does not factorize, but in the limit r -> oo the velocity distribution of each molecule is Gaussian. The equivalence of the various ensembles in statistical mechanics is based on this fact. 

The equilibrium state for a gas of monoatomic particles is described by a spatially unifonn, time independent distribution fiinction whose velocity dependence has the fomi of the Maxwell-Boltzmaim distribution, obtained from equilibrium statistical mechanics. That is,/(r,v,t) has the fomi/" (v) given by... 

We now proceed to develop a specific expression for the rate constant for reactants where the velocity distributions /a( )(va) and /B(J)(vB) for the translational motion are independent of the internal quantum state (i and j) and correspond to thermal equilibrium.4 Then, according to the kinetic theory of gases or statistical mechanics, see Appendix A.2.1, Eq. (A.65), the velocity distributions associated with the center-of-mass motion of molecules are the Maxwell-Boltzmann distribution, a special case of the general Boltzmann distribution law ... 

For a gas mixture at rest, the velocity distribution function is given by the Maxwell-Boltzmann distribution function obtained from an equilibrium statistical mechanism. For nonequilibrium systems in the vicinity of equilibrium, the Maxwell-Boltzmann distribution function is multiplied by a correction factor, and the transport equations are represented as a linear function of forces, such as the concentration, velocity, and temperature gradients. Transport equations yield the flows representing the molecular transport of momentum, energy, and mass with the transport coefficients of the kinematic viscosity, v, the thermal diffirsivity, a, and Fick s diffusivity, Dip respectively. 

Let us look at the physical conditions which make local equilibrium a valid assumption. First we must look at the concept of temperature. From statistical mechanics it can be seen that temperature is well defined when the velocity distribution is Maxwellian. According to the Maxwell distribution of velocities, the probability P y) that a molecule has a velocity v is given by... 

The temperature is identified through relation (15.1.4), in which m is the mass of the molecule and is the Boltzmann constant. In practice, only under very extreme conditions do we find significant deviations from the Maxwell distribution. Any initial distribution of velocities quickly becomes Maxwellian due to molecular collisions. Computer simulations of molecular dynamics have revealed that the Maxwell distribution is reached in less than 10 times the average time between collisions, which in a gas at a pressure of 1 atm is about 10 s [1]. Consequently, physical processes that perturb the system significantly from the Maxwell distribution have to be very rapid. A detailed statistical mechanical analysis of the assumption of local equilibrium can be found in [2]. 

Maxwell-Boltzmann distribution A mathematical function used in the kinetic theory of gases. It is derived on the basis of statistical mechanics and gives the distribution of particle velocities in a gas at a particular temperamre. 

The Andersen thermostat is very simple. After each time step Si, each monomer experiences a random collision with a fictitious heat-bath particle with a collision probability / coll = vSt, where v is the collision frequency. If the collisions are assumed to be uncorrelated events, the collision probability at any time t is Poissonian,pcoll(v, f) = v exp(—vi). In the event of a collision, each component of the velocity of the hit particle is changed according to the Maxwell-Boltzmann distribution p(v,)= exp(—wv /2k T)/ /Inmk T (i = 1,2,3). The width of this Gaussian distribution is determined by the canonical temperature. Each monomer behaves like a Brownian particle under the influence of the forces exerted on it by other particles and external fields. In the limit i —> oo, the phase-space trajectory will have covered the complete accessible phase-space, which is sampled in accordance with Boltzmann statistics. Andersen dynamics resembles Markovian dynamics described in the context of Monte Carlo methods and, in fact, from a statistical mechanics point of view, it reminds us of the Metropolis Monte Carlo method. 

---