Maxwell-Boltzmann Distribution, Transition Probability and Large Number of Particles

 In (1) it is suggested that a number of systems in nature have a total number of states with energy less than or equal to energy E proportional to E to the power of k (where k is of the order of the number particles). It is then shown that this leads to a power law distribution  (E-ei) power k-1  where ei is the energy of a single particle. In (2) it is shown that the power law distribution follows from a Fokker-Planck equation with appropriate friction and diffusion coefficients. It is also stated in (2) that reaction balance requires:  P(q) w(q,q1) = P(q1) w(q1,q) where P(q) is the probability to be in state q and w(q,q1) is the transition rate from q to q1. 

    If the power exponent of the power law distribution (which from (1) is proportional to the number of particles) becomes very large (i.e. approaches infinite), it becomes a Maxwell-Boltzmann (MB) distribution (proportional to exp(-ei/T)). Then, P(q)/P(q1) = exp(-(E(q)-E(q1))/T) =  w(q1,q)/w(q,q1) which is not the result which holds for a power law. 

     Traditionally, the MB distribution may be obtained (3) by considering the total number of arrangements (distributions) of a total energy E among N particles. This is written as N!/ Product over i n(ei)!. It is then assumed that N is very large and Stirling’s approximation applied to the ln of the total number of arrangements. The underlying idea is that each arrangement carries the same weight. ln(Number of arrangements) in the high N limit is then maximized with respect to n(ei) subject to the constraint: Sum over i  ei P(ei) to obtain the MB distribution. Thus the MB distribution again appears because of a high N limit.

     In this note, we examine the assumption that all energy arrangements carry the same weight. Reaction balance for two body scattering is: P(e1)P(e2)=P(e3)P(e4) with e1+e2 = e3+e4. Thus there is no transition probability linked to say velocity. If there were, then different energy distribution arrangements would not necessarily carry the same weight. For example, P(ei) may be very low for a high ei, but such a particle has a high velocity. If particle density is low, such a particle would move through a large region in space compared with a particle with low velocity. One might expect a factor linked to space covered to be multiplied by P(ei) when considering reaction balance, but this is not the case in the MB formulation. In the MB case, one considers only P(ei). This seems to suggest that each small region of space has a large number of particles so that the velocity of a particle does not really affect the probability to scatter. Thus assuming equal weights of variations distributions of energy (or arrangements) seems to automatically assume large numbers of particles. One then does not need to introduce the assumption of a high N a second time in order to justify using Stirling’s approximation. In fact, it seems one does not need to use Stirling’s approximation, because the derivative of ln(n(ei)!+1) subject to the average energy constraint yields the MB distribution for n(ei)>>1 which is already assumed by considering equal weights for all arrangements.