Scaling of the Reduced Liouville Equation

In order to nondimensionalize the reduced Liouville equation, we introduce the following dimensionless variables denoted by an 

For most problems, the characteristic velocity would be the molecular velocity and, thus, mv kTo, where To is a characteristic temperature. Also, the characteristic intermolecular potential (po is usually on the 

Interlude 6.1 What Exactly is Scaling In general, the reduced Liouville equation represents a complex, multidimensional partial differential equation that must be solved subject to the boundary and initial conditions of a particular problem. In many cases, however, a number of terms in this equation are negligible, small, or insignificant and, thus, they can be discarded 

The answer to this question is the subject oiscaling and dimensional analysis. In general, scaling involves the nondimensionalization of the conservation equations where the characteristic variables used for nondimensionalization are selected as their maximum values, e.g., the maximum values of velocity, temperature, length, and the like, in a particular problem. Let s see specifically how this method works and why it can often lead to a simplification of partial differential equations. 

The first step in scaling involves nondimensionalization of the partial differential equation (PDE). To accomplish this we introduce dimensionless variables (denoted by an asterisk) for every dependent and independent variable of the PDE, such as 

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