Distribution function and probability density in phase space

1 Distribution function and probability density in phase space 

Consider an arbitrarily large collection M of systems with N particles (A/ and N are not to be confused), each pertaining to a separate point X = (/7, q) in 6iV-dimensional phase space, F. This manifold of points is called an ensemble. 

Consider the points AA/ in a given region of phase space (Fig. 4.1). Assuming that all points are equally probable, the probabiUty of finding 

In the limit of an infinitesimal phase space region dX = dpdq, from X to A + dX, the probability of finding a point in dX is 

We can choose AA to be sufficiently large and populate the phase space densely enough to consider dJ f a continuously varying quantity. 

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Distribution function and probability density in phase space 67 a point within this region is... 

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