By Paolo Pasini; Slobodan Žumer; Claudio Zannoni
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Extra info for Computer simulations of liquid crystals and polymers
Particle N in dR0 P(1)(R0 ) dR0 P(1)(R0 ) dR0 .. P(1)(R0 ) dR0 Note that dR 0 is an element of volume dx 0 dy 0 dz0 at R 0 . The quantity r(1)(R 0 ) is referred to as the singlet molecular distribution function. The meaning of r(1)(R 0 ) as a local density will prevail in all our applications. However, in some cases one may also assign to r(1)(R 0 ) the meaning of probability density. This must be done with some caution, as will be shown below. 8). This is the probability of ﬁnding a speciﬁc particle, say 1, in dR 0 at R 0 .
Since the particles of a given species are indistinguishable, only the generic MDF has physical meaning. However, the speciﬁc MDF is an important step in the deﬁnition of MDFs. One ﬁrst ‘‘labels’’ the particles to obtain the speciﬁc MDF, then ‘‘un-labels’’ them to obtain the generic MDF. THE SINGLET DISTRIBUTION FUNCTION 27 simple fashion. It is for this reason that the meaning of r(1)(R 0 ) as a local density at R 0 should be preferred. 18) is the average number of particles in S. The meaning of r(1)(R 0 )dR 0 as an average number of particles is preserved upon integration; the probabilistic meaning is not.
However, since we specify the arguments of the functions there should be no reason for confusion as to this notation. 3 The pair correlation function We now introduce the most important and most useful function in the theory of liquids: the pair correlation function. , this is the intersection of the two events. Two events are called independent whenever the probability of their intersection is equal to the product of the probabilities of the two events. 36) are not independent; the occurrence of one of them may inﬂuence the likelihood, or the probability, of the occurrence of the other.
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