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itaischles

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I'm reading an article titled "statistical mechanics of rigid spheres" by Reiss, Frisch and Lebowitz (1959). It tries to capture the equation of state of rigid spheres.

In their article they write the probability that at a random point in the fluid the center of the nearest molecule is a distance ##\lambda ## away and in the range ##[\lambda,\lambda+d\lambda]##. They separate this probability to the product of the probability of finding a cavity of radius ##\lambda##, denoted ##p_0(\lambda)## times the conditional probability of finding at least one molecular center in the ##[\lambda,\lambda+d\lambda]## shell given that there is a cavity of radius ##\lambda## there. So far so good.

My problem is that they assign things that doesn't look like probabilities to the two product terms. First, they claim that ##p_0(\lambda)=e^{-W(\lambda)/kT}## where ##W(\lambda)## is the reversible work needed to create the cavity. Since we don't know anything about this ##W(\lambda)## why do we assume that ##p_0(\lambda)## is normalized? Then they say that the conditional probability of finding at least one molecular center in the ##[\lambda,\lambda+d\lambda]## shell is ##4\pi \lambda^2 \rho G(\lambda) d\lambda## where ##\rho## is the average fluid density and ##G(\lambda)## is the radial distribution function at the cavity's surface. This too is not normalized since if we look for example at an ideal gas (where ##G(\lambda)=1##) and integrate over ##\lambda## between 0 and ##\infty## we get ##\rho V=N## where ##V## is the volume and ##N## the number of particles in the system. Can anybody comment on that please? Thanks!