One-dimensional hard-rod fluid

It. is also instructive to consider fluctuations in the one-dimensional hard-rod fluid. Focusing on density fluctuations one realizes from Eqs. (1.81) and (2.75) that the isothermal compressibility is a quantitative measure of such fluctuations. For the ono-dimonsional fluid considered in this section, we may dcfiiu ... 

Despite the absence of capillary condensation, the one-dimensional hard-rod fluid is still so useful because we have an analytic expression for its partition function [see Eq. (3.12)] that permits us to derive closed expressions for any thormophysical property of interest. One such quantity that is closely related to the Isothermal compressibility discussed in the preceding section is the particle-number distribution P (N), whidi one may also employ to compute thermomechanical properties [see, for example, Eqs. (3.65) and (3.68)]. Moreover, in a three-dimensional system P ) is useful to investigate the sj stem-size dependence of density fluctuations as we shall demonstrate in Section 5.4.2 [see Eq. (5.80)]. 

We now turn to a microscopic treatment of the. Toule-Thomson effect and begin with the limit of vanishing density. Th(j treatment below is very sinrilar to the one presented in Section 3.2.2 where we derived molecular expressions for the first few virial coefficients of the one-dimensional hard-rod fluid. Here it is important to realize that a mechanical expression for the grand potential exists for a fluid confined to a slit-pore with chemically structured substrate surfaces as we demonstrated in Section 1.6.1 [see Eq. (1.65)]. Combining this cxpres.sion with the tnolocular expression given in Eq. (2.81) we may write... 

We immediately apply the concepts of statistical thermodynamics in Chapter 3 to a class of systems that can be handled analytically, namely one-dimensional hard-rod fluids confined between hard walls. Despite its simplicity, this system exhibits key features of confined fluids as we shall demonstrate by comparing the results obtained in Chapter 3 with those for more realistic systcmis in later c,haj)ters. 

Hence, as we arc given an analytic expression, it seems worthwhile to apply the analysis detailed in Appendix C.3 to the one-dimensional hard-rod fluid. Our starting point is the probability to find iV hard rods in a (one-dimensional) bulk system of volume L given by [see Eq. (3.12)]... 

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