By G.H. A. Cole
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Extra resources for An Introduction to the Statistical Theory of Classical Simple Dense Fluids
It will be found that for equilibrium conditions / (iV) has the canonical form of Gibbs; for non-equilibrium the appropriate distribution is to be determined from equations that are still under construction. 10) are of the first order, so that a knowledge of the phase at any time t is sufficient to determine the phase at any later (or earlier) time. 11b) that the y-phase distribution behaves as an incompressible fluid. e. 22) can be rewritten in the form: —ΒΓ- A \Έ>'-5ΪΓΓ& (« ~*r)· ( } This is an alternative statement of Liouville's theorem and is called the Liouville equation.
But before entering this discussion it is convenient here to consider a generalisation of the canonical form. 4. 11) applies for a mechanical system having a fixed number of particles. The more general situation is that where the number of particles is not fixed4, and is described by the grand canonical ensemble. 11) is denoted by f^ and is to involve N as well as the particle phase. 23) where μ is a parameter yet to be indentified, and S is a normalising constant called the grand partition function.
The general range of validity of the equations involving the pair distribution for the various types of fluid is likely to be wider for a gas where the particle number density is lower than for a liquid, and the interparticle force potential is less likely to involve more than particle pairs simultaneously. Other expressions for the thermodynamic functions can be expressed in terms of the pair distribution. As an example, the specific heat at constant volume Cv is given by Cv 3nk n2 f ~ , x dgi2\r) A 9 Λ — y t o ^ W 4jrr2 dr.
An Introduction to the Statistical Theory of Classical Simple Dense Fluids by G.H. A. Cole