ABSTRACT
Let us consider a system with volume V in contact with a heath reservoir at temperature T and with a molecule reservoir with chemical potential μ, and suppose that each particle is subjected to an external potential V (r), which depends on the position r of the particle and is the responsible for the inhomogeneity of the system. It can be proved (see Evans 1979) that, for a given intermolecular potential, there is a unique external potential V (r) compatible with a given grand canonical equilibrium single-particle density ρ(1)(r), Equation 2.50 for n= 1, which we will denote by ρ(r), for simplicity, from now on. Therefore, all the thermodynamic functions may be expressed as functionals of the single-particle density, whence the name of density functional theory (DFT) given to this formalism. In particular, defining a local potential ψ(r) = μ − V (r), the grand potential = F − μN = −kBT ln will take the form
[ρ (r)] = F [ρ (r)] −
ρ (r)ψ (r) dr, (9.1)
where F [ρ (r)] is a free energy functional, which is related to the Helmholtz free energy of the system by means of the relationship
F [ρ (r)] = F [ρ (r)] +
V (r) ρ (r) dr, (9.2)
where ρ(r) is the equilibrium nonuniform density, which will be the one minimizing the grand potential (Equation 9.1). This yields (Lebowitz and Percus 1963a)
δF [ρ (r)] δρ (r)
= ψ (r) , (9.3)
where δ means a functional derivative. On the other hand, Lebowitz and Percus (1963a) also showed that
δ
δψ (r) ≡ −kBTδ ln
δψ (r) = −ρ (r) , (9.4)
as can be easily seen from the definition of ρ(r) in the grand canonical ensemble, Equation 2.50 for n = 1, and the properties of the functional derivative (see Evans 1979 for a detailed derivation). For an ideal gas, the intermolecular potential is u(r) = 0 and, from Equation 2.49, one easily obtains
id [ρ (r)] = −kBT−3 eβψ(r)dr, (9.5)
where is the thermal wavelength. Then, Equation 9.4 yields for the equilibrium single-particle density of the ideal gas
ρ (r) = −3eβψ(r). (9.6)
Taking into account that=−PV and, therefore, for an ideal gaswith average number of particles 〈N〉 we have =−〈N〉kBT , and that
ρ(r)dr=〈N〉, from Equation 9.1
the ideal gas free energy functional is
Fid [ρ (r)] =
ρ (r) fid (ρ (r)) dr = kBT
ρ (r) { ln
( 3ρ (r)
) − 1} dr, (9.7)
which is a quite obvious extension to an inhomogeneous ideal gas of expression (6.42) for the free energy of the homogeneous ideal gas, where fid(ρ) is the free energy per particle of an ideal gas with density ρ.
