This special issue on density functional theory is dedicated to the memory of Yaakov (Yasha) Rosenfeld. The Liquid State community lost an exceptional scientist and a special friend on 21 July 2002. Yasha had intended to contribute to this volume. Sadly, he died from lung cancer at the age of 54.

Yasha made outstanding contributions to the statistical mechanics of liquids and dense plasmas. He was the author of more than 100 papers, several of which are regarded as `classics'. His early work was on the theory of freezing in simple fluids and on high pressure equations of state. In 1979, together with N W Ashcroft, he developed the idea of universality of the bridge function for simple fluids. The key publication [1] proved very influential in the theory of bulk liquids and carries >400 citations. At about the same time Yasha began his lifelong studies of the properties of classical plasmas, focussing on the equation of state in the strong-coupling limit. He remains a major figure in the theory of Coulomb fluids to which he successfully applied the ideas of Onsager `charge smearing', providing a lower bound to the electrostatic energy. In the mid-80s Yasha turned his attention to the properties of fluids in the asymptotic high density limit (AHDL), arguing that at sufficiently high densities hard-core liquids and the classical one-component plasma (OCP) share common structural and thermodynamic features and that it is possible to perform a perturbative analysis about the AHDL, i.e. about packing fraction η = 1 for hard spheres and plasma parameter Γ = ∞ for the OCP. Whilst some of the concepts introduced in the AHDL papers appeared esoteric to many researchers, these provided a basis for much of Yasha's later work. An example of his skilful combination of classical and quantum DFT for dense ion-electron plasmas is contained in [2]. This paper provided inspiration for later Thomas-Fermi molecular dynamics simulations developed in France and the USA (Livermore).

In a remarkable paper [3] he developed a new graphical expansion for the pair direct correlation functions c_ij⁽²⁾ of uniform mixturesof hard particles. The expansion involves pair overlap volumes as basis functions and pair excluded volumes as basic variables. Yasha showed that the lowest order version of the theory, the `scaled field particle approximation' yields c_ij⁽²⁾ that are the same as those of Percus Yevick (PY) theory for hard spheres in D (odd) dimensions. The paper, which constitutes a synthesis of scaled particle ideas and PY theory, provided geometric approximations for c_ij⁽²⁾ based on expansions in terms of the fundamental measures of hard particles. It was an important forerunner for Yasha's first contribution [4] to hard-sphere DFT, a Letter which continues to have an enormous influence on the development and on the applications of classical DFT.

We recall some of the history. Independently, Tarazona [5] and Curtin and Ashcroft [6] had constructed weighted-density approaches to DFT, primarily for the hard-sphere fluid. Their strategy was to write the excess free energy functional as F_ex[ρ]=∫ dr ρ(r) Ψ_ex((r)) where the excess free energy per particle Ψ_ex is evaluated at some coarse-grained or weighted density (r). The weight function w(r;), which determines (r), was obtained by requiring the second functional derivative of -β F_ex[ρ] to yield an accurate c⁽²⁾ for the uniform fluid and in practice this meant enforcing the known PY result for hard spheres. Although the approaches proved very successful in a wide variety of applications (including hard-sphere freezing) they are somewhat ad hoc in character and extensions to mixtures are not straightforward. Yasha [4] started from a very different perspective, motivated by his earlier work [3]. His is foremost a theory for hard-sphere mixtures, and it is based upon a deconvolution of the Mayer f functions f_ij(r) in terms of four scalar and two vector weight functions. The functional is written as βF_ex[{ρ_i}]= ∫ dr Φ({n_α(r)}) where Φ is a function of the weighted densities {n_α(r)} with α labelling the six weights. In the uniform fluid the weighted densities reduce to the usual scaled particle variables. Φ is constructed so that F_ex recovers the exact low densitiy limit and satisfies a certain thermodynamic requirement (a brief summary can be found in [7]). The resulting functional generates c_ij⁽²⁾ which are identical to those of PY theory. Since Yasha's approach is based on the geometrical properties of the spheres he termed it fundamental measure theory (FMT). Unlike the earlier approaches, where the density dependent weight has a range equal to the hard sphere diameter, in FMT the range of the weights is equal to the radius. FMT has now become the theory of choice for most DFT practitioners and the number of applications to inhomogeneous, and, indeed, bulk systems is growing very rapidly-as can be gleaned from several papers in the present volume. In subsequent papers Yasha proposed extensions of his FMT to nonspherical, hard convex bodies.

Another important development of FMT was made in [8]. As the title suggests, this paper encompasses many topics in the theory of liquids; one of us had the (pleasurable!) task of refereeing it-the original version was longer and broader in its scope than the published version of 23 pages in J. Chem. Phys. A key feature is a self-consistent approach for determining the density profiles for general fluid mixtures in external potentials. Crudely speaking, F_ex[{ρ}]is expanded about a uniform reference fluid and terms beyond second order are approximated by a `bridge functional', assumed to be universal. Self-consistency is imposed by the test-particle procedure for the uniform fluid where the method is equivalent to the modified HNC theory. The approach enabled the ideas of FMT to be taken over to a very wide class of liquids, including Coulomb fluids, and has proved very successful for determining the one-body structure of highly inhomogeneous fluids, and bulk fluid pair structure.

The theory of freezing was a favourite topic of Yasha. The original FMT [4] could not account for the freezing transition of the hard sphere fluid. For some time this lead Yasha to ponder whether freezing could/should be described by DFT. (He was not too impressed by results from the earlier DFTs!) However, during a sabbatical year spent in Bristol, Düsseldorf, Lyon, Madrid, München, ... he enquired how the FMT could be modified in order to incorporate freezing. He and his co-workers argued that an accurate DFT should account for dimensional crossover, i.e. situations for which the fluid is confined to effectively lower dimensionality such as in narrow slits or cylindrical pores. In particular they focused on the zero-dimensional limit, which pertains to a cavity that can contain at most one sphere mimicking the situation of a sphere restricted to a site of the crystal lattice, and constructed empirical modifications to the original result for the reduced free energy function Φ({n_α(r)}] that could recover the exact free energy in this limit [9]. The resulting functionals provided a good description of the freezing transition for the hard sphere fluid and found applications in other situations of extreme confinement. In subsequent work with Tarazona [10], Yasha developed a systematic FMT approach based entirely on knowledge of the exact free energy in the zero dimensional limit. Such an approach has provided an excellent account of the properties of the hard-sphere solid and, as discussed by M. Schmidt in this volume, has provided a powerful prescription for generating DFTs for other types of fluids. Moreover it enabled Yasha to make contact with and shed new light on his earlier work on the AHDL for fluids and solids. Among other recent extensions of FMT, Yasha and co-workers made one of the first attempts to include the solvent explicitly in DFT of electric double-layers [11].

Yasha continued working until the end. Bravely he attended the Les Houches Meeting for JPH in April 2002 and his contribution appeared in the special issue [12]. Another paper [13] appeared on July 22, one day after his death.

Of course our brief description does not do justice to Yasha's contributions to physics nor does it reflect properly his inimitable style. Reading his papers (not an easy challenge but one that is ultimately rewarding!) one could not fail to be impressed by Yasha's remarkable insight, ingenuity and technical mastery of the subject. He thought about problems in a unique, sometimes near mystical, way. He made connections where other researchers could not. He was inspirational in discussion-even if one was not really sure where his ideas originated. He enlivened the subject. It was a privilege to work with Yasha and to know him personally. Each of us has his own recollections of times with Yasha. RE remembers walking in the countryside outside of Bristol, discussing details of the AHDL. We got lost and ended up in the grounds of a mental hospital. Fortunately we obtained a ride from a doctor and his patient. RE persuaded Yasha, or was it the other way round, that it was better not to restart the discussion. HL experienced Yasha's ingenuity and `good nose' in fields removed from physics. Yasha persuaded him to buy a particular stock (TEVA) in March 2000 when the high tech stocks were booming (a detailed list of all Yasha's recommendations is available on request). The TEVA stock continues to flourish contrary to the general Nasdaq crash. In February 2000 JPH experienced Yael and Yasha's boundless hospitality when they proudly showed him around many parts of Israel, including the Dead Sea, Masada and Jerusalem. Rapidly science gave way to spirited discussions on archaeology, history and the state of world affairs!

We shall miss a gentleman of science.

R Evans, J-P Hansen and H Löwen

References

[1] Rosenfeld Y and Ashcroft N W 1979 Theory of simple classical fluids-universality in the short-range structure Phys. Rev. A 20 1208

[2] Ofer D, Nardi E and Rosenfeld Y 1988 Interionic correlations in plasmas- Thomas-Fermi hypernetted-chain density-functional theory Phys. Rev. A 38 5801

[3] Rosenfeld Y 1988 Scaled field particle theory of the structure and the thermodynamics of isotropic hard particle fluids J. Chem. Phys. 89 4272

[4] Rosenfeld Y 1989 Free energy model for the inhomogeneous hard-sphere fluid mixture and DFT of freezing Phys. Rev. Lett. 63 980

[5] Tarazona P 1984 Mol. Phys. 52 81
Tarazona P and Evans R 1984 Mol. Phys. 52 847
Tarazona P 1985 Phys. Rev. A 31 2672

[6] Curtin W A and Ashcroft N W 1985 Phys. Rev. A 32 2909

[7] Roth R, Evans R, Lang A and Kahl G 2002 J. Phys.:Condens. Matter 14 12063

[8] Rosenfeld Y 1993 Free energy model for inhomogeneous fluid mixtures: Yukawa-charged hard spheres, general interactions, and plasmas J. Chem. Phys. 98 8126

[9] Rosenfeld Y, Schmidt M, Löwen H and Tarazona P 1996 Fundamental measure free energy density functional for hard spheres: dimensional crossover and freezing Phys. Rev. E 55 4245

[10] Tarazona P and Rosenfeld Y 1997 From zero-dimensional cavities to free-energy functionals for hard disks and hard spheres Phys. Rev. E 55 R4873

[11] Biben T, Hansen J-P and Rosenfeld Y 1998 Generic density functional for electric double layers in a molecular solvent Phys. Rev. E 57 R 3727

[12] Rosenfeld Y 2002 Structure and effective interactions in multi-component hard-sphere liquids: fundamental-measure density functional approach J. Phys.: Condens. Matter 14 9141

[13] Juranek H, Redmer R and Rosenfeld Y 2002 Fluid variational theory for pressure dissociation in dense hydrogen: multicomponent reference system and non-additivity effects J. Chem. Phys. 117 1768