We discuss two-point correlations of the actions of classical periodic orbits in chaotic systems. For systems where the semiclassical trace formula is exact and the spectral statistics follow random matrix theory, there exist nontrivial correlations between actions, which we express in a universal form. We illustrate this result with the analogous problem of the pair correlations between prime numbers. We also report on numerical studies of three chaotic systems where the semiclassical trace formula is only approximate, but nevertheless these unexpected action correlations are observed.PACS numbers: 05.45.+b, 03.65.Sq Semiclassical trace formulas relate the quantum spectral density to classical periodic orbits [1]. In this Letter we use this link to express a classical two-point correlation function, involving the actions and stabilities of the periodic orbits of chaotic systems, in terms of a two-point statistic of the quantum energy spectrum. We first consider systems for which the trace formulas are exact. Assuming that the spectral fluctuations follow the predictions of random matrix theory (RMT) [2,3], we derive a universal expression for the classical correlation function. This expression represents in some cases a tendency towards action repulsion. Studying in exactly the same way the correlations between pairs of prime numbers, i.e., assuming that the zeros of the Riemann zeta function follow RMT, we get a correlation which is consistent with the Hardy-Littlewood conjecture of number theory [4,5]. Surprisingly, we find that such correlations also occur in other chaotic systems which we studied numerically, and for which the semiclassical approximation (SCA) is not expected to be valid in the limit of long times. For definiteness we consider Hamiltonian flows with 2 freedoms and maps with 1 degree of freedom.Trace formulas express the spectral density by the Selberg-Gutzwiller sum [1], d{E) ^diE)-^dosM) -d{E)-\--^^SpTp h^r |det(M;-/)|'/2 exp / ^ 171 ^rS,--rv,where d{E) is the mean level density, and the summation is over primitive periodic orbits p and repetitions r (both positive and negative). Sp, Tp, gp, Mp, and Vp are the action, period, multiplicity, monodromy matrix, and Maslov index of the pth orbit, respectively. We label the preexponential factor in (1) by Aj, where j signifies the pair ip,r), and define Sj=rSp, Vj~rvp, and Tj=rTp,

Density Functional Theory (DFT) is one of the most widely used methods for "ab initio" calculations of the structure of atoms, molecules, crystals, surfaces, and their interactions. Unfortunately, the customary introduction to DFT is often considered too lengthy to be included in various curricula. An alternative introduction to DFT is presented here, drawing on ideas which are well-known from thermodynamics, especially the idea of switching between different independent variables. The central theme of DFT, i.e. the notion that it is possible and beneficial to replace the dependence on the external potential v(r) by a dependence on the density distribution n(r), is presented as a straightforward generalization of the familiar Legendre transform from the chemical potential µ to the number of particles N . This approach is used here to introduce the Hohenberg-Kohn energy functional and to obtain the corresponding theorems, using classical nonuniform fluids as simple examples. The energy functional for electronic systems is considered next, and the Kohn-Sham equations are derived. The exchange-correlation part of this functional is discussed, including both the local density approximation to it, and its formally exact expression in terms of the exchange-correlation hole. A very brief survey of various applications and extensions is included.

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