-Recently developed techniques allow for simultaneous measurements of the positions of all ultra cold atoms in a trap with high resolution. Each such single shot experiment detects one element of the quantum ensemble formed by the cloud of atoms. Repeated single shot measurements can be used to determine all correlations between particle positions as opposed to standard measurements that determine particle density or two-particle correlations only. In this paper we discuss the possible outcomes of such single shot measurements in case of cloud of ultra-cold noninteracting Fermi atoms. We show that the Pauli exclusion principle alone leads to correlations between particle positions that originate from unexpected spatial structures formed by the atoms.Introduction. -Tremendous progress in experimental techniques of preparing, manipulating and probing ultra-cold gases have opened new possibilities of optical methods of monitoring atomic systems. Atomic fluorescence microscopes with resolution in the range of hundreds of nanometers became accessible [1][2][3][4][5][6][7]. The microscopes allow for observation of both boson and fermion atoms with resolution comparable to the optical wavelength. Single shot pictures of such systems correspond to a single realization of the N -body probability density as opposed to a one-particle probability distribution. Difference between the two is tremendous, they differ by N body correlations. The seminal work of [8] shows how interference fringes, visible in a simultaneous single shot picture of N atoms, arise in the course of measurement. No fringes are observed in a single particle detection instead. In a similar way the solitons emerge in a process of detection of N -particles prepared in a type II excited state of a 1D system of bosons interacting via short-range potential described by the Lieb-Linger model [9]. Single shot time-dependent simulations of many-body dynamics showing appearance of fluctuating vortices and center-ofmass fluctuations of attractive BEC have been reported recently [10].

A systematic approach on collective diffusion in an interacting lattice gas adsorbed on a non-homogeneous substrate is formulated. It is based on a variational Ritz procedure of determining a diffusive eigenvalue of a transition rate matrix describing microscopic kinetics of particle migration processes in the system. Form of a trial eigenvector and a choice of variational parameters are discussed and justified on physical grounds. Reed-Ehrlich factorization of the collective diffusion coefficient into the thermodynamic and kinetic factors is explicitly shown to emerge naturally from the variational approach, and closed expressions for both factors are derived. Validity of the approach is tested by applying it to the simplest case of diffusion of noninteracting adparticles across steps on a stepped substrate ͑modeled by a Schwoebel barrier͒. The coverage dependence of collective diffusion coefficient, obtained in an algebraic form, agrees very well with the results of Monte Carlo simulations. It is demonstrated that the results obtained provide a substantial improvement over the mean-field theory results for the same system. Generalizations necessary to include interparticle interactions are listed and discussed.

A variational approach to microscopic kinetics of an interacting lattice gas is presented. It accounts for the equilibrium correlations in the system and allows one to derive an algebraic expression for the particle density (coverage) dependent chemical diffusion coefficient for a wide variety of interaction models. Detailed derivation is presented for a one dimensional case for which the results are compared with the results of Monte Carlo simulations. Generalization and an application to the simplest case of the two dimensional lattice gas is briefly described.

Exact solutions to many-body interacting systems of both bosonic and fermionic particles confined to harmonic potential in an arbitrary number of dimensions are given. Energy levels and their degeneracies for trapped identical particles interacting via harmonic forces are calculated. This specific form of the interaction allows for analytical solutions. The mutual interaction, attractive or repulsive, modifies significantly the properties of the considered system. For a large number of particles the interaction essentially results in a frequency shift. Statistical properties ͑e.g., microcanonical and grand canonical partition functions͒ as well as some illustrative, physically relevant examples are discussed. Our results give an unusual opportunity for further studies of interacting systems in the framework of the exactly soluble model.