Asymptotic behavior of many-particle distribution functions

Adrian Patrascioiu

Hiroaki Hara

The behavior of the one-particle reduced density matrix ρ(1)(r) and the two-particle distribution function ρ(2)(r) of the ideal Fermi gas are investigated for various limiting cases. These cases are classified by the relative magnitude of the following parameters; the distance r, the thermal wave length λT and an “effective length” R0 = (6π2ρ/2S+1)-1/3, where ρ is the density and S is spin. At T = 0, R0-1 is reduced to the Fermi wave number k0 and the obtained long distance behaviors are well known; for instance, R0-2r-4[1+cos (2k0r)] for ρ(2)(r), whereas for T ≠0, this well-known behavior is shown to be valid only when r is smaller than λT and the asymptotic behavior is found to be λT-4r-2[1+cos (2k0r)]exp (-arλT), where a is a constant. This is a sum of terms which shows exponential decay and oscillatory exponential decay, respectively. An explicit form of the three-particle distribution function ρ(3)(r12, r23, r31) of the ideal Fermi gas is also given and its behaviors are discussed.

A. Isihara

André Ahlbrecht

In this thesis we study the propagation behavior of quantum particles which perform discrete-time quantum walks on spatial lattices.By definition, quantum walks are discrete-time evolutions such that a single time step couples only neighboring lattice sites.Additionally, we assume in this thesis that the dynamical law commutes with all translations of the spatial lattice, that is we consider translationally invariant quantum walks.The results presented in the following are separated into two parts: The first part is devoted to the study of single-particle quantum walks, and in the second part we investigate quantum walks of several particles.Single-particle quantum walks: Our aim is to study the asymptotic behavior of particles undergoing general translationally invariant quantum walks.Employing perturbation theory, we establish a direct method for computing the asymptotic distribution of the particle's position rescaled by an appropriate time-dependent factor.If the time evolution is implemented by a unitary operator commuting with all lattice translations and with empty pure point spectrum, we find linear scaling of the asymptotic position distribution.In this case, the asymptotic distribution of position divided by time is determined by the analog of the group velocity.If, on the other hand, the quantum walk is subject to decoherence, we expect that it behaves like a classical random walk.We provide explicit criteria under which this intuitive statement is true, that is, the asymptotic scaling of the particle's position distribution is that of a diffusion process.We find that the asymptotic distribution of position divided by the square root of time is a mixture of Gaussians in this case.However, we also provide examples of decoherent quantum walks which violate these criteria, and exhibit linear scaling of the asymptotic position distribution. Many-particle quantum walks:We investigate unitary quantum walks of two particles with collisional interactions.That is, the particles perform independent single-particle quantum walks if they are at distinct lattice sites, and an interaction takes place if both particles occupy the same lattice site.We prove that such quantum walks generically exhibit a molecular binding effect between the particles.More precisely, for a measurement of the particles' positions we have a bound on the probability to obtain a certain result which decays exponentially with the particles' distance.As a preparatory step for this analysis, we consider single-particle quantum walks with point defects, and provide a method to compute their eigenvalues in the spectral gap of the quantum walk without defect.We further generalize our model to arbitrary numbers of particles and study the behavior of local excitations in many-particle quantum walks.

Calvin Stubbins

V. R. Struletskaya