Experimental determinations of correlation functions typically involve events of differing particle number. In order to interpret such data, it is important to relate these measurements to fixed-number correlations. We exhibit the total correlation function in terms of fixed n correlations and fluctuations with respect to the average. We note that moments, constructed as integrals of the appropriate correlation functions, can be dominated in the case of broad distributions by cumulant moments constructed from single-particle density fluctuations.
Experimental determinations of correlation functions typically involve events of differing particle number. In order to interpret such data, it is important to relate these measurements to fixed-number correlations. We exhibit the total correlation function in terms of fixed n correlations and fluctuations with respect to the average. We note that moments, constructed as integrals of the appropriate correlation functions, can be dominated in the case of broad distributions by cumulant moments constructed from single-particle density fluctuations.
The study of electrical currents in a regime where a description as a continuous flow fails and the particle nature becomes important, but where notions such as particle density or currents continue to be meaningful, will be at the heart of the present dissertation. Given such a mesoscopic system through which electrons flow, either actively through quantum pumping or passively due to a potential difference, we investigate quantities such as transferred charge, noise, current fluctuations, and relations between them. More generally we shall examine the statistics of particle transport. All discussions will concentrate on compact devices transferring charge between two one-dimensional reservoirs (the leads) of independent electrons. Firstly, we consider the generating function of the moments of transferred charge and we prove that it is given by the determinant of a single-particle operator. It is formally equivalent to a formula of Lesovik and Levitov, but is shown to be well-defined in the situations under consideration, whereas the former is not, in general. The mechanism leading to regularization is typical for mesoscopic systems: we consider a limiting case that reveals the fundamental structure of the problem. In fact, as suited for an open system, the leads contain an infinite number of charges. It follows mathematically that the second quantization ought to be carried out upon the Fermi sea and not upon the vacuum, a procedure known as the GNS construction. We shall rely heavily on operator algebraic techniques. The set of observables, the C*-algebra of canonical anticommutation relations, admits different representations for systems in the thermodynamic limit, which are described by their reduced one-particle density matrix. An important question will be the existence of operators such as the charge or the time evolution in a given representation, which induce so-called Bogoliubov transformations. These prerequisites at hand, we shall pose the problem of counting statistics clearly, prove that the regularized determinant emerges naturally, and show that it describes the transport of integer charges with non-negative probabilities. Secondly, we introduce a model of instantaneous scattering in which various examples can be computed explicitely. In particular, we show that the above theorem applies for scattering through a smoothly operating pump over a finite time interval. We also exhibit situations where the propagator does exist but the generating function is ambiguous. In a more computational approach, we then turn to the relation of individual moments of transferred charge to current correlators. It turns out that the former can indeed be expressed as integrals of the latter over time, provided the current operators are ordered according to Matthews’ T∗ ordering. It is given by first arranging the charge operators according to their time indices, and only then taking the time derivatives, which induce contact terms whose singular support lies along time coincidences. We shall give two derivations of this result, because the first one is transparent, and the second one is related to contradictory computations in the literature where the system is considered to be measured by an external spin. In the model of constant instantaneous scattering, we then argue from the Lesovik-Levitov determinant that the transport can be interpreted in terms of a binomial process in the long time limit. In a parallel computation, we concentrate on the third cumulant as defined by integration of current correlators and show how the contact terms of the T∗ ordering play an essential role in providing the expected binomial result, too. To close this thesis, we suggest an algebraic framework that lays the foundations for a rigorous discussion of the Casimir effect. The problem of infinites arising in the comparison of two physical situations for a bosonic field is interpreted as the non-equivalence of states on the algebra of canonical commutation relations. We exhibit sufficient conditions on the one-particle Hamiltonians which ensure that the Casimir energy is finite and give an expression for it. We discuss its formal equivalence with the result obtained from the standard mode summing method.
Abstract One of the fundamental problems of statistical mechanics and its quantum field theory formulation is the characterization of the order parameters and the computation of their correlation functions. Besides the intrinsic interest of this problem, the correlation functions are the key quantities in the determination of the universal ratios of the renormalization group and therefore they can have direct experimental confirmation (see Chapter 8). In general, the computation of correlation functions is a difficult task, usually achieved with partial success through perturbative methods.
The article formulates the concept of a cumulant representation for distribution functions that describe the state of many-particle systems with topological interaction, i.e., using the interaction potential determined by the proximity rank of particles. Cumulants of probability distribution functions are interpreted as correlations of particle states and are defined as solutions of the corresponding cluster expansions of probability distribution functions. We emphasize that the correlations that arise during the evolution of a system of particles with topological interaction naturally differ from the structure of correlations of many-particle systems, the state of which is traditionally described by symmetric probability distribution functions. The article establishes a hierarchy of evolution nonlinear equations for correlation functions (hierarchy of recursive Liouville equations). In the space sequences of integrated functions, a non-perturbative solution of the Cauchy problem for a hierarchy of such nonlinear evolution equations is constructed. Typical properties of the expansion of such a solution, which are generated by the properties of its generating operators, namely, cumulants of groups of operators of Liouville equations, are investigated. Based on the dynamics of correlations, the structure of series expansions is established, which determines reduced distribution functions, as well as reduced correlation functions, which, in particular, made it possible to substantiate the structure of generating operators of the solution of the Cauchy problem for the hierarchy of BBGKY equations (Bogolyubov–Born–Green–Kirkwood–Yvon). It is proved that the structure of expansions for correlation functions, the generating operators of which are cumulants of groups of operators of the corresponding order of Liouville equations, induces the cumulant structure of series expansions for reduced distribution functions and reduced correlation functions. Thus, the dynamics of state correlations generate the dynamics of many-particle systems with topological interaction.
We investigate the probability distribution of the quantum fluctuations of thermodynamic functions of finite, ballistic, phase-coherent Fermi gases. Depending on the chaotic or integrable nature of the underlying classical dynamics, on the thermodynamic function considered, and on temperature, we find that the probability distributions are dominated either (i) by the local fluctuations of the single-particle spectrum on the scale of the mean level spacing, or (ii) by the long-range modulations of that spectrum produced by the short periodic orbits. In case (i) the probability distributions are computed using the appropriate local universality class, uncorrelated levels for integrable systems and random matrix theory for chaotic ones. In case (ii) all the moments of the distributions can be explicitly computed in terms of periodic orbit theory, and are system-dependent, non-universal, functions. The dependence on temperature and number of particles of the fluctuations is explicitly computed in all cases, and the different relevant energy scales are displayed. . .