N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

A quantum dot is a sub-micron-scale conducting device containing up to several thousand electrons. Transport through a quantum dot at low temperatures is a quantum-coherent process. This review focuses on dots in which the electron's dynamics are chaotic or diffusive, giving rise to statistical properties that reflect the interplay between one-body chaos, quantum interference, and electron-electron interactions. The conductance through such dots displays mesoscopic fluctuations as a function of gate voltage, magnetic field, and shape deformation. The techniques used to describe these fluctuations include semiclassical methods, random-matrix theory, and the supersymmetric nonlinear \ensuremath{\sigma} model. In open dots, the approximation of noninteracting quasiparticles is justified, and electron-electron interactions contribute indirectly through their effect on the dephasing time at finite temperature. In almost-closed dots, where conductance occurs by tunneling, the charge on the dot is quantized, and electron-electron interactions play an important role. Transport is dominated by Coulomb blockade, leading to peaks in the conductance that at low temperatures provide information on the dot's ground-state properties. Several statistical signatures of electron-electron interactions have been identified, most notably in the dot's addition spectrum. The dot's spin, determined partly by exchange interactions, can also influence the fluctuation properties of the conductance. Other mesoscopic phenomena in quantum dots that are affected by the charging energy include the fluctuations of the cotunneling conductance and mesoscopic Coulomb blockade.

The Statistical theory of quantum dots

Critical point explosions in two-dimensional wave fields

This is a brief review of some recent developments inexperiments and theory of the Coulomb blockade phenomenon. Weconcentrate on electron tunneling through semiconductorquantum dots and normal-metal nanoparticles. The linearconductance and non-linear I–V characteristics are sensitivenot only to the charging energy, but also to smaller energyscales. These are the spacing between discrete energy levelsof intra-dot electron excitations, and the width these levelsacquire due to the electron escape to leads through thejunctions. At sufficiently low temperatures, or at not-so-small junction conductances, the many-body phenomena, such asKondo effect, become observable in the dot-lead system. Kondoeffect yields an additional fine energy scale to the problem.We discuss a simple theoretical model incorporating theseenergy scales, and its relation to the experimentalobservations.

Single Electron Tunneling

In quantum/wave systems with chaotic classical analogues, wavefunctions evolve in highly complex, yet deterministic ways. A slight perturbation of the system, though, will cause the evolution to diverge from its original behaviour increasingly with time. This divergence can be measured by the fidelity, which is defined as the squared overlap of the two time evolved states. For chaotic systems, two main decay regimes of either Gaussian or exponential behaviour have been identified depending on the strength of the perturbation. For perturbation strengths intermediate between the two regimes, the fidelity displays both forms of decay. By applying a complementary combination of random matrix and semiclassical theory, a uniform approximation can be derived that covers the full range of perturbation strengths. The time dependence is entirely fixed by the density of states and the so-called transition parameter, which can be related to the phase space volume of the system and the classical action diffusion constant, respectively. The accuracy of the approximations is illustrated with the standard map.

A uniform approximation for the fidelity in chaotic systems

We analyse the ground state of spinless fermions on a lattice in a weakly disordered potential, interacting via a nearest neighbour interaction, by applying the self-consistent Hartree-Fock approximation. We find that charge density modulations emerge progressively when r_s >1, even away from half-filling, with only short-range density correlations. Classical geometry dependent "magic numbers" can show up in the addition spectrum which are remarkably robust against quantum fluctuations and disorder averaging.

Density modulations and addition spectra of interacting electrons in disordered quantum dots

We give a theoretical treatment of magnetic dipole absorption of electromagnetic radiation in small conducting particles, at photon energies which are large compared to the single-particle level spacing, and small compared to the plasma frequency. We discuss both diffusive and ballistic electron dynamics for particles of arbitrary shape. The conductivity becomes non-local when the frequency is smaller than the frequency characterizing the transit of electrons from one side of the particle to the other, but in the diffusive case plays no role in determining the absorption coefficient. In the ballistic case, the absorption coefficient is proportional to for , but is a decreasing function of for .

/pdf/magnetic-dipole-absorption-of-radiation-in-small-conducting-e3tl2h0s2h.pdf

Magnetic dipole absorption of radiation in small conducting particles

We give a theoretical treatment of magnetic dipole absorption of electromagnetic radiation in small conducting particles, at photon energies which are large compared to the single particle level spacing, and small compared to the plasma frequency. We discuss both diffusive and ballistic electron dynamics for particles of arbitrary shape. 
The conductivity becomes non-local when the frequency is smaller than the frequency \omega_c characterising the transit of electrons from one side of the particle to the other, but in the diffusive case \omega_c plays no role in determining the absorption coefficient. In the ballistic case, the absorption coefficient is proportional to \omega^2 for \omega > \omega_c.

https://arxiv.org/pdf/cond-mat/9709075.pdf

Magnetic Dipole Absorption of Radiation in Small Conducting Particles

Chern integers describing quantized transport change (typically by $\ifmmode\pm\else\textpm\fi{}1$) when the energy levels or bands with which they are associated become degenerate. We give a statistical treatment of these degeneracies and the consequent fluctuations of the Chern integers. The density of degeneracies is calculated exactly for a parametrization of the Gaussian unitary ensemble of matrices, and we present numerical results indicating that the ``gas'' of degeneracies has a ``charge neutrality'' property. The results apply to a broad class of complex systems after rescaling the parameter space.

Universal Fluctuations of Chern Integers.

We consider singularities of the set of energy levels En(X) of a quantum Hamiltonian, obtained by varying a set of d parameters X=(X1,..,Xd). Singularities such as minima, degeneracies, branch points, and avoided crossings can play an important role in physical applications. We discuss a general method for counting these singularities, and apply it to a random matrix model for the parameter dependence of energy levels. We also show how the density of avoided crossing singularities is related to a non‐analyticity of a correlation function describing the energy levels.

/pdf/singularities-in-the-spectra-of-random-matrices-2xvl5mogdr.pdf

Singularities in the spectra of random matrices

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