More is the Same; Phase Transitions and Mean Field Theories

This paper looks at the early theory of phase transitions. It considers a group of related concepts derived from condensed matter and statistical physics. The key technical ideas here go under the names of "singularity", "order parameter", "mean field theory", and "variational method". In a less technical vein, the question here is how can matter, ordinary matter, support a diversity of forms. We see this diversity each time we observe ice in contact with liquid water or see water vapor, "steam", come up from a pot of heated water. Different phases can be qualitatively different in that walking on ice is well within human capacity, but walking on liquid water is proverbially forbidden to ordinary humans. These differences have been apparent to humankind for millennia, but only brought within the domain of scientific understanding since the 1880s. A phase transition is a change from one behavior to another. A first order phase transition involves a discontinuous jump in a some statistical variable of the system. The discontinuous property is called the order parameter. Each phase transitions has its own order parameter that range over a tremendous variety of physical properties. These properties include the density of a liquid gas transition, the magnetization in a ferromagnet, the size of a connected cluster in a percolation transition, and a condensate wave function in a superfluid or superconductor. A continuous transition occurs when that jump approaches zero. This note is about statistical mechanics and the development of mean field theory as a basis for a partial understanding of this phenomenon.Comment: 25 pages, 6 figure

Extension of the sum rule for the transition rates between multiplets to the multiphoton case

The sum rule for the transition rates between the components of two multiplets, known for the one-photon transitions, is extended to the multiphoton transitions in hydrogen and hydrogen-like ions. As an example the transitions 3p-2p, 4p-3p and 4d-3d are considered. The numerical results are compared with previous calculations.Comment: 10 pages, 4 table

Stochastic motion of test particle implies that G varies with time

The aim of this letter is to propose a new description to the time varying gravitational constant problem, which naturally implements the Dirac's large numbers hypothesis in a new proposed holographic scenario for the origin of gravity as an entropic force. We survey the effect of the Stochastic motion of the test particle in Verlinde's scenario for gravity\cite{Verlinde}. Firstly we show that we must get the equipartition values for $t\rightarrow\infty$ which leads to the usual Newtonian gravitational constant. Secondly,the stochastic (Brownian) essence of the motion of the test particle, modifies the Newton's 2'nd law. The direct result is that the Newtonian constant has been time dependence in resemblance as \cite{Running}.Comment: Accepted in International Journal of Theoretical Physic

Analytic theory of ground-state properties of a three-dimensional electron gas at varying spin polarization

We present an analytic theory of the spin-resolved pair distribution functions $g_{\sigma\sigma'}(r)$ and the ground-state energy of an electron gas with an arbitrary degree of spin polarization. We first use the Hohenberg-Kohn variational principle and the von Weizs\"{a}cker-Herring ideal kinetic energy functional to derive a zero-energy scattering Schr\"{o}dinger equation for $\sqrt{g_{\sigma\sigma'}(r)}$. The solution of this equation is implemented within a Fermi-hypernetted-chain approximation which embodies the Hartree-Fock limit and is shown to satisfy an important set of sum rules. We present numerical results for the ground-state energy at selected values of the spin polarization and for $g_{\sigma\sigma'}(r)$ in both a paramagnetic and a fully spin-polarized electron gas, in comparison with the available data from Quantum Monte Carlo studies over a wide range of electron density.Comment: 13 pages, 8 figures, submitted to Phys. Rev.

Universal features of the order-parameter fluctuations : reversible and irreversible aggregation

We discuss the universal scaling laws of order parameter fluctuations in any system in which the second-order critical behaviour can be identified. These scaling laws can be derived rigorously for equilibrium systems when combined with the finite-size scaling analysis. The relation between order parameter, criticality and scaling law of fluctuations has been established and the connexion between the scaling function and the critical exponents has been found. We give examples in out-of-equilibrium aggregation models such as the Smoluchowski kinetic equations, or of at-equilibrium Ising and percolation models.Comment: 19 pages, 10 figure

Probing Ion-Ion and Electron-Ion Correlations in Liquid Metals within the Quantum Hypernetted Chain Approximation

We use the Quantum Hypernetted Chain Approximation (QHNC) to calculate the ion-ion and electron-ion correlations for liquid metallic Li, Be, Na, Mg, Al, K, Ca, and Ga. We discuss trends in electron-ion structure factors and radial distribution functions, and also calculate the free-atom and metallic-atom form-factors, focusing on how bonding effects affect the interpretation of X-ray scattering experiments, especially experimental measurements of the ion-ion structure factor in the liquid metallic phase.Comment: RevTeX, 19 pages, 7 figure

Active Brownian Particles. From Individual to Collective Stochastic Dynamics

We review theoretical models of individual motility as well as collective dynamics and pattern formation of active particles. We focus on simple models of active dynamics with a particular emphasis on nonlinear and stochastic dynamics of such self-propelled entities in the framework of statistical mechanics. Examples of such active units in complex physico-chemical and biological systems are chemically powered nano-rods, localized patterns in reaction-diffusion system, motile cells or macroscopic animals. Based on the description of individual motion of point-like active particles by stochastic differential equations, we discuss different velocity-dependent friction functions, the impact of various types of fluctuations and calculate characteristic observables such as stationary velocity distributions or diffusion coefficients. Finally, we consider not only the free and confined individual active dynamics but also different types of interaction between active particles. The resulting collective dynamical behavior of large assemblies and aggregates of active units is discussed and an overview over some recent results on spatiotemporal pattern formation in such systems is given.Comment: 161 pages, Review, Eur Phys J Special-Topics, accepte