Integrable Systems: An Overview Preamble. The following pages
... intimately related to the n-soliton solutions of a host of soliton equations. Indeed, the existence of these relativistic systems was first established (by Ruijsenaars and Schneider) with an eye on replacing the interaction of n sine-Gordon solitons by that of an equivalent interaction between n rel ...
... intimately related to the n-soliton solutions of a host of soliton equations. Indeed, the existence of these relativistic systems was first established (by Ruijsenaars and Schneider) with an eye on replacing the interaction of n sine-Gordon solitons by that of an equivalent interaction between n rel ...
String Theory
... but not the path integral that they approximate. • That is good enough at weak coupling for most things but even then not entirely eg. Tunneling is ...
... but not the path integral that they approximate. • That is good enough at weak coupling for most things but even then not entirely eg. Tunneling is ...
Euler Lagrange Equation
... - Gluons do carry colour charge,so to solve the QCD theory, approximations are proposed (e.g. Lattice calculation method ). - There is a fermion doubling problem in lattice which can be solved by various methods. - In order to obtain light quark properties, we need bigger computers and the calculati ...
... - Gluons do carry colour charge,so to solve the QCD theory, approximations are proposed (e.g. Lattice calculation method ). - There is a fermion doubling problem in lattice which can be solved by various methods. - In order to obtain light quark properties, we need bigger computers and the calculati ...
Clustering of Particles in Turbulent Flows
... Particles move in a less dense fluid with velocity field Particles do not affect the velocity field, or interact (until they make contact). The equation of motion is assumed to be ...
... Particles move in a less dense fluid with velocity field Particles do not affect the velocity field, or interact (until they make contact). The equation of motion is assumed to be ...
Scale invariance
In physics, mathematics, statistics, and economics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor. The technical term for this transformation is a dilatation (also known as dilation), and the dilatations can also form part of a larger conformal symmetry.In mathematics, scale invariance usually refers to an invariance of individual functions or curves. A closely related concept is self-similarity, where a function or curve is invariant under a discrete subset of the dilatations. It is also possible for the probability distributions of random processes to display this kind of scale invariance or self-similarity.In classical field theory, scale invariance most commonly applies to the invariance of a whole theory under dilatations. Such theories typically describe classical physical processes with no characteristic length scale.In quantum field theory, scale invariance has an interpretation in terms of particle physics. In a scale-invariant theory, the strength of particle interactions does not depend on the energy of the particles involved.In statistical mechanics, scale invariance is a feature of phase transitions. The key observation is that near a phase transition or critical point, fluctuations occur at all length scales, and thus one should look for an explicitly scale-invariant theory to describe the phenomena. Such theories are scale-invariant statistical field theories, and are formally very similar to scale-invariant quantum field theories.Universality is the observation that widely different microscopic systems can display the same behaviour at a phase transition. Thus phase transitions in many different systems may be described by the same underlying scale-invariant theory.In general, dimensionless quantities are scale invariant. The analogous concept in statistics are standardized moments, which are scale invariant statistics of a variable, while the unstandardized moments are not.