Cygwin Setting only for X11

Following module should be installed :

X11>xorg-server (required, the Cygwin/X X Server)

X11>xinit (required, scripts for starting the X server: xinit, startx, startwin.sh, startxwin.bat (and a shortcut on the Start Menu to run it), startxdmcp.bat )

NET> openssh

Material Derivative(Lagrangian)

This comes from Wiki. I just liked this description for distinction between Eulerian and Lagrangian.

—————————————————————–

This static position derivative is called the Eulerian derivative.

An example of this case is a swimmer standing still and sensing temperature change in a lake early in the morning: the water gradually becomes warmer due to heating from the sun.

If, instead, the path x(t) is not a standstill, the (total) time derivative of φ may change due to the path. For example, imagine the swimmer is in a motionless pool of water, indoors and unaffected by the sun. One end happens to be a constant hot temperature and the other end a constant cold temperature, by swimming from one end to the other the swimmer senses a change of temperature with respect to time, even though the temperature at any given (static) point is a constant. This is because the derivative is taken at the swimmer’s changing location. A temperature sensor attached to the swimmer would show temperature varying in time, even though the pool is held at a steady temperature distribution.

That is, the path follows the fluid current described by the fluid’s velocity field v. So, the material derivative of the scalar φ is:

Method of characteristics (from wiki)

For a first-order PDE, the method of characteristics discovers curves (called characteristic curves or just characteristics) along which the PDE becomes an ordinary differential equation (ODE). Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE.

For the sake of motivation, we confine our attention to the case of a function of two independent variables x and y for the moment. Consider a quasilinear PDE of the form

		( 1)

Suppose that a solution u is known, and consider the surface graph z = u(x,y) in R³. A normal vector to this surface is given by

As a result, equation (1 ) is equivalent to the geometrical statement that the vector field

is tangent to the surface z = u(x,y) at every point. In other words, the graph of the solution must be a union of integral curves of this vector field. These integral curves are called the characteristic curves of the original partial differential equation.

The equations of the characteristic curve may be expressed invariantly by the Charpit-Lagrange equations^[1]

or, if a particular parametrization t of the curves is fixed, then these equations may be written as a system of ordinary differential equations for x(t), y(t), z(t):

These are the characteristic equations for the original system.

2D turbulence

2D turbulence – Inverse cascade, self-organization

ref)

http://www.fluid.tue.nl/WDY/2Dturb/2Dturb.html

 

 

 

Resistivity and Plasma

Sometimes, I thought that plasma is a kind of metal. And when I asked a question to my friend who were studying liquid Lithium metal that why you did experiment with not plasma but lithium, he said, liquid metal is a kind of conducting material like plasma and shows some similar behavior like plasma with regard to MagnetoHydroDynamics(MHD).

I don’t know lithium well, and I’m not sure whether lithium is an exception or not about tendency of resistivity for metal, anyway, but there is a difference of dependence on temperature about resistivity between metal and plasma. 

Before we deal with the subject, if we look at only results, Interestingly, as temperature goes to high, resistivity of metal increases, but resistivity of plasma decreases.

Just want to check the difference of tendency on resistivity between plasma and metal in terms of temperature. In addition, will historically review improvement of developing theory for resistivity of plasma following some lecture notes and books ( Greg Hammett’s Lecture notes , etc… )

 

Resistivity ( Reference : Wikipedia )

============================

In general, electrical resistivity of metals increases with temperature, while the resistivity of semiconductors decreases with increasing temperature. In both cases, electron-phonon interactions can play a key role. At high temperatures, the resistance of a metal increases linearly with temperature. As the temperature of a metal is reduced, the temperature dependence of resistivity follows a power law function of temperature. Mathematically the temperature dependence of the resistivity ρ of a metal is given by the Bloch-Gruneissen formula:

where ρ(0) is the residual resistivity due to defect scattering, A is a constant that depends on the velocity of electrons at the fermi surface, the Debye radius and the number density of electrons in the metal. Θ_R is the Debye temperature as obtained from resistivity measurements and matches very closely with the values of Debye temperature obtained from specific heat measurements. n is an integer that depends upon the nature of interaction:

1. n=5 implies that the resistance is due to scattering of electrons by phonons (as it is for simple metals)
2. n=3 implies that the resistance is due to s-d electron scattering (as is the case for transition metals)
3. n=2 implies that the resistance is due to electron-electron interaction.

As the temperature of the metal is sufficiently reduced (so as to ‘freeze’ all the phonons), the resistivity usually reaches a constant value, known as the residual resistivity. This value depends not only on the type of metal, but on its purity and thermal history. The value of the residual resistivity of a metal is decided by its impurity concentration. Some materials lose all electrical resistivity at sufficiently low temperatures, due to an effect known as superconductivity.

An even better approximation of the temperature dependence of the resistivity of a semiconductor is given by the Steinhart-Hart equation:

where A, B and C are the so-called Steinhart-Hart coefficients.

This equation is used to calibrate thermistors.

In non-crystalline semi-conductors, conduction can occur by charges quantum tunnelling from one localised site to another. This is known as variable range hopping and has the characteristic form of , where n=2,3,4 depending on the dimensionality of the system.

==============================

Adiabatic Invariant

It seems adiabatic invariant is important to analyze and understand physics of nature heuristically like magnetic moment which is usually assumed as a conserved quantity in plasma physics field. To understand adiabatic invariant in the structure of classical mechanics, we need to deal with some theory.

This article is written to understand more deeply where adiabatic quantities come from. When I solve some problems of a particle motion in slowly varing field( Electric or Magnetic field), I did a mistake that I thought energy of particle is a conserved quantity, and the problem required adiabatic moment conserving flux of mangtic field in the circle of drift motion of the particle. To exactly figure out this kind of conserved quantities is a final object of this article. Following is a very similar example of above problem,

<Excerpt from Classical Mechanics by Goldstein>

“At the first Solvay Conerence 1911, which grappled with the problems of introducing quantum notions into physics, a deceptively simple problem in classical mechanics was raised. Consider a bob on a string oscillating as a plane pendulum, with the string pssing through a small hole in the ceiling. Now imagine that the string is either pulled up or let down slowly, so slowly that there is little change in the length of the pendulum during one period of oscillation. What happens to the frequency of oscillation during this process?”

 

To figure out how it works, I’ll cover following subjects.

1. Hamilton-Jacobi equation

2. Action-Angle Principle

3 Canonincal Perturbation Theory.

 

Example )

1. A counterintuitive problem by Greg Hammet

http://w3.pppl.gov/~hammett/courses/gpp1/counter-intuitive.pdf

 

MHD and Turbulence

http://www.damtp.cam.ac.uk/user/as629/PartIIIL05/course_blogL05.html

 

Energy Principle Ref.

The original famous paper on the MHD energy principle is I. B. Bernstein, E. A. Frieman, M. D. Kruskal & R. M. Kulsrud, Proc. Roy. Soc. LondonA244, 17 (1958)
Lagrangian formulation of MHD and the action principle are discussed in the excellent original paper by Newcomb:
W. A. Newcomb, Nucl. Fusion: 1962 Supplement, Part 2, p. 451 (distributed in class)
A more recent useful reference is D. Pfirsch & R. N. Sudan, Phys. Fluids B 5, 2052 (1993)

My favorite songs

Pullback

<Wiki>

The notion of pullback in mathematics is a fundamental one. It refers to two different, but related processes: precomposition and fiber-product.

Precomposition with a function probably provides the most elementary notion of pullback: in simple terms, a function f of a variable y, where y itself is a function of another variable x, may be written as a function of x. This is the pullback of f by the function y(x).

Let φ:M→ N be a smooth map between (smooth) manifolds M and N, and suppose f:N→R is a smooth function on N. Then the pullback of f by φ is the smooth function φ^*f on M defined by (φ^*f)(x) = f(φ(x)). Similarly, if f is a smooth function on an open set U in N, then the same formula defines a smooth function on the open set φ^-1(U) in M. (In the language of sheaves, pullback defines a morphism from the sheaf of smooth functions on N to the direct image by φ of the sheaf of smooth functions on M.)

More generally, if f:N→A is a smooth map from N to any other manifold A, then φ^*f(x)=f(φ(x)) is a smooth map from M to A.

<Frankel> p.57

The presence of the Poincare’ 1-form field on T*M and the capability of pulling back 1-form fields under mappings endow T*M with a powerful tool that is not available on TM.

싸이월드에 대한 단상

모든 홈페이지나 블로그 등이 여러 역할이 있겠지만, 무엇인가를 누군가에게 보여주는 역할이 강하다는 것에 대해서는 부인할 수 없을 것이다. (물론, 이 블로그는 자료의 저장과 내 기억을 위해서 만들어지긴 했다만 이 글은 누군가를 위해서 쓰는지도 모르겠다.) 자신을 다른 사람에게 알리고 싶은 욕구에 대한 표출일 수 있겠다. 달리 말하면 사랑받고자 하는 욕망인가.

싸이월드를 했었고, 그리고 마지막으로 탈퇴를 하면서 느낀 바는, 내가 항상 일기를 남기고 있다는 것이다. 일기는 관념적으로 남에게 보여주기 보다는 주로 사적인 영역으로 감추어지는 경향이 크다.  그럼에도 불구하고 계속적으로 일기를 남기고자 했었던 것이, 어떻게 보면 내가 다른 사람에게 말하고 싶었지만 하지 못했던 것들을 해결하려고 했었던 것이 아닌가 싶다.  또한 내가 다른 사람들의 여러 일기들을 들여다보면서, 그들의 생활에 관심을 가지게 되고 궁금증을 가지게 되었었다. 이런 반응들을 목적으로 적혀져 있는 일기들이 있음을 부인하지는 못할 것이다.

 어쨌든, 과거의 싸이월드가 아닌 오프라인에서 적던 일기가 누군가에게 보여주기 위해서 적던 일기가 있었는지 궁금하다.  과거의 일기는 비밀 일기에 가깝겠다. 하지만 싸이월드에서의 일기들은 비밀 일기라기 보다는 읽고 상대의 반응을 바라는, 비밀을 가장한 목적 일기에 가까운 것이 아닌가라는 생각이 든다.

이러한 알게 모르게 인간 심리를 움직이는 것이 나를 열어 놓음으로 상대와 더욱 가까워 질 수 있다는 순기능도 있겠지만, 상대의 마음을 이용하여 원하는 바를 얻으려 하는 역기능이 될 수도 있겠다. 점점 사람의 마음에 교묘함을 심는 일기가 순수하게 하루를 회고하고 반성하던 일기의 역할을 대체하게 됨을 느껴 싸이월드를 그만 둘 수 밖에 없었다.