Causality principle for the klein- gordon equation, Physics

Assignment Help:

Suppose Fluid (say, water) occupies a domain D 230_causality principle for the Klein- Gordon equation.png and has velocity field V=V(x, t). A substance (say, a day) is suspended into the fluid and will be transported by the fluid as well as diffused within it; let u= u(x,t) be the concentration of the substance( mass per unit volume). Let 826_causality principle for the Klein- Gordon equation1.png be the concentration flux (mass per unit area per unit time, analogous to heat flux). Let B(x) be a ball of radius r > 0 contained in D. Derive the conservation law.

995_causality principle for the Klein- Gordon equation2.png

Fick's law for diffusion states  that the concentration flux due to diffusion is proportional to the gradient of the concentration flux due to diffusion is proportional to the gradient of the concentration. Deduce that 1363_causality principle for the Klein- Gordon equation3.png.

Apply the divergence theorem to the conservation law and substitute the flux formula to arrive at the diffusion- transport equation:

783_causality principle for the Klein- Gordon equation4.png

this is the higher- dimensional transport equation. If the fluid is motionless, it is called the diffusion equation.

Determine the causality principle for the Klein- Gordon equation in one dimension. Deduce that the speed of propagation is at most c.


Related Discussions:- Causality principle for the klein- gordon equation

Process of neodymium-yttrium-aluminium garnet laser, Explain the exact proc...

Explain the exact process of Neodymium-Yttrium-Aluminium-Garnet Laser. While the flash lamp is switched start, the intense beam of white light through the lamp excites the Nd3+

Cylindrical volume at a distance x, A long cylindrical volume haves a unifo...

A long cylindrical volume haves a uniformly distributed charge of density ρ and radius R. Determine the electric field at a point P inside the cylindrical volume at a distance x (x

Circular orbit at this height, One  scientist wants to put an 100kg experim...

One  scientist wants to put an 100kg experimental package in orbit around the Earth. The cost of deployment depends on the amount of more extra energy it takes to get it into the r

Estimate the amount of heat removed, A cooling drum consists of a (horizont...

A cooling drum consists of a (horizontally) rotating, large stainless steel cylinder. The cylinder has a diameter D and a width of W. The cooling at the inside of the drum is so ef

Atwoods machine, Atwood's machine A weight-and-pulley system is used t...

Atwood's machine A weight-and-pulley system is used to measure the acceleration because of gravity at Earth's surface through measuring the net acceleration of a set of weight

Calculate the temperature of phase transition, An element exists in two sol...

An element exists in two solid phases (labelled P1 and P2). The standard molar entropies of P1 and P2  are 21.9 J mol -1 K -1  and  23.3 J mol -1 K -1  respectively. The

State the law of conservation of electric charges, Q. State the law of cons...

Q. State the law of conservation of electric charges? The entirety charge in an isolated system always remains constant. For instance Uranium ( 92 U 238 ) can decay by emittin

Determine the q-factor of damped oscillator, Determine the Q-factor of damp...

Determine the Q-factor of damped oscillator with resonant frequency 500Hz and damping coefficient 0.5 per second.

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd