Poisson Equation

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  eBook - PDF

  Introduction to Stellar Dynamics

  • Luca Ciotti(Author)
  • 2021(Publication Date)
  • Cambridge University Press

    (Publisher)

  The formula becomes particularly simple in the case of systems for which the limit M(r 0 )/r 0 is finite for r 0 → ∞, so that the sum of the first two quantities on the right-hand side of Eq. (2.5) can be set equal to zero by considering r 0 = ∞ and φ(∞) = lim r 0 →∞ GM(r 0 )/r 0 . Finally, it can be useful to remind the student that Newton’s second theorem applies to the force, not to the potential (i.e., the potential at distance r from the center of a spherical mass distribution is not the potential of a point of mass M(r) placed at the origin unless ρ(r) is truncated and r is outside the mass distribution itself). 2.1.1 The Poisson Equation We can now combine Eqs. (1.9) and (2.1) and obtain one of the most important equations in physics, the Poisson Equation φ(x) = 4π Gρ (x), (2.6) where  ≡ div grad is the so-called Laplacian (Appendix A.4); incidentally, by con- sidering the special case of the point mass, from Eqs. (1.10) and (2.6) we recover the correspondent of Eq. (1.8); in other words, we can prove that in  3  x 1 x − y = −4πδ(x − y), (2.7) a fundamental identity in physics and mathematics. We will come back to this identity in Section 2.4. As we will see in the next section, from Helmholtz’s theorem the Poisson Equation, when supplemented with the appropriate boundary conditions, 2 encapsulates all of the properties of the gravitational field, and it can be considered as the field equation of classical gravity 2 It is elementary to realize the importance of boundary conditions: for a given pair (ρ,φ), the Poisson Equation remains unchanged if the harmonic function Ax, x + b, x + c (with A being a 3 × 3 matrix with TrA = 0, b being a constant vector, and c being a constant scalar) is added to φ, but g = −∇φ would be different! 18 The Gravitational Potential (and of electrostatics). In an empty space, Eq. (2.6) is called the Laplace equation, and its solutions are called harmonic functions.

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  Numerical Methods in Astrophysics

  An Introduction

  • Peter Bodenheimer, Peter Bodenheimer, Gregory P. Laughlin, Gregory P. Laughlin, Michal Rozyczka, Tomasz Plewa, Harold. W Yorke, Michal Rozyczka, Harold W. Yorke(Authors)
  • 2006(Publication Date)
  • CRC Press

    (Publisher)

  7 Poisson Equation The Poisson Equation is an example of an elliptic equation in which the solution is not explicitly time-dependent. A distribution of density ρ as a function of position in three-dimensional space is given, and it is desired to find the gravitational potential at each point, from which the gravitational force can be derived. Specification of the boundary conditions is a key element of the problem. In Cartesian coordinates, the equation can be written ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2 = 4 π G ρ ( x , y , z ) , (7.1) where G is the gravitational constant and where boundary values B are specified. In a cylindrical coordinate system ( R , Z ,φ ), where R is the distance to the axis of the cylinder, the equation is: 1 R ∂ ∂ R R ∂ ∂ R + 1 R 2 ∂ 2 ∂φ 2 + ∂ 2 ∂ Z 2 = 4 π G ρ. (7.2) In spherical polar coordinates ( r ,θ,φ ), where r is the distance to the origin, the equation is: ∂ 2 ∂ r 2 + 2 r ∂ ∂ r + 1 r 2 1 sin 2 θ ∂ 2 ∂φ 2 + 1 sin θ ∂ ∂θ sin θ ∂ ∂θ = 4 π G ρ. (7.3) The formal solution to this equation (in Cartesian coordinates) is: ( x , y , z ) = − τ G ρ ( x , y , z ) | x − x | dx dy dz (7.4) where x = ( x , y , z ) is the point at which the gravity is being calculated, and τ is a finite volume containing all the mass contributing to the potential (Margenau and Murphy, 1956, Section 7.17). Implicit in this solution is the boundary condition that vanish at infinity at least as fast as 1 / | x | . Thus, it is usual to set the potential equal to zero at infinity. With the standard definition that the gravitational force per unit mass g = − ∇ , it is easily shown that the potential at the surface of a sphere of radius R and mass M is = − GM R (7.5) representing the work done by the forces of gravity on a particle of unit mass as it moves, in an otherwise empty universe, from an infinite distance to the surface of the sphere. 223

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  Classical Electromagnetic Radiation

  • Jerry Marion(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  CHAPTER 3

  The Equations of Laplace and Poisson

  Publisher Summary

  The Laplacian operator occurs in many different types of physical problems probably the most important of which is that of wave propagation. This chapter discusses some of the mathematical functions that arise in the solution of wave equations that are the same as those that result from the solution of Laplace’s equation. It discusses some of the important properties of harmonic functions—that is, functions that satisfy Laplace’s equations. The chapter discusses the solutions of Laplace’s equation in rectangular coordinates, in spherical coordinates, and in cylindrical coordinates. It describes the parallel-plate diode that is a simple example of Poisson’s equation.

  3.1 Introduction

  In Chapter 1 we found that the general problem of the electrostatic field is described by Poisson’s equation, Eq. (1.9) :

  (3.1a)

  In regions not containing charge, this reduces to Laplace’s equation:

  (3.1b)

  The Laplacian operator occurs in many different types of physical problems, * probably the most important of which is that of wave propagation. Although we are interested in this book primarily in electromagnetic wave phenomena rather than in electrostatics, some of the mathematical functions which arise in the solution of wave equations are the same as those that result from the solution of Laplace’s equation. It is somewhat easier to introduce these harmonic functions (Legendre functions, spherical harmonics, and Bessel functions) in connection with electrostatic problems.

  We shall study such problems in some detail in order to become familiar with the functions that will be of use later in discussions of radiation phenomena. This will be the extent of the treatment of electrostatics; we will not discuss the method of images nor the use of conjugate functions in the solution of problems in electrostatics. The interested reader is referred to the list of Suggested References for sources of such material.

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  A Modern Course in Transport Phenomena

  • David C. Venerus, Hans Christian Öttinger(Authors)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  2 The Diffusion Equation One of the most famous equations in the field of transport phenomena is the diffusion equation. Its wide-ranging importance is underlined by the fact that, depending on the context, it is known by various names. In a proba-bilistic interpretation, it is usually referred to as a Fokker–Planck equation, which is a special type of Kolmogorov’s forward equation for memoryless stochastic processes. 1 In the context of Brownian motion, the name Smolu-chowski equation is most appropriate. The variety of names nicely indicates that this equation is not only useful for describing the transport phenomenon of mass diffusion; we will actually encounter it many times, in particular, also in the description of momentum and heat transport and in polymer kinetic theory. In the present chapter, we introduce it to describe the flow of probability. In doing so, we present the basic theme of transport phenomena, including some important concepts, tools, and results. 2.1 A Partial Differential Equation By a first glance at the Fokker–Planck or diffusion equation in one space dimension, ∂p ( t, x ) ∂t = − ∂ ∂x A ( t, x ) p ( t, x ) + 1 2 ∂ 2 ∂x 2 D ( t, x ) p ( t, x ) , (2.1) one recognizes a second-order partial differential equation for the evolution of some function p ( t, x ) of two real arguments involving coefficient functions A ( t, x ) and D ( t, x ). Here we simply assume D ( t, x ) ≥ 0, but later will show that this follows from the second law of thermodynamics (see Exercise 2.4 ). In many cases, the given coefficient functions A and D are independent of their first argument, t . The goal of this chapter is to bring the reader 1 See, for example, Chapter 3 of Gardiner, Handbook of Stochastic Methods (Springer, 1990). 8 The Diffusion Equation from this superficial perspective to a deep understanding of the physical meaning and implications of the diffusion equation ( 2.1 ).

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  Stochastic Processes for Physicists

  Understanding Noisy Systems

  • Kurt Jacobs(Author)
  • 2010(Publication Date)
  • Cambridge University Press

    (Publisher)

  (7.74) The stochastic equation is the same for the y - and z-coordinates as well. If we denote the vector of the position of the particle in three dimensions as x = (x, y, z), then we can write the joint probability density for the position of the parti- cle in these three dimensions as P (x, t ). The Fokker–Planck equation for P is then ∂P ∂t =  D 2   ∂ 2 P ∂x 2 + ∂ 2 P ∂y 2 + ∂ 2 P ∂z 2  =  D 2  ∇ 2 P. (7.75) 7.8 Chemistry: reaction–diffusion equations 117 The sum of the three second spatial derivatives is called the Laplacian. To make the notation more compact, one usually writes this as ∇ 2 ≡ ∂ 2 ∂x 2 + ∂ 2 ∂y 2 + ∂ 2 ∂z 2 , (7.76) so that the equation becomes ∂ ∂t P (x, t ) =  D 2  ∇ 2 P (x, t ). (7.77) Now imagine that we have many particles and that the particles are very small. If we divide the volume of liquid up into tiny cubes of size V , then we have a large number of particles in each cube. In this case, to good approximation, the density of the particles in each cube is simply the value of the probability density P (x, t ) for a single particle, multiplied by the total number of particles, N . The equation of motion for the density of particles is therefore simply the equation of motion for the single-particle probability density P (x, t ). So if we denote the density of particles by D(x, t ), the equation for D(x, t ) is ∂ ∂t D(x, t ) =  D 2  ∇ 2 D(x, t ). (7.78) This equation describes not only Brownian particles suspended in liquid, but also the density of molecules of a chemical that is dissolved in a liquid (or indeed the molecules of the liquid itself). In the context of density, Eq. (7.78) is referred to as the diffusion equation. (This equation also models the flow of heat, and because of this is also referred to as the heat equation.) The parameter D is called the diffusion coefficient, and determines how fast the particles diffuse through the liquid.

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