Relativistic brownian motion and related diffusion equation

Texto completo

(1)Relativistic Brownian Motion and Related Diffusion Equation. Santiago Vélez Ferro Departamento de Fı́sica Universidad de los Andes. A thesis submitted for the degree of Fı́sico December2010.

(2) 1. Reviewer:. 2. Reviewer:. Day of the defense:. Signature from head of the committee:. ii.

(3) Abstract. In this thesis the problem of relativistic diffusion and it’s consequences in relativistic Brownian motion are addressed. In order to do so, since the diffusion processes are modeled with second-order partial differential equations, the first thing we do is to exploit the characteristics of these equations. Within this, we outline the problem of classical diffusion in the frame of special relativity and the several attempts to solve this problem as eg. the one done by Cattaneo. In the next chapter further attempts to solving the relativistic diffusion model are exposed. Afterwards, Brownian motion and it’s relativistic, or Lorentz invariant form, are developed and analyzed with the help of the diffusion theory developed before..

(4) iv.

(5) To my parents....

(6) Acknowledgements. I would like to acknowledge my supervisor, prof. Marek Nowakowski, for all his support and sharing with me his expertise..

(7) Contents List of Figures. vii. 1 Introduction. 1. 2 Objectives. 3. 2.1. General Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 2.2. Specific Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 3 Basic Partial Differential Equations in Physics 3.1. 3.2. 5. Characteristics of Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. 3.1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. 3.1.2. Types of Second-Order PDEs . . . . . . . . . . . . . . . . .. 6. Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8. 3.2.1. Classical Derivation of the Wave Equation . . . . . . . .. 8. 3.2.2. Derivation of the Wave Equation from Electromagnetism 11. 3.2.3. Solution of the Wave Equation . . . . . . . . . . . . . . . 12. 3.2.4. Solution of the Inhomogeneous Wave Equation in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14. 3.2.5 3.3. Properties of the Wave Equation . . . . . . . . . . . . . . 15. Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . 15 3.3.1. Classical Derivation of the Diffusion Equation. . . . . . 15. 3.3.2. Derivation of the Diffusion Equation from Thermodynamics (3D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16. 3.3.3. Derivation of the Diffusion Equation from Random Walk 17. 3.3.4. Solution of the Diffusion Equation . . . . . . . . . . . . . 18. iii.

(8) CONTENTS. 3.3.5 3.4. Properties and Problems of the Diffusion Equation . . . 21. Telegraph Equation . . . . . . . . . . . . . . . . . . . . . . 22 3.4.1. Telegraph Equation Derivation from Thermodynamics. 3.4.2. Telegraph Equation Historical Derivation. 3.4.3. Telegraph Equation Derivation from Random Walk Mod-. . 22. . . . . . . . . 23. els . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25. 3.4.4. Other derivations . . . . . . . . . . . . . . . . . . . . . . . . 27. 3.4.5. Solution to the Telegraph Equation . . . . . . . . . . . . 28. 3.4.6. Problems of the Telegraph Equation . . . . . . . . . . . . 29. Bibliography. 31. 4 Attempts at a Relativistic Diffusion Equation. 33. 4.1. Lorentz Transformation . . . . . . . . . . . . . . . . . . . 33 4.2 Propagator Manipulation . . . . . . . . . . . . . . . . . . 35 4.3 Linearization of the Diffusion Equation . . . . . . 36 Bibliography. 41. 5 Non-Relativistic Brownian Motion. 43. 5.1. Langevin Equation . . . . . . . . . . . . . . . . . . . . . . . . 43 5.2 Derivation of the Langevin Equation from Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.3 Fokker-Planck Equation and Discretization Rules 46 5.4 Non-relativistic Ornstein-Uhlenbeck Process . . 49. Bibliography. 53. 6 Relativistic Brownian Motion. 55. 6.1. Relativistic Langevin Equation . . . . . . . . . . . . . 55 6.2 Relativistic Fokker-Plank Equation and Discretization rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.3 Relativistic Ohrstein Uhlenbeck Process . . . . . 58. Bibliography. 63. iv.

(9) CONTENTS. 7 Conclusions. 65. v.

(10) CONTENTS. vi.

(11) List of Figures 3.1 3.2. Plot of a Traveling Wave . . . . . . . . . . . . . . . . . . . . . . . . . . 15. 3.3. Electrical Sketch of the Telegraphers Circuit . . . . . . . . . . . . . . . . 24. Plot of a diffusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21. vii.

(12) LIST OF FIGURES. viii.

(13) 1 Introduction In this thesis the discrepancy between the classical diffusion equation and the special theory of relativity is examined. In order to do this, we will begin by reviewing the literature concerning partial differential equations, because the diffusion equation belongs to a special type of partial differential equation called parabolic. Being of such a type permits the diffusion process, eg. for temperature T, to spread throughout the space infinitely fast, which is physically not acceptable. Therefore, we would prefer the diffusion equation to be of a type called hyperbolic. This is another type of partial differential equation and to whom the wave equation belongs to. In doing so we arrive at the problem concerning Cattaneo’s solution, arriving at the telegraph equation, which is hyperbolic. Comparison between wave, diffusion and telegraph equation and a further explanation of why this last one is not a satisfactory solution to the problem posed is achieved. Secondly, as all basic equations of physics, the diffusion equation, to be found, must be relativistically covariant. After this, other attempts of finding the solution to the problem are reviewed and extended. The attempt to find the solution to the problem via Lorentz’s transformation of the diffusion equation is reviewed and extended to the telegraph equation. Also, the attempt of finding a relativistic propagator is discussed. Finally, the attempt of linearizing the diffusion equation is done in analogy to the Schrodinger equation, which is analytically the same as the diffusion equation. At the end, a review of the literature concerning the Brownian motion and its relativistic generalization is done and contrasted with the aim of the development of the diffusion equation developed before.. 1.

(14) 1. INTRODUCTION. 2.

(15) 2 Objectives 2.1. General Objective. To study the relativistic generalization of Brownian motion and its connection to generalized hyperbolic diffusion equation.. 2.2. Specific Objectives. • Formulation of the problem (in more detail). • Study of Cattaneo’s telegraph equation as a first suggestion for a hyperbolic diffusion equation. • Investigation of different suggestions of relativistic diffusion equation. • Comparison between the different approaches. • Study of relativistic Brownian motion.. 3.

(16) 2. OBJECTIVES. 4.

(17) 3 Basic Partial Differential Equations in Physics 3.1. Characteristics of Partial Differential Equations. Since the diffusion equation is a partial differential equation(PDE), the first step to understand the characteristic of this and other equations of this type in physics is to develop the theory of PDE’S.. 3.1.1. Introduction. To begin with, in order to have a PDE we need two or more independent variables relating with a dependent one. Once we have this, a PDE is an identity in which dependent, independent variables and the partial derivatives of the dependent variable with respect to the independent variable combine such that: F (u,. ∂u ∂u ∂ 2u ∂ 2u , , ..., 2 , 2 , ...) = 0 ∂x ∂t ∂x ∂t. (3.1). For example, an important PDE in physics, the wave equation, is: 2 ∂ 2u 2∂ u = c ∂ 2t ∂ 2x. As usual, in this thesis, the notation. 5. ∂u ∂ 2u = ux and 2 = utt is employed. ∂x ∂ t. (3.2).

(18) 3. BASIC PARTIAL DIFFERENTIAL EQUATIONS IN PHYSICS. Therefore, the wave equation is a PDE of two independent variables x, t of second order. The order of a PDE is the highest derivative you may find in it. This is why the wave equation is of second order, both utt and uxx are double derivatives with respect to the dependent variable. Notice that an equation containing a term, following the notation of the wave equation, given by uxt is at least of second order. Another important characteristic of PDEs is linearity. To understand this property, lets start by writing the PDE in homogeneous form. In this way the wave ∂2 ∂2 equation looks like utt −c2 ukk = 0. Now we write it as Lu = 0 with L = 2 −c2 2 . ∂t ∂x L is the PDE’s operator. We say that a PDE is linear if the operator associated with it satisfies: L(u + cv) = Lu + cLv. (3.3). for any dependent variables u,v and constant c. The last PDE we will introduce in this section is the diffusion equation, given by: ut = kuxx. (3.4). It’s important to notice that as well as the wave equation, the diffusion equation is second-order linear differential equation.. 3.1.2. Types of Second-Order PDEs. In this section, the different types of second-order differential equations are shown explicitly the way done by Strauss[6]. To do this, assume the most general form with only two independent variables: α11 uxx + α12 uxt + α22 utt + α1 ux + α2 ut + α0 u = 0. (3.5). The discriminant D, of these equations, is defined by: 2 D = α12 − 4α11 α22. (3.6). If D = 0, by means of a linear transformation it’s possible to transform (3.5) to: uxx + ux ..... = 0. 6. (3.7).

(19) 3.1 Characteristics of Partial Differential Equations. these equations are called parabolic. On the other hand, if D > 0 it’s possible, by means of the appropriate linear transformation, to reduce (3.5) to: uxx − utt ..... = 0. (3.8). these equations are called hyperbolic. Finally, if D < 0 it’s possible to transform (3.5) to: uxx + utt ..... = 0. (3.9). these equations are called elliptic. It’s easy to see now what type of second-order PDE the wave equation is, since ut = ux = 0, utt = −1 and uxx = c2 the discriminant yields D = (0)2 − 4(−1)(c2 ) = 4c2 . Hence, the wave equation is hyperbolic. In the other hand, for the diffusion equation utt = ux = 0, ut = −1 and uxx = k so (0)2 − 4(k)(0) = 0 so it’s parabolic. Now we consider the case in which more than two, say n, independent variables are taken into account. In such a case, the general form of the PDE takes the form: n X i,j=1. aij uxi xj +. n X. ai uxi + a0 u = 0. (3.10). i=1. Assumed that the coefficients aij , ai and a0 real. Since crossed derivatives are equal uxi xj = uxj xi so are the coefficients aij = aji . Therefore, consider a linear change of independent variables: (γ1 , ..., γn ) = ~γ = B~x. Clearly B is an n × n matrix. By means of the chain rule, proceed to change the partial derivatives we have before to obtain them in terms of the new ones: X ∂γk ∂ ∂ = ∂xi ∂xi ∂γk k Notice that. ∂γk ∂xi. (3.11). = bki . This means the entry (k, i) of the transformation matrix. B. With this, the double derivative is expressed as: X ∂ X ∂ u xi xj = ( bki )( blj )u ∂γk ∂γl k l. 7. (3.12).

(20) 3. BASIC PARTIAL DIFFERENTIAL EQUATIONS IN PHYSICS. Inserting this in the second order equation: X. aij uxi xj =. i,j. XX ( bki aij blj )uγi γj k,l. (3.13). i,j. Note that a second order equation with new variables ~γ which have a new coefficient matrix is obtained. This new matrix, the one in parenthesis, may be expressed as BAB t . At this point it’s necessary to recall a theorem from linear algebra that says that whenever there is a real symmetric matrix, in this case A, an orthogonal matrix with determinant 1 such that BAB t is a diagonal matrix, may be achieved. The diagonal entries of this matrix are the eigenvalues of A. Therefore, by a change of scale, this diagonal matrix may be converted into a matrix of +10 s, 00 s or −10 s. Having this final matrix, the PDE with whom we began with is: • parabolic if at least one of the entries is 0 • elliptic if all the eigenvalues are not zero and have the same sign • hyperbolic if they are nonvanishing and there is at least one eigenvalue with different sing from the rest.. 3.2. Wave Equation. Wave equation is one of the most important equations in physics and in this chapter the development of this equation is going to be done in an environment without boundaries. Although in most physical applications neglecting the boundaries is unreal, this is done because if we are located far away from the boundaries the effect of them may be neglected. On the other hand the properties we are looking for of PDE’s are easily obtained without boundaries.. 3.2.1. Classical Derivation of the Wave Equation. In order to derive the wave equation it is very important to establish clearly the assumptions made in order to develop the physical problem correctly. Throughout this section, the derivation found in Guenther and Lee [1] of small vibrations of an elastic string is going to be followed.. 8.

(21) 3.2 Wave Equation. The following things are assumed: 1. The string is thought as a continuum media that is perfectly elastic so that when pulled out from equilibrium the force or tension acts tangentially. 2. The vibrations are small enough for the displacement of the particles taken to be vertical. To begin with, lets analyze what happens with a string from the point of view of density. If we take a part of the string, say between x1 and x2 , we know, according with our second assumption that the mass between these points will be constant. So let’s assume that the density before the displacement is ρ0 and after it is ρ. Setting the vertical position of the string as a function of x and t, say u(x, t), due to the conservation of mass and the arc length formula we obtain: Z. x2. Z. x2. ρ0 (x)dx = x1. ρ(x, t)[1 + u2x (x, t)]1/2 dx. (3.14). x1. Since x1 and x2 are arbitrary and the integrands continuous: ρ0 = ρ[1 + u2x (x, t)]1/2. (3.15). Now, making use of the fact that there is no horizontal motion, if a infinitesimal piece of the string is taken, the horizontal component of the string must balance. Therefore, denoting Tx as the tension of the string in the position x and αx its angle formed with the horizon, we have: Tx+δx cos(αx+δx ) − Tx cos(αx ) = 0. (3.16). By simply dividing by ∆x, making ∆x → 0 and applying the definition of derivative we obtain: d (Tx cos(αx )) = 0 dx. (3.17). Tx cos(αx ) = τ. (3.18). So we get:. 9.

(22) 3. BASIC PARTIAL DIFFERENTIAL EQUATIONS IN PHYSICS. In order to analyze the vertical motion, the fact that time rate of change of linear momentum is equal to the sum of the forces acting in the vertical direction must be employed. Taking a small string element we know that the momentum is given, with the help of (3.15), by: Z. x+δx. Z. ρ[1 +. ρut dx =. u2x (x, t)]1/2 ut. Z. x+δx. dx =. x. ρ0 ut dx. (3.19). x. And the time rate of change will be given by: d dt. Z. x+δx. x+δx. Z ρ0 ut dx =. ρ0 utt dx. x. (3.20). x. Let’s start now by developing the forces that act on the string. The tension force acting vertically in the string will be worked with the help and in the same way as the horizontal component derivation: Tx+δx sin(αx+δx ) − Tx sin(αx ) = τ. sin(αx ) sin(αx+δx ) − cos(αx+δx ) cos(αx ). . = τ (tan(αx+δx ) − tan(αx )). = τ (ux (x + δx, t) − ux (x, t)). (3.21). Again dividing by δx and making δx → 0 we obtain: d (Tx sin(αx )) = τ uxx Tx sin(αx ) = dt. Z. x+δx. τ uxx dx. (3.22). x. After this we may obtain the term containing all external forces. If we have and external force with force density f (x, t) and we include gravity, the external force term would be given by: Z. x+δx. ρ0 (f − g) dx. (3.23). x. Finally, assuming linear friction, we obtain a friction force given by: Z. x+δx. F =−. ρ0 ut dx x. 10. (3.24).

(23) 3.2 Wave Equation. With all this terms and applying Newton‘s second law: Z. x+δx. Z. x+δx. x+δx. τ uxx dx −. ρ0 utt dx = x. Z. x. Z. x+δx. ρ0 (f − g) d. ρ0 ut dx + x. (3.25). x. This takes us to: ρ0 utt = τ uxx − kρ0 ut + ρ0 (f − g). (3.26). And finally we arrive, by setting c2 = τ /ρ0 and F = f − g to the damped one dimensional wave equation: utt = c2 uxx − kut + F. (3.27). If there is a negligible friction, negligible weight of the string and no external forces we obtain the typical 1 + 1dimensional wave equation: utt = c2 uxx. 3.2.2. (3.28). Derivation of the Wave Equation from Electromagnetism. In this section the derivation of electromagnetic waves in vacuum done by Griffiths[2] is followed. Let’s begin by stating two of Maxwell’s laws: ~ ~ = − ∂B ∇×E ∂t ~ ~ = µ0 0 ∂ E ∇×B ∂t Let’s take the curl on each side of these equations:. (3.29). (3.30). ~ ~ = ∇ × (− ∂ B ) ~ = ∇(∇ · E) ~ − ∇2 E ∇ × (∇ × E) ∂t =−. 2~ ∂ ~ = −µ0 0 ∂ E (∇ × B) ∂t ∂t2. ~ ~ = ∇(∇ · B) ~ − ∇2 B ~ = ∇ × (µ0 0 ∂ E ) ∇ × (∇ × B) ∂t. 11. (3.31).

(24) 3. BASIC PARTIAL DIFFERENTIAL EQUATIONS IN PHYSICS. 2~ ∂ ~ = −µ0 0 ∂ B (∇ × E) (3.32) ∂t ∂t2 So basically doing this we have decoupled the first two differential equations. ~ = ∇·E ~ = 0, we Now using the other two Maxwell’s equations in vacuum, ∇ · B. = µ0 0. finally obtain: ~ = µ0 0 ∇2 E. ~ ∂ 2E ∂t2. (3.33). ~ = µ0 0 ∇2 B. ~ ∂ 2B ∂t2. (3.34). ~ and B. ~ which are clearly 1 + 3-dimensional wave equations for E. 3.2.3. Solution of the Wave Equation. The objective in this section is to solve the differential equation, the solution done by Strauss[6] is developed: utt = cuxx ; −∞ < x < +∞. (3.35). This equation has a very special characteristic, it factors up nicely: ∂ ∂ ∂ ∂ − c )( + c )u = 0 (3.36) ∂t ∂x ∂t ∂x The easiest way to solve this equation is by means of a change of variables. Define utt − cuxx = (. ζ = x + ct and η = x − ct, by the chain rule we have ∂x = ∂ζ + ∂η and ∂t = c(∂ζ − ∂η ). With this ∂t − c∂x = −2c∂η and ∂t + c∂x = +2c∂ζ so the wave equation turns to be: −4c∂ζ ∂η u = 0 and the solution would be given by: u(ζ, η) = f (ζ) + g(η) = f (x − ct) + g(x + ct) = u(x, t). (3.37). Now let’s solve the same problem including the initial conditions: u(x, 0) = φ(x), ut (x, 0) = ψ(x). 12. (3.38).

(25) 3.2 Wave Equation. According to this we know that: φ(x) = g(x) + f (x). (3.39). Applying the chain rule to (3.37) in order to be able to differentiate with respect to t and then setting t = 0 to get: 1 ψ(x) = g(x) − f (x) c. (3.40). Differentiating (3.39) we get: φ0 (x) = g 0 (x) + f 0 (x). (3.41). Solving the last two equations for g 0 and f 0 we obtain: 1 ψ 1 ψ g 0 = (φ0 + ) and f 0 = (φ0 − ) 2 c 2 c. (3.42). After integrating its obtained that: x. 1 1 g(x) = φ(x) + 2 2c. Z. 1 1 f (x) = φ(x) − 2 2c. Z. ψ+A. (3.43). ψ+B. (3.44). 0. and x. 0. So the general solution to the wave equation obtained is: 1 1 u(x, t) = [ψ(x + ct) + ψ(x − ct)] + 2 2c. Z. x+ct. 0. 1 1 = [ψ(x + ct) + ψ(x − ct)] + 2 2c. 1 ψ− 2c Z. Z. x−ct. ψ. (3.45). 0. x+ct. ψ. (3.46). x−ct. In order to solve the wave equation in three dimensions it’s important to notice first that under this condition (3D) the wave equation is invariant under rotations.. 13.

(26) 3. BASIC PARTIAL DIFFERENTIAL EQUATIONS IN PHYSICS. So it may be expected that it depends only in the radial distance from a given point. There for the problem initially expressed as: utt − c2 ∆u = 0. (3.47). may be expressed as: 2 utt − c2 (urr + ur ) = 0 r This last equation may be written as:. (3.48). (ru)tt − c2 (ru)rr = 0. (3.49). So the quantity ru satisfies the original wave equation and so the solution is: 1 1 u(r, t) = F (r − ct) + G(r + ct) r r. 3.2.4. (3.50). Solution of the Inhomogeneous Wave Equation in Three Dimensions. In this section the solution of the wave equation in 1 + 3-dimensions as done in Jackson[7] is achieved. The aim of this section is to solve the problem: 1 ∂t u(~x, t) = −4πf (~x, t) (3.51) c2 with c being the velocity of propagation in the medium and f (~x, t) the source ∆u(~x, t) −. distribution. The Green’s function derivation is not exposed here but it yields: . 0. 0. G(±) (~x, t; ~x , t ) =. h δ t + t± 0. 0. |~ x−~ x| c. i (3.52). |~x − ~x0 |. Recall the solution of the wave equation was the sum of two solutions. We have here two Green’s functions, G+ and G− , that give each of this solutions: (±). u. ZZ (~x, t) =. 0. 0. 0. 0. G(±) (~x, t; ~x , t )f (~x, t)d3 x dt. (3.53). The Green function G+ is called retarded Green function because it exhibits the causal behavior given by the wave disturbance. It’s easily seen, from the delta. 14.

(27) 3.3 Diffusion Equation. wave.png. Figure 3.1: Plot of a Traveling Wave. function, that an effect observed at a point ~x at a time t is caused by the action 0. 0. of a source a distance R = |~x − ~x | away in an earlier time t = t − R/c. So R/c is simply the propagation time of a disturbance. With the same argument, G− is called advanced Green function.. 3.2.5. Properties of the Wave Equation. In order to understand the properties of the wave equation let’s have a look first at Figure 1 which shows the graph of its solution and its evolution in time: There are two basic properties that waves have that are intrinsic of hyperbolic PDE’s that must be outlined here. First we may see it travels at a constant speed. With the classical wave equation that has been used here, this speed is c. The second property that is very important is the fact that we may see the information isn’t lost while time evolves, it simply travels.. 3.3 3.3.1. Diffusion Equation Classical Derivation of the Diffusion Equation. To begin this section, a first light derivation of the diffusion equation, developed by Strauss[6] is done. The starting point of this derivation is Fick’s law. It says substances move from places of high concentration to lower concentration ones. Say for example we are willing to describe the process in which some substance spreads throughout a tube filled with water. Consider a region of the tube in which x0 is the starting point and x1 is the end point. Say u(x, t) is the concentration of the substance in the liquid in a point x, between x0 and x1 , in a time t. Therefore, the. 15.

(28) 3. BASIC PARTIAL DIFFERENTIAL EQUATIONS IN PHYSICS. total mass of the substance within the considered region is going to be: x1. Z. u(x, t)dx. M (t) =. (3.54). x0. therefore, differentiating with respect to t we obtain: ∂M (t) = ∂t. x1. Z. ut (x, t)dx. (3.55). x0. Now we use Fick’s law to obtain: ∂M (t) = k(ux (x0 , t) − ux (x1 , t)) (3.56) ∂t where k is a proportionally constant. This last equation means simply that the change in the mass of the substance, in the considered region of the pipe, equals the amount of substance flowing in minus the amount of substance flowing out. By simply making equal equations (3.55) and (3.56) and differentiating with respect x1 in both sides the diffusion equation, in 1 + 1 dimensions is obtained: ut = kuxx. 3.3.2. (3.57). Derivation of the Diffusion Equation from Thermodynamics (3D). In this section the derivation of the heat equation is developed via the problem of heat flow[2]. Let’s define c as the heat capacity of the media, ρ the media’s density, u(x, y, z, t) the temperature. With this we may define the amount of heat H(t) in a region D of the material as: ZZZ H(t) =. cρudxdydz. (3.58). D. By introducing Fourier’s law, which says the heat may only be lost from the region D by leaving through the boundary. Therefore, using the conservation of energy, it may be stated that the change of heat in the region D is equal to the heat flux through the boundaries yielding: ∂H(t) = ∂t. ZZ κ(n · ∇u)dxdydz. surf aceD. 16. (3.59).

(29) 3.3 Diffusion Equation. In the last expression κ is a proportionality constant. Applying the divergence theorem the last equation is transformed to: ZZ. ZZZ ∇ · (κ∇u)dxdydz. κ(n · ∇u)dxdydz =. (3.60). D. surf aceD. In the other hand, deriving H(t) by t on its definition at the begining the change of heat is: ∂H(t) = ∂t. ZZZ cρut dxdydz. (3.61). D. So comparing the integrands of the last two expressions: cρut = ∇ · (κ∇u). (3.62). And due to the fact that κ is also a constant we may rewrite this as: κ 2 ∇u cρ This equation is a diffusion equation in 1 + 3 dimensions. ut =. 3.3.3. (3.63). Derivation of the Diffusion Equation from Random Walk. The derivation done by Gardiner[8] of a one dimensional random walk with fixed time steps τ of magnitude δ, is developed next. To begin with, suppose that at each step the random walker has a probability: • r of moving right • l of moving left • 1 − r − l of staying is the same place naturally r + l < 1. If there is a constant number of random walkers the probability density function and a position x at a time t is given by:. p(x, t) = p(x, t − τ )[1 − r − l] + p(x − δ, t − τ )r + p(x + δ, t − τ )l. 17. (3.64).

(30) 3. BASIC PARTIAL DIFFERENTIAL EQUATIONS IN PHYSICS. Expanding as a Taylor series up to O(x2 ) and O(t):. p(x, t) =. dp p−τ dt. . 2 2 dp dp dp dp 2d p 2d p [1−r−l]+ p − τ −δ −δ r+ p − τ +δ +δ l dt dx dx2 dt dx dx2 (3.65). Rearranging: δ(l − r) dp δ 2 (l + r) d2 p dp = + dt τ dx τ dx2. (3.66). Defining: D=. δ 2 (l + r) δ,(l+r),τ →0 τ. (3.67). u=. δ(r − l) δ,(r−l),τ →0 τ. (3.68). dp d2 p dp = −u + D 2 dt dx dx. (3.69). lim. And lim. The final result is:. This is the diffusion equation with a drift, the parameter u will make the diffusion tend to the right or to the left. The first thing to be noticed is that if r = l then we get the classical diffusion equation (u = 0). The other thing to be noticed is we have arrived again to a parabolic PDE. The solution of this equation may be achieved the same way as in the wave with a change of variable γ = x + ut. In the next section the solution of the diffusion equation is achieved.. 3.3.4. Solution of the Diffusion Equation. During this section the solution to the diffusion equation developed by Strauss[6] is going to be followed. In it, the derivation is based on the conditions expected by the solution of the diffusion equation to satisfy. This is a very different approach from classical separation of variables one, which is mathematically more rigorous,. 18.

(31) 3.3 Diffusion Equation. but we selected the other approach because it exploits the physical properties of the diffusion equation nicely. The aim here is to solve the following problem: ut = kuxx with − ∞ < x < ∞, 0 < t < ∞. (3.70). with initial condition: u(x, 0) = φ(x). (3.71). In order to achieve this it’s necessary to exploit the following properties of the solution of the diffusion equation, assuming that u(x, t) is a solution: • The translate of the solution u(x − y, t) is again a solution • The derivative of a solution is also a solution • If we have another solution, say g(x, t), a linear combination of u and g is also a solution • The integral of a solutions is a solution. • The dilated function u(ax, a2 t) is also a solution. To check this simply 2. 2. a u(ax, a t) and. ∂ 2 u(ax,a2 t) ∂t. 2. 2. ∂ 2 u(ax,a2 t) ∂x. 2. = a u(ax, a t) so u(ax, a t) is clearly a solution.. Now let’s begin to seek our solution function say q(x, t). Due to conditions we have up to now we may say write a special initial condition in which: q(x, 0) = 1 for x > 0, q(x, 0) = 0 for x < 0. (3.72). Due to the property (5) we may make the ansatz that q(x, t) has the special form: x q(x, t) = g(p) for p = √ 4kt. (3.73). The factor 4k is introduced because it makes the solution easier later. We may √ check that if we multiply x by a and t by a then p is unchanged. With this we. 19. =.

(32) 3. BASIC PARTIAL DIFFERENTIAL EQUATIONS IN PHYSICS. may now change, via chain differentiation, our partial differential equation in to an ordinary one: ∂q 1 = − pg 0 ∂t 2t ∂q 1 = √ q0 ∂x 4kt 2 ∂ q 1 00 g = 2 ∂x 4kt So we obtain the following ordinary differential equation: 1 1 1 0 = qt − kqxx = − [ pg 0 + g 00 ] t 2 4 We finally arrive to the system:. (3.74). (3.75). g 00 + 2pg 0 = 0. (3.76) R This ordinary differential equation is easily solved by an integration factor exp( 2pdp) = exp(p2 ). From this we get that g 0 (p) = c1 exp(−p2 ) so we get that: Z. 2. c1 e−p dp + c2. q(x, t) = g(p) =. (3.77). Now, via the special initial condition, we may find the explicit formula for q because: √. +∞. Z if x > 0 → limx→0+ q = c1. −p2. e. dp + c2 = c1. 0. Z. (3.78). √. −∞. if x < 0 → limx→0− q = c1. π + c2 = 1 2. −p2. e. dp + c2 = −c1. 0. π + c2 = 0 2. (3.79). √ Solving for the coefficients we obtain c1 = 1/ π and c2 = 1/2. At this point, because of property (4), we know that the integral of q(x, t) is also going to be a solution so let’s define S =. ∂q ∂x. and exploiting the announced property it’s obtained. that: Z. +∞. S(x − y, t)φ(y)dy. u(x, t) = −∞. 20. (3.80).

(33) 3.3 Diffusion Equation. diffusionc.png Figure 3.2: Plot of a diffusion. Clearly this is a solution if t > 0. A careful reader should notice that with this, S is the Green’s function of the one-dimensional diffusion operator. Now lets have a look to the solution for the three dimensional problem. In this case we want to solve:. ut −. 3 X. kuxi xi (~x, t) = 0 with − ∞ < xi < ∞, 0 < t < ∞. (3.81). i=1. with initial condition: u(~x, t) = δ(~x). (3.82). S(~x, t) = (4kt)−3/2 exp(−~x · ~x/4kt). (3.83). Hence the Green’s function:. With the solution for three dimensions is: Z u(~x, t) =. 3.3.5. S(~x − ~y , t)g(~y )d~y. (3.84). Properties and Problems of the Diffusion Equation. In order to understand the properties of the diffusion equation let’s proceed in the same way done with the wave equation, having a look first at figure 2 in which a graph that shows diffusions equation solution and its evolution in time some properties may be pointed out. The first property of the diffusion equation that may be seen from the graph is the gradual lost of information as time goes on. It’s worth to note that there’s nothing wrong with this property and we might expect that further modifications of the diffusion equation lead to solutions containing this property.. 21.

(34) 3. BASIC PARTIAL DIFFERENTIAL EQUATIONS IN PHYSICS. The other property, the one with whom the conflict begins, is that it isevident that the solution to the diffusion equation with a delta function as initial condition is a Gaussian. Therefore, the solution u(x, t|x0 )dx gives the probability to find something diffused to the interval [x, x + dx] in a time t > 0.Due to the functional form of the propagator it’s known that for each t > 0, there is a small but nevertheless non-vanishing probability that [x, x + dx] > ct. Having shown this, the demonstration of why diffusion equation violates Einstein’s relativity principle is on the table. The first attempt of solving this problem is the Telegraph equation and it’s going to be exposed in the next section.. 3.4. Telegraph Equation. During this section, the telegraph equation(TE) is explored deeply. Historical motivations for the development of this equation are exposed. Afterward, the solution to it via Fourier-Laplace transform will be shown and in the end of the chapter, new problems that arise from it will be outlined.. 3.4.1. Telegraph Equation Derivation from Thermodynamics. The Telegraph equation was the first proposed solution of the infinitely fast propagation of the diffusion equation. It was developed by Cattaneo but here the development followed by Muller[3] is outlined here. Energy conservation and Fourier’s law are the basis for the development of this i ) and ρ being the theory. If we have temperature T , heat(energy) qi , T = ( ∂q ∂T g. density, the energy conservation equation of the system is: ρT Ṫ +. ∂qi =0 ∂xi. (3.85). If on the other hand we have κ being the heat conductivity. The stationary Fourier law is: qi = −κ. ∂T ∂xi. 22. (3.86).

(35) 3.4 Telegraph Equation. Inserting the Fourier law in the energy conservation outlined first we get:. ρT Ṫ =. ∂T ∂(κ ∂x ) i. ∂xi. = κ∆T. (3.87). We have arrived then to the classical heat diffusion equation. As we outlined last chapter, this equation has the problems of relativistic limits violation of all the classic diffusion process. This is why, in 1948 Cattaneo[4], in order to fix the problem, made a slight change to the stationary Fourier law. He argued that due to a time lag, generated by the fact that if the temperature changes in time the heat flux in a certain point depends on the temperature gradient in an earlier time, the Fourier law changes. Therefore Cattaneo stated the instationary Fourier law: qi + τ q˙i = −κ. ∂T ∂xi. (3.88). and again inserting this into the energy balance equation we finally get the Cattaneo equation: τ T̈ + Ṫ = D∆T. (3.89). We have reached a hyperbolic differential equation, if τ > 0, the solution (as will become evident later when the solution is achieved) has two diffusion fronts with traveling speed: V =±. p D/τ. (3.90). So at least partially the infinitely fast diffusion is solved. In the next sections, alternative models for deriving the Telegraph equation are exposed.. 3.4.2. Telegraph Equation Historical Derivation. Here the informal development of TE done by Guenther and Lee[1] is followed. It is said to be informal because the average value of variables is employed rather than the integrals over the space. In this case our objective is to understand the flow of electricity in a transmission line. Therefore the goal here is to determine the voltage v(x, t) and the current i(x, t) in the inner wire of a short piece of it, say beginning at. 23.

(36) 3. BASIC PARTIAL DIFFERENTIAL EQUATIONS IN PHYSICS. cable1.png. Figure 3.3: Electrical Sketch of the Telegraphers Circuit. x and ending at x + ∆x . At this moment it is worth to say the following electrical properties of the wire will be employed: • R, Resistance per unit length • L, Inductance per unit length • G, Conductance per unit length • C, Capacitance per unit length In fig. [3] a electrical sketch of the wire is done. It may be seen from the figure that using Kirchhoff’s loop law, the sum of potential drops, we get: v(x, t) − (R∆x)ĩ − v(x + ∆x, t) − (L∆x)i˜t = 0. (3.91). where ĩ is the average value of the current which is nearly constant from x to x + ∆x. In the other hand, Kirchhoff’s current law in the upper left point of the diagram gives us this equation: i(x, t) − i(x + ∆x, t) − (C∆x)v˜t − (G∆x)ṽ = 0. (3.92). again, ṽ is the average value of the voltage which is nearly constant from x to x + ∆x. After dividing by ∆x both equations and taking the limit ∆x → 0 we arrive to the telegraphers system: ix = Cvt + Gv. 24. (3.93).

(37) 3.4 Telegraph Equation. vx = Lit + Ri. (3.94). Differentiating the first equation with respect to x, the second one with respect to t and multiplying it by C, it’s possible, by means of a substitution, to insert the second equation in the first one to obtain: ixx = CLitt + CRit + Gvx. (3.95). If we replace vx by the form found for it in (3.94) we finally obtain: ixx = CLitt + (CR + GL)i(t) + RGi. (3.96). If either R or G are neglected, the generalized TE is obtained: ixx = CLitt + (CR + GL)it. (3.97). It’s a remarkable fact that if G and L are negligible we obtain a diffusion equation: ixx = CRit. (3.98). and if G and R are negligible we obtain a wave equation: ixx = LCitt. (3.99). So this example shows how simply the diffusion, wave and telegraph equation are connected.. 3.4.3. Telegraph Equation Derivation from Random Walk Models. Now, a derivation of Goldstein[4] developed by Masoliver and Weiss[5] of the TE from a random walk model is going to be followed. To begin with let’s suppose we are in a lattice in which the random walker has a certain probability α of going to the next step in the same direction he is going. This means if he is right now in the position j, with a probability α we are going to find the particle in the position j + 1 in the next step. Let’s now define an (j) as the probability of the random walker being at j, in step n and coming from j − 1. In the other hand, bn (j) is defined as. 25.

(38) 3. BASIC PARTIAL DIFFERENTIAL EQUATIONS IN PHYSICS. the probability that the random walker is at j at step n and coming from j + 1. At last, let’s define β = 1 − α. So we may establish Goldstein’s recurrence relations: an+1 (j) = αan (j − 1) + βbn (j − 1). (3.100). bn+1 (j) = αbn (j + 1) + βan (j + 1). (3.101). We face now Goldstein’s key assumptions. The first thing he realized is that α and β have a certain relation with momentum, both show the tendency to keep moving in the same direction. In the other hand, he realized that only two initial conditions are needed in order to get any an (j),bn (j) and these could be called say an (0) and bn (0). In order to change the recurrence relations stated above for partial differential equations only a simple scaling is needed. Let’s now define x = j∆x and t = n∆t. The two scaling relations needed are simply: ∆t ∆x = vα = 1 − ∆x,∆t→0 ∆t 2T with T being a parameter with time dimensions. lim. (3.102). Thanks to the scaling relations, the Goldstein’s recurrence relations may be stated as a system of partial differential equations given by: ∂a 1 ∂b ∂b 1 ∂a = −v + (b − a) = v + (a − b) (3.103) ∂t ∂x 2T ∂t ∂x 2T To solve this system start by solving for b in the first equation to obtain: ∂a ∂a +v ]+a ∂t ∂x Inserting this b in the second equation we obtain: b = 2T [. 2T [. ∂ 2a ∂ 2a ∂a ∂ 2a ∂ 2a ∂a ∂a ∂a + v ] + = 2T v[ + v ]+v − −v 2 2 ∂t ∂x∂t ∂t ∂t∂x ∂x ∂x ∂t ∂x. (3.104). (3.105). By simply arranging we finally obtain the TE from the random walk model: 2 ∂ 2a 1 ∂a 2∂ a + = v ∂t2 T ∂t ∂x2. 26. (3.106).

(39) 3.4 Telegraph Equation. The difference of this process and the one that yield the diffusion one is here the ”walker’s” are constantly moving and change their direction rather than change their position (as happened with the other process). This is why the need of keeping track of the position and direction of movement while in the other process just position was taken into account.. 3.4.4. Other derivations. Up to now, three derivations of the TE have now been accomplished: thermodynamics, transmission line and random walk model. Here two other derivations are going to be made. The first one follows a similar procedure as in the derivation of the diffusion equation in chap. 1, by simply changing Fick’s law, we will arrive to TE, and the second one by Maxell’s electrodynamics law in homogeneous space we will also arrive to TE for the electric field. This two derivations follow the ones done by Masoliver and Weiss[5]. Let’s begin with the derivation of the TE that is similar to one of the diffusion equation of chapter 1. In that section, we simply used the conservation of mass law ∂M/∂t + ∂J/∂x = 0 in which M is the mass and J is the diffusion flux given by Fick’s law J = D∂M/∂x. Now, if instead of using this form of Fick’s law we use its non-local generalization that may be written as: ∂J ∂M J = −v 2 − ∂t ∂x T. (3.107). Solving in this equation for J and differentiating with respect to x and inserting into the conservation law we arrive again to the TE: 2 ∂ 2M 1 ∂M 2∂ M + = v ∂t2 T ∂t ∂x2. (3.108). A last derivation is easily seen in electrodynamics. Take Maxwell’s equations in an homogeneous media:. µ. ~ ~ ∂H ~ σE ~ + ε ∂ E = ∇ × H∇ ~ ·E ~ = ρ/ε, ∇ · H ~ =0 = −∇ × E, ∂t ∂t. 27. (3.109).

(40) 3. BASIC PARTIAL DIFFERENTIAL EQUATIONS IN PHYSICS. The second equation may be written as: ~ σ ~ ∂E 1 ~ = ∇×H E+ ε ∂t ε. (3.110). and after being differentiated with respect to time we get: ~ ~ ~ ~ ∂ 2E 1 ∂H σ ∂E 1 ∂∇ × H + 2 = = ∇× ε ∂t ∂t ε ∂t ε ∂t. (3.111). So by inserting the first of Maxwell’s equation in the last one obtained we get: ~ ~ σ ∂E ∂ 2E 1 2~ + 2 = ∇E ε ∂t ∂t εµ. (3.112). ~ And this is again a TE for E.. 3.4.5. Solution to the Telegraph Equation. In this section the TE,. ∂2p ∂t2. +. 1 ∂p T ∂t. 2. ∂ p = v 2 ∂x 2 , solution is found by means of double. Fourier-Laplace transform[7]. It’s worth to notice that since the TE is a second order differential equation we need two boundary conditions in order to obtain a unique solution. The first condition that is going to be employed here is a pulse at an arbitrary point x0 at time t = 0. Therefore: p(x, 0) = δ(x − x0 ). (3.113). The general solution to the TE with this condition is going to be denoted as p(x, t|x0 ). This way the solution found for the TE this way will be the propagator or Green’s function. The second condition that will be employed here because it simplifies the solution is: ∂p(x, 0|x0 ) |t0 = 0 ∂t. (3.114). The left hand side of the TE under a Laplace transform is: s2 P − sδ(x − x0 ) +. s 1 P − δ(x − x0 ) T T. 28. (3.115).

(41) 3.4 Telegraph Equation. and after the Fourier transform is employed we have: 1 s )P̂ − (s + )eiwx0 (3.116) T T It’s easily seen that the rigt hand side of TE under the double transform is: (s2 +. v 2 w2 P̂. (3.117). s + T1 exp(iwx0 ) P̂ = 2 s s + T + v 2 w2. (3.118). Solving for P̂ the result is:. To get the solution to the TE under the initial conditions stated above we first do the inverse Fourier transform and obtain: P =. 1 s + 1/T |x − x0 | p 2 [p s + s/T ) ]exp(− 2v v s2 + s/T. (3.119). Finally, the inverse Laplace transform is applied to obtain: 1 e−t/(2T ) p(x, t|x0 ) = e−t/(2T ) (δ(x−x0 −vt)+δ(x−x0 −vt))+ [I0 (ξ)+I1 (ξ)]H(vt−|x−x0 |) 2 8vT (3.120) Here H is the heaviside function (H(x) = 1 if x > 0, H(x) = 0 if x < 0), In (ξ) is the nth Bessel function and p v 2 t2 − (x − xo )2 ξ= 2vt . In the next section the characteristics and problems of the TE are outlined.. 3.4.6. Problems of the Telegraph Equation. As it may be seen from the form of the solution of the TE derived in last section, the TE equation has two δ−peaked diffusion fronts, one heading in the positive direction and the other one in the opposite. This is a severe challenge because this could rarely be the solution for the diffusion of massive particles. This because it would mean that a finite fraction of the particles carries a great amount of the kinetic energy.. 29.

(42) 3. BASIC PARTIAL DIFFERENTIAL EQUATIONS IN PHYSICS. This opens us the door for further exploration within relativistic diffusion and it’s why in the next chapter other solutions are explored.. 30.

(43) Bibliography [1] Guenther R. B. and Lee John Partial differential equations of mathematical physics and integral equations Prentice-Hall, c1988 chap. 1 [2] Griffiths, D. J. Introduction to electrodynamics Prentice Hall, c1999. 3rd edition. Chap. 8 [3] Muller I. and Ruggeri T. 1993 Extended Thermodynamics Springer-Verlag Chap.1 [4] Goldstein S. On diffusion by discontinuous movements, and on the telegraph equation, Q. J. Mech. Appl. Math. 4, 129 (1951). [5] Masoliver J. and Weiss G. H. Finite Velocity Diffusion Eur. J. Phys. 17 (1996) 190â196 [6] Strauss, Walter A. Partial differential equations : an introduction New York : John Wiley Sons, c1992 chap 1-3 [7] Jackson, John D. Classical Electrodynamics New York : John Wiley Sons, c1999, 3rd edition sec. 6.6 [8] Gardiner, C.W. 1983 Handbook of Stochastic Methods New York : SpringerVerlag sec. 8.2. 31.

(44) BIBLIOGRAPHY. 32.

(45) 4 Attempts at a Relativistic Diffusion Equation 4.1. Lorentz Transformation. In this section the derivation of the diffusion equation in a general Lorentz frame done by Kostat an Liu[1], in 1 + 1-dimension, is followed. In this derivation, the diffusion equation’s variables x, t are changed by the Lorentz invariant ones t = γ t̃ − γvx̃ and x = γ x̃ − γ cv2 t̃. With this, naturally the derivatives with respect to x, t are given by: ∂x = γ∂x̃ − γ. v ∂ c2 t̃. ∂t = γ∂t̃ − γv∂x̃. (4.1). (4.2). Inserting this in the diffusion equation we obtain: v2 2 v ∂ ∂ + ∂ )ϑ = 0 (4.3) x̃ t̃ c2 c4 t̃ If we take a deep look at this equation we realize it is still of parabolic type γ(∂t̃ − v∂x̃ )ϑ − αγ 2 (∂x̃2 − 2. because (2 cv2 )2 −4(1)(α2 γ 4 ) = 0. This is the expected result because the discriminant is invariant under linear transformations such as the Lorentz one. Now let’s see if it’s possible to make this last result at least covariant. If we establish ∆µν = η µν +. uµ uν , c2. uµ = (−γv, γc), ∂µ = (∂x̃ , ∂ct̃ ) and η µν = diag(−1, 1). 33.

(46) 4. ATTEMPTS AT A RELATIVISTIC DIFFUSION EQUATION. we obtain the Lorentz invariant, but still hyperbolic, form of the diffusion equation given by:. uµ ∂µ ϑ − α∆µν ∂µ ∂ν ϑ = 0. (4.4). The question now is to see what happens if this same procedure is done with the telegraph equation. Using the telegraph equation: τ ∂t2 + ∂t = D∂x2 and repeating the same Lorentz transformations done before:. τ γ 2 (∂t̃2 − 2v∂x̃ ∂t̃ + v 2 ∂x̃2 )ϑ − γ(∂t̃ − v∂x̃ )ϑ = Dγ 2 (∂x̃2 − 2. v v2 2 ∂ ∂ + ∂ )ϑ x̃ t̃ c2 c4 t̃. (4.5). It’s 4to realize that this equation has a discriminant D= τ D + 2 important 2 2 τ D vc2 − 2τ D vc4 . Making some rearrangements we obtain: D = τ D 1 − vc2 . So obviously D > 0, because τ, D > 0, hence the equation is hyperbolic. Therefore the first step is on the table. Now, as in the last case, we seek for the covariant form of this equation that may be achieved by stating Λµν = −Dη µν + (τ − D)c−2 uµ uν and we obtain the final resultlt:. Λµν ∂µ ∂ν ϑ − uµ ∂µ ϑ = 0. (4.6). It’s evident that both resulting equations are very similar, this reflects the fact that one of them may be parabolic and the other one is in fact parabolic. A similar approach done by Kazinski [5] gave as result:. uµ ∂µ ϑ − αη µν ∂µ ∂ν ϑ = 0. (4.7). So comparing the two approaches, Kazinski and ours, we realize the equations are similar except because for an extra term (τ − D)c−2 uµ uν ∂µ ∂ν ϑ. But this isn’t so challenging because by means of linear transformations in the variables, as shown in the first chapter, we should be able to reproduce Kazinski’s result.. 34.

(47) 4.2 Propagator Manipulation. 4.2. Propagator Manipulation. In this section we are going to explain the solution to the relativistic diffusion propagator done by Dunkel et.al.[2], in 1 + 1-dimension (easily extended to 1 + 3dimensions). The method followed is to manipulate the propagator of the classic diffusion to obtain from it a relativistic one. The way to accomplish it is by means of obtaining the propagator from an action approach rather than from a position approach. Let’s begin by doing the approach for the non-relativistic case in order to proceed in a similar way in the relativistic one. Recall that the propagator of the diffusion equation, with Λ being the normalization constant, is: (x − xx )2 ) 4D(t − t0 ) We know that the action per mass is given by:. (4.8). p(x̄|x̄0 ) = Λ exp (−. Z a= Now we define the velocity v =. dt0 v(t0 )2. x−x0 . t−t0. (4.9). Inserting this into the definition of action. we get: (x − x0 )2 (4.10) 2(t − t0 ) To continue, it’s necessary that we know the limits of the action in the nona=. relativistic case. Since supraluminal velocities are allowed we may establish that the maximum limit of the action is a+ = +∞. In the other hand, the minimum action is achieved when the particle encounters no collisions at all in which we may say v=. x−x0 , t−t0. so applying the definition of the action we get that: a− =. (x−x0 )2 . 2(t−t0 ). The key part of the demonstration is to show that if we establish the propagator to be: Z. a+. p(x̄|x̄0 ) = Λ. da exp (− a−. a ) 4D. (4.11). we obtain the same propagator as for the classical diffusion equation. With this, now let’s change the action for the relativistic for of it given by: Z a=. dt0. p 1 − v(t0 )2. 35. (4.12).

(48) 4. ATTEMPTS AT A RELATIVISTIC DIFFUSION EQUATION. Taking the same generalizations as before we establish the upper limit the one in which the particle travels with speed c. In such a case for the upper limit we have a+ = 0.In the other hand, for the lower limit of the action we have that: a− = −. p (t − t0 )2 − (x − x0 )2. (4.13). Applying the same formula for the propagator applied within the action approach developed before we get that the relativistic propagator is given by: p(x̄|x̄0 ) = Λ(exp (−. a− ) − 1) 2D. (4.14). This has been a great approach of the problem because it achieves a propagator for relativistic diffusion process. This because the propagator satisfies all the conditions that a priori we were seeking it to satisfy. Even though, it’s quite disturbing the fact that the equation having this propagator hasn’t yet been found.. 4.3. Linearization of the Diffusion Equation. In this section we perform the linearization of the diffusion equation similar to what Levy-Leblanc[3] has done for the Schrodinger equation. This may be done because the Schrodinger equation and the diffusion are very similar, they differ only in the constants. Indeed, Schrodinger equation and diffusion equation are connected to one another by analytic continuation[6]. The diffusion operator may be written, having d being the diffusion constant, as: Ď ≡. i d ∂ − d∆ = − Ê + 2 p̂~2 ∂t ~ ~. (4.15). This may be done like this because: i ∂ ~ ∂t. (4.16). p̂~2 = −~2 ∆. (4.17). Ê = − and. 36.

(49) 4.3 Linearization of the Diffusion Equation ~ What we want now to do is to find operators Â, B̂ and Ĉ such that the equation now is linear in Ê and p̂~ in order to get: ~ Θ̂ψ = (ÂÊ + B̂ · p̂~ + Ĉ)ψ = 0. (4.18). Therefore the operators Ď and Θ̂ must be simultaneously valid. If this happens then another operator: ~ Θ̂0 ψ = (Â0 Ê + B̂ 0 · p̂~ + Ĉ 0 )ψ. (4.19). must exist that satisfies again the diffusion equation: ~2 Θ̂ Θ̂ψ = Ďψ d 0. with. ~2 d. (4.20). being introduced by convenience.. Therefore, 0. (Â Ê +. 3 X i=1. Bi0 pi. 0. + Ĉ )(ÂÊ +. 3 X i=1. 3 X ~ Bi pi + Ĉ) = −i Ê + pk d k=1. (4.21). With this, the following conditions are obtained: Â0 Bi + Bi0 Â = 0 Â0 Â = 0 ~ Â0 Ĉ + Ĉ 0 Â = −i d Bi0 Bj + Bj0 Bi = −2δij Ĉ 0 Ĉ = 0 Ĉ 0 Bi + Bi0 Ĉ = 0 To simplify these conditions the following operators are introduced: B4 = i(Â + ~d Ĉ) B40 = i(Â0 + ~d Ĉ 0 ) B5 = Â + ~d Ĉ B50 = Â0 + ~d Ĉ 0 With all this, the conditions above may be written as[4]: Bµ0 Bν + Bν0 Bµ = −2δµν with µ, ν = 1, 2, 3, 4, 5. (4.22). This condition is satisfied by the following matrices: Bi = M γi with i = 1, 2, 3, 4. 37. (4.23).

(50) 4. ATTEMPTS AT A RELATIVISTIC DIFFUSION EQUATION. Bi0 = −γi M̂ −1 with i = 1, 2, 3, 4. (4.24). B50 = −iM̂ −1 , B5 = −iM̂. (4.25). with γi being the Dirac representation matrices given by: . 0 σ̂i γi = σ̂i 0. with i = 1, 2, 3. (4.26). and . 1̂ 0 γ4 = 0 −1̂. (4.27). being with σ̂i being the Pauli matrices and 1̂ the identity 2 × 2 matrix. Finally we have M being the matrix:  0 0 M̂ =  1 0. 0 0 0 1. 1 0 0 0.  0 1  0 0. (4.28). At this moment the only thing left to find are Â and Ĉ that may be obtained from the definition of B4 and B5 : . 0 0 B5 − iB4  0 0 Â = =  −i 0 2 0 −i. 0 0 0 0.  0 0  0 0. (4.29). and:  0 −~(B5 + iB4 )  0 Ĉ = =  0 2d 0.  0 −i~/d 0 0 0 −i~/d  0 0 0  0 0 0. (4.30). ~ we already know that B̂ is given by: " # ~σ̂ 0 ~ B̂ = 0 ~σ̂. 38. (4.31).

(51) 4.3 Linearization of the Diffusion Equation with ~σ̂ being the Pauli matrices vector Since all of the operators obtained are 4 × 4 matrices it’s clear that the state vector ψ must have four components also. Therefore let’s define the state vector as two two-vectors: φ ψ= χ. (4.32). Therefore the resulting system may be written as: ~σ̂ · p̂φ ~ − i~ χ = 0 d. (4.33). ~ =0 −iÊφ + ~σ̂ · p̂χ. (4.34). and. At this moment it’s evident that the diffusion equation has been decomposed into two first order PDE’s (recall it was of second order). Since the Dirac equation is linear, one can always restrict ourself to the real part of the solution of the linearized diffusion equation. The connection between Schrodinger equation and diffusion equation can be exploited also relativistically by studying the Dirac equation and the analytic connection. The result of such investigation gives again the telegraph equation[6].. 39.

(52) 4. ATTEMPTS AT A RELATIVISTIC DIFFUSION EQUATION. 40.

(53) Bibliography [1] Kostadt P. and Liu M. Causality and stability of the relativistic diffusion equation PHYS. REVIEW D, VOLUME 62, (2000) [1] Dunkel J. Talkner P. and Hanggi P. Relativistic diffusion processes and random walk models PHYS. REVIEW D 75, (2007) [3] Levy-Leblond J.M. Comm. Math. Phys. 6, 286 (1967) [4] Greiner W. 2001 Quantum mechanics : an introduction Springer, 4 ed. Chap. 13 [5] Kazinski,P.O. Relativistic Diffusion Equation from Stochastic Quantization ARXIV eprint arXiv:0704.3877 [6] Gaveau, B. and Jacobson, T. and Kac, M. and Schulman, L. S. Relativistic Extension of the Analogy between Quantum Mechanics and Brownian Motion Phys. Rev. Lett., 53 (1984). 41.


(55) 5 Non-Relativistic Brownian Motion 5.1. Langevin Equation. The objective of this section is to develop the theory and equations that describe the Brownian motion, the development we follow here is the one done by Dunkel et.al.[1]. In order to do this, consider a point-like Brownian particle of mass M. This particle is surrounded by a stationary homogeneous heat bath. To understand this heat bath suppose it consists of smaller liquid particles at constant temperature T and mass m (with m M ). In this heat bath the development from now on is going to be done in a frame such that the mean velocity of the heat bath particles vanish, this frame is called the inertial rest frame Σ. Within this frame, the position of the Brownian particle is denoted by X(t), the velocity by V (t) =. dX(t) dt. and the. nonrelativistic momentum by P (t) = M V (t). In order to proceed, we develop the forces interacting with this Brownian particle. The first force taken into account here is the external force F (t, X). Many times, for simplicity, this force is taken to be zero, but when the Ornstein-Uhlenbeck process is developed in next sections, we will consider it to be a conservative force. Another force taken into account is the friction force α(P )P , being α(P ) the friction coefficient. The last force taken into account is the one involving the interaction of the heat bath and the Brownian Motion, this forced is called the stochastic Langevin force and is defined to be: L(P (t), t) =. p. 43. 2D(P ). ζ(t). (5.1).

(56) 5. NON-RELATIVISTIC BROWNIAN MOTION. This equation reflects the fluctuations in the surrounding heat bath.. Here. D(P ) > 0 reflects the amplitude of the fluctuating force, ζ(t) is the Gaussian white noise that drives Langevin’s force and has the properties:. and. hζ(t)i = 0,. (5.2). hζ(t)ζ(s)i = δ(t − s). (5.3). shows the different possible discretization rules. This rules will be dis-. cussed later on this chapter. Taking all this together the stochastic dynamics of the Brownian motion may be described by a system of coupled differential equations given by: P dX = dt M. (5.4). p dP = F (x, t) − α(P )P + 2D(P ) dt. ζ(t). (5.5). Usually this system of equations is expressed in differential notation, which yield: dX =. P dt M. dP = F (x, t)dt − α(P )P dt +. (5.6). p 2D(P ). dB(t). (5.7). The function B(t) is a simple standardized one-dimensional Brownian motion. By simple comparison of the two different representations the relation dB(t) = ζ(t)dt is evident. In the differential representation, dX(t) = X(t + dt) − X(t) and dP (t) = P (t + dt) − P (t) respectively denote position and momentum change. In the same way, dB(t) = B(t + dt) − B(t) but B(t + dt) and B(t) are taken to be stochastically independent and follow a Gaussian probability distribution: PdB(t) ∈ [y, y + dy] = (2πdt)−1/2 exp[−y 2 /2dt]dy. 44. (5.8).

(57) 5.2 Derivation of the Langevin Equation from. Harmonic Oscillator With this we may establish, by means of an average h·i with respect to the Gaussian probability density, that:. 5.2. hdB(t)i = 0,. (5.9). hdB(t)dB(s)i = δts dt,. (5.10). Derivation of the Langevin Equation from Harmonic Oscillator. Before searching the solution of a Brownian process, let’s have a look to a process that leads us into Langevin equation, the harmonic oscillator.In order to develop the Langevin equation from the harmonic oscillator, besides the mass M, the position X and the momentum P of the Brownian particle we need the same quantities for the heat bath particles: mass mr , the position xr and the momentum pr . In this problem an external potential Φ(x) is also taken into account. With this we may write the Hamiltonian function as: H=. X p2 cr mr ωr2 P2 [ + Φ(x) + (xr − X)2 ] + 2 2M 2m 2 m ω r r r r. (5.11). with cr and ωr being the coupling constant and the oscillator frequency of the heat bath particle. The following Hamilton equations of motion are developed from the Hamiltonian: Ẋ = (P/M ),. Ṗ = F (x) +. X. cr (xr −. r. (5.12) cr X) mr ωr2. (5.13). x˙r = (pr /mr ),. (5.14). p˙r = −mr ωr2 xr + cr X. (5.15). 45.

(58) 5. NON-RELATIVISTIC BROWNIAN MOTION. being F (x) = − dφ(x) the conservative force acting on the Brownian particle. In dx order to solve this system of equations the last two equations are solved by formally integrating. Then, the solution to this equations is inserted in the first two in order to obtain: Ẋ = (P/M ), Z. t. ν(t − s)P (s)ds + L(t). Ṗ = F (x) −. (5.16). (5.17). 0. If as usual we have deterministic initial values for X(0), P (0) and initial distribution conditions for x0 and p0 the following results are achieved for ν(t − s) (the memory friction kernel) and the Langevin noise force L(t): ν(t − s) =. L(t) =. X. cr [[xr (0) −. r. 1 X cr cos[ωr (t − s)], M r mr ωr2. (5.18). cr pr (0) X(0)]cos(ωr t) + sin(ωr t)] 2 mr ωr mr ωr. (5.19). With this, it’s evident that equations (5.16) and (5.17) describe a Brownian process. 5.3. Fokker-Planck Equation and Discretization Rules. When developing the stochastic differential equations (SDEs) of Brownian motion, the final objective is to find the probability of finding the Brownian particle at a time t in the infinitesimal phase space interval [x, x + dx][p, p + dp]. The phase space probability density function (PDF) is defined by f (t, x, p) and must be non-negative everywhere. Also it must be normalized at all times, say: Z f (t, x, p)dxdp = 1. (5.20). This equation must be satisfied for every positive time and the integral ranges over the full position and momentum space, respectively. As a matter of convenience,. 46.

(59) 5.3 Fokker-Planck Equation and Discretization Rules. we define, relative to the phase space PDF, the momentum PDF and the position PDF by: Z f (t, x, p)dx = φ(t, p). (5.21). f (t, x, p)dp = ρ(t, x). (5.22). Z As usual, we take the deterministic initial conditions, which in this case are: f (0, x, p) = δ(x − x0 )δ(p − p0 ). (5.23). φ(0, p) = δ(p − p0 ). (5.24). ρ(0, x) = δ(x − x0 ). (5.25). Here is where the possible discretization rules play an important role because different discretization rules lead to different results. The three most common discretization rules are discussed here: • Pre-point discretization of Ito[1,2], denoted by ” = ∗” is developed by comp puting the function A(P ) = 2D(P ) at P (t) so: A(p) ∗ dB(t) ≡ A(P (t))dB(t). (5.26). This yields a very important result and is that the conditional expectation with respect to the Gaussian measure of the Wiener process B(t) will be given by: hA(P (t))dB(t)|P (t) = pi = 0. (5.27). Finally, with Ito’s discretization method, the Fokker-Plank equation for the phase space PDF would give: ∂f P ∂f ∂f ∂ ∂ + + F (t, x) = [α(P )P f + (D(p)f )] ∂t M ∂x ∂p ∂p ∂p. 47. (5.28).

(60) 5. NON-RELATIVISTIC BROWNIAN MOTION. • Post-point discretization or Backward Ito[1,3], denoted by ” = ∗” is develp oped by computing the function A(P ) = 2D(P ) at P (t + dt) so:. A(p) ∗ dB(t) ≡ A(P (t) + dP (t))dB(t). (5.29). In this case the conditional expectation with respect to the Gaussian measure of the Wiener process B(t) yields:. hA(P (t))dB(t)|P (t) = pi = A(P )A0 (P )dt with A0 (P ) =. (5.30). dA . dP. Finally, with the backward-Ito’s discretization method, the Fokker-Plank equation for the phase space PDF would give: ∂f P ∂f ∂f ∂ ∂ + + F (t, x) = [α(P )P f + D(p) (f )] ∂t M ∂x ∂p ∂p ∂p • mid-point discretization or Stratonovich and Fish[1,4], denoted by ”. (5.31) = ◦” is. developed by computing the mean value of the Ito and the Backward Ito, so: 1 A(p) ◦ dB(t) ≡ [A(p) 2. dB(t) + A(p) ∗ dB(t)]. (5.32). In this case the conditional expectation with respect to the Gaussian measure of the Wiener process B(t) yields: 1 hA(P (t))dB(t)|P (t) = pi = A(P )A0 (P )dt 2. (5.33). Finally, with the mid-point discretization method, the Fokker-Plank equation for the phase space PDF would give:. ∂f P ∂f ∂f ∂ 1p ∂ p + + F (t, x) = [α(P )P f + 2D(P ) ( 2D(P )f )] (5.34) ∂t M ∂x ∂p ∂p 2 ∂p. 48.

(61) 5.4 Non-relativistic Ornstein-Uhlenbeck Process. The choice of the discretization rule reduces to a matter of convenience. From the Fokker-Plank equations (FPEs) stated before, it may be seen that the most simple solution is obtained with the post-point rule. This is why we take this to solve. If there are no external force then the FPE for the momentum PDF φ(t, p) is: ∂ ∂φ ∂φ = [α(P )P φ + D(P ) ] ∂t ∂p ∂p. (5.35). The reader should notice this last equation is parabolic since its discriminant yields D = (0)2 − 4(D)(0) = 0. Therefore we expect the solution to this equation to be very similar to the diffusion equation. In fact the stationary solution of this equation is: Z. p. φ(t → ∞, p) = N exp[− −p∗. α(p0 ) 0 0 p dp ] D(p0 ). (5.36). with N being the normalization constant and p∗ a constant such that the integral exist. This last equation means we are able to generate any arbitrary momentum distribution, the only thing needed is a relation between α(p) and D(p) and even then we are still free to choose one of those parameters. For example if one uses the generalized Einstein fluctuation dissipation relation, relevant whenever the classical Brownian particle is in thermal equilibrium with the surrounding heat bath, the relationship between α(p) and D(p) is given by: D(p) = α(p)M kB T. (5.37). with kB being Boltzmann constant and T being the temperature of the heat bath, the Maxwell distribution is obtained: φ(t → ∞, p) =. 5.4. p 2B T exp[−p2 /(2M kB T )]. (5.38). Non-relativistic Ornstein-Uhlenbeck Process. The simplest possible nonrelativistic Brownian motion is the Ornstein-Uhlenbeck process. This process consists of a FPE with F (t, x) = 0 and constant coefficients. 49.

(62) 5. NON-RELATIVISTIC BROWNIAN MOTION. D(P ) ≡ D0 and α(P ) ≡ α0 . This would yield the following SDE: P dt M. dX =. dP = F (x, t)dt − α(P )P dt +. (5.39). p 2D(P ). dB(t). (5.40). Since for this particular case the choice of the discretization rule is not relevant when integrating the momentum equation it was left as initially. .. With this, the FPE governing the momentum PDF φ(t, p) for this process would be: ∂ ∂φ ∂φ = (α0 pφ + D0 ) ∂t ∂p ∂p. (5.41). Using the deterministic initial condition (5.24), the time-dependent solution of this process gives:. φ(t, p) = (. α0 α0 (p − p0 exp(−α0 t))2 )1/2 exp(− ) 2πD0 [1 − exp(−2α0 t)] (2D0 (1 − exp(−2α0 t)). (5.42). In the limit t → ∞: φ(t → ∞, p) = (. α 0 p2 α0 1/2 ) exp(− ) 2πD0 2D0. (5.43). which is the stationary Gaussian distribution. The objective now is to produce another approach to this problem, the aim here is to find the asymptotic spatial diffusion constant D∞ . This quantity gives us the notion of mean square displacement of a diffusive process and it’s defined by: 1 h[X(t) − X(0)]2 i (5.44) t→∞ t So in order to be able to find this quantity we need the second moment of the 2D∞ ≡ lim. position. Let’s begin by solving equations (5.6) and (5.7), we assumed F (x, t) = 0 and Ito’s discretization rule, explicitly: Z X(t) = X(0) +. t. dsP (s)/M 0. 50. (5.45).

(63) 5.4 Non-relativistic Ornstein-Uhlenbeck Process. P (t) = P (0)e. −α0 t. +. p. Z. t. 2D0. eα0 s ∗ dB(s). (5.46). 0. Now let’s proceed to find the first to moments of the momentum coordinate with the help of the Gaussian probability defined before (5.8): hP (t)i = P (0)e−α0 t. hP (t)2 i = P (0)2 e−2α0 t +. D0 1 − e−2α0 t α0. (5.47). (5.48). With this the two centered moments of the position coordinate yield: hX(t) − X(0)i =. P (0) 1 − e−α0 t α0 M. (5.49). 2 P (0) −α0 t D0 2D0 −α0 t −2α0 t t+ e + −3 + 4e − e (5.50) h[X(t)−X(0)] i = (α0 M )2 α0 M α03 M 2 2. So we get the asymptotic spatial diffusion to be: D∞ = D0 /(α0 M )2. (5.51). Moreover, if we assume we are in a situation in which the Einstein relation, D0 = α0 kB T holds, the asymptotic spatial diffusion would give: D∞ = kB T /(α0 M ). (5.52). This last approach might seem useless because the explicit solution of the problem may be achieved but when the explicit solution is difficult to find this approach is very useful. Now lets face the case in which a non-zero conservative external force F (t, x) = ∂ F (x) = − ∂x Φ(x) is applied. In this case the FPE describing the stochastic process. gives: P ∂f ∂Φ(x) ∂f ∂ 1p ∂ p ∂f + − = [α(P )P f + 2D(P ) ( 2D(P )f )] ∂t M ∂x ∂x ∂p ∂p 2 ∂p. 51. (5.53).

(64) 5. NON-RELATIVISTIC BROWNIAN MOTION. Proceeding in a similar way as we did before, the stationary solution (applying the limit t → ∞) obtained by imposing the Einstein relation gives a MaxwellBoltzmann distribution: p2 + Φ(x))] 2M with β = 1/(kB T ) and N being the normalization constant. f (x, p) = N exp[−β(. 52. (5.54).

(65) Bibliography [1] Dunkel J. Talkner P. Relativistic Brownian Motion Phys.Report 471, (2009), 1-73 [2] Ito, K. On Stochastic Differential Equations Mem. Amer. Mathem. Soc., 4:51-89, 1951 [3] Hanggi, P. Stochastic Processes I: Asymptotic Behavior and Symmetries Helv. Phys. Acta, 51:183-201, 1978 [4] Fisk, D. Quasimartingales Trans. Amer. Math. Soc., 120:369-389, 1965. 53.


(67) 6 Relativistic Brownian Motion The objective of this section is to perform the development of the theory within the relativistic Brownian motion[1]. In order to do so, we consider a stochastic process in which the absolute velocity V(t) = dX(t)/dt does not exceed the speed of light at any time. Therefore, in order to solve this problem, the process must satisfy the conditions specified in last chapter and also: |V(t)| =. |P| |P| = ≤1 P0 (M 2 + P2 )1/2. (6.1). at any time, having M > 0 as the rest mass of the Brownian particle.. 6.1. Relativistic Langevin Equation. In order to consider the stochastic motion of a Brownian particle, the first thing we consider is the characteristic time parameters. In contrast with the non-relativistic Langevin problem in which only the universal time t could be considered, in this problem there are two naturally arising time parameters can be distinguished: the proper time of the heat bath (the coordinate time of inertial lab frame Σ) denoted by t or the proper time of the Brownian particle denoted by τ . In principle, either time parameters could be used to formulate the SDEs for the spatial components of the particle momentum, P = (P i ). Therefore, choosing between those two is a matter of convenience. Since the friction and noise terms are considered to be externally imposed forces that act within the heat bath frame, the heat bath time. 55.

(68) 6. RELATIVISTIC BROWNIAN MOTION. parameter t is more natural because the statistical properties of them are naturally described by this parameter. Here we follow the lab-time approach[1] in which the objective is to construct a t-parametrized 2d -dimensional stochastic process X(t), P(t) with respect to the lab frame Σ. As in the non-relativistic case, the position and spatial momentum coordinates are connected by: dX i (t) = V i dt = (. Pi )dt P0. (6.2). with P 0 = E(t) = (M 2 + P2 )1/2 being the relativistic energy of the particle and V i dt = P i /P 0 being the velocity the velocity components in Σ. Again as before the coupling momentum components of P i yield a second SDE: dP i (t) = Fi dt − αji P j dt + cij. dB j (t). (6.3). with • Fi is an external force; • −αji P j is a friction term • Several discretization rules are considered. ∈ ∗, ◦, ◦. • Bi is a d -dimensional standard Wiener Process • Fi , −αji P j and cij are in general functions of (t,X,P). 6.2. Relativistic Fokker-Plank Equation and Discretization rules. The relativistic Fokker-Plank equations are obtained[1,2] in a similar way to the one done in the previous chapter. It should be taken into account the fact that the external force, in this case, is a function of P also. Recall it wasn’t a function of it. 56.