ANOMALOUS TRANSPORT IN RANDOM MEDIA: A ONE-DIMENSIONAL GAUSSIAN MODEL FOR ANOMALOUS DIFFUSION

Transcription

1 THESIS FOR THE DEGREE OF MASTER OF SCIENCE ANOMALOUS TRANSPORT IN RANDOM MEDIA: A ONE-DIMENSIONAL GAUSSIAN MODEL FOR ANOMALOUS DIFFUSION ERIK ARVEDSON Department of Theoretical Physics CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden, 5

2 Anomalous transport in random media: a one-dimensional Gaussian model for anomalous diffusion ERIK ARVEDSON c ERIK ARVEDSON, 5 Department of Theoretical Physics Complex Adaptive Systems Chalmers University of Technology SE Göteborg Sweden Telephone: +46 () Chalmers Reproservice Göteborg, Sweden, 5

3 Anomalous transport in random media: a one-dimensional Gaussian model for anomalous diffusion ERIK ARVEDSON Department of Theoretical Physics Chalmers University of Technology Abstract This thesis describes a model that shows how particles suspended in a Gaussian random force field can exhibit anomalous diffusion. The dynamics of the particles are examined both with analytical calculations and numerical simulations. A program written for visualisations of particle simulations is described and a few examples are shown. Key words: Fokker-Planck, anomalous diffusion, random force field i

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5 Acknowledgements I would like to thank Bernhard Mehlig for his support and never ending enthusiasm, especially in times when the errors in the simulations seemed impossible to find. Erik Arvedson Göteborg, 4 October 5 iii

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7 Table of Contents 1 Introduction Overview Anomalous diffusion Formulation of the problem Possible applications The Model 5.1 Introduction Equations of motion The random force The unbounded case The bounded case Parameters Correlations of the force A particle at a fixed position The unbounded case The bounded case The diffusion model The Focker-Planck equation Distribution of the momentum The unbounded case The bounded case Variance of the momentum The unbounded case The bounded case v

8 vi TABLE OF CONTENTS 3 Implementation Language considerations The random force The unbounded case The bounded case Source code Visualisation Play Results of numerical simulation Correlations of the force Distribution of the momentum Variance of the momentum Anomalous diffusion The unbounded case The bounded case Discussion and future work A simple model explains a lot A new system with a ladder spectra What to do next A Calculations 33 A.1 Normalization constants of the random force A.1.1 Time dependent part A.1. Space dependent part, unbounded case A.1.3 Space dependent part, bounded case A. Linear approximation of a particle trajectory A..1 The momentum of a particle A.. The position of a particle A.3 Force correlations A.3.1 Fixed position in space A.3. On a trajectory in the unbounded case A.4 The diffusion model A.4.1 Assuming equal jump probabilities A.4. Assuming different jump probabilities A.5 Distribution of the momentum

9 TABLE OF CONTENTS vii A.5.1 The unbounded case A.5. The bounded case A.6 Variance of the momentum A.6.1 The unbounded case A.6. The bounded case References 45

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11 Chapter 1 Introduction 1.1 Overview At first sight, it seems as if particles suspended in a randomly stirred fluid should become evenly distributed. In other words it is expected that a uniform distribution should stay uniform over time. However, the opposite, i.e. clustering or aggregation, has been observed in many cases, for instance in particles suspended in homogeneously turbulent fluids [1], in atmospheric clouds [], the aggregation of interstellar dust [3], and particles floating on a turbulent fluid [4]. It is important to note that the apparent clustering is not a consequence of an attractive force between the particles. Instead the attraction originates from the smoothness of the random force field, i.e. the randomly moving fluid, and has been discussed in [5, 6, 7], and references cited therein. Figure 1.1: Evolution of a uniform particle distribution (image taken from [6]) If two particles are separated by a distance longer than the correlation length of the force, ξ, the forces they experience are statistically independent, and there is a high probability that they will be kicked in different directions. However, if two particles are separated by a distance smaller than the correlation length, the will experience roughly the same force, and there is a 1

12 CHAPTER 1. INTRODUCTION high probability that they will receive a kick in the same direction. That means that statistically, over time particles will cluster together. This thesis discusses the dynamics of particles suspended in an incompressible, randomly moving fluid, or more precisely, a one-dimensional caricature of that problem, which is however, still able to model some of the underlying physics in an interesting way. A smooth Gaussian random force field f(x, t) is assumed, with typical size σ, correlation length ξ and correlation time τ: ṗ = γp + f(x, t), mẋ = p. (1.1) Here x and p are particle position and momentum, respectively, m is the mass, and γ is the rate at which the particle momentum is damped due to viscous drag. This model was first considered by Deutsch [5] whose theoretical analysis and numerical simulations indicated the existence of a phase transition in this model: if the damping rate exceeds a critical value γ c all particles will eventually follow the same trajectory. In [7] this path-coalescence transition was solved exactly in the limit of weak damping and weak forcing. Both this path-coalescence effect as well as the dynamics of an individual particle are very well understood in the limit of weak forcing. The latter is is diffusive with diffusion constant D x σ τ/(mγ) in the under- and overdamped regimes. But what happens at large forcing? At least two important questions are of interest in this regime: one concerns the singleparticle behaviour and the other the path-coalescence transition. First, what is the nature of the dynamics of one given particle? Is it also diffusive? If so, what is the spatial diffusion constant D x in terms of the microscopic parameters of the model? The second question addresses the path-coalescence transition: what determines the the critical damping at large forcing? Only the first question will be addressed in this thesis. A phase diagram of the single particle dynamics, determined by two dimensionless parameters: a dimensionless forcing (χ = στ /mξ) and a dimensionless damping (ω = γτ), is fully derived in [8]. It shows that individual particles can exhibit at least five different dynamic behaviours, fig. 1., depending on the values of ξ and ω. In all cases the long-time dynamics are diffusive, but the stationary distribution of momentum may be strongly non-gaussian. The diffusion constant D x depends in different ways on the microscopic parameters. This thesis focuses on the under-damped non-gaussian regime III. 1. Anomalous diffusion The variance of particle positions in diffusion increase as V ar(x) t α (1.) where α equals one in the usual case. In anomalous diffusion, α is different from one, but in general smaller than two. The case where α = is called ballistic, and corresponds to the case where all particles are moving away from each other at a constant rate.

13 1.3. FORMULATION OF THE PROBLEM 3 logω Ib Ia λ < ω χ /3 ω χ ω χ 1/ λ > IIIb IIIa logχ Ic ω χ II Figure 1.: Phase diagram summarising the different dynamical behaviours of eqn. (1.1): single-particle behaviour (red) and path-coalescence transition (blue). Dynamics of a single particle: Ia overdamped advective, Ib underdamped advective, Ic underdamped Gaussian, III underdamped non-gaussian, IIa overdamped minimum tracking, IIb underdamped minimum tracking. Phases Ia-c and III are ergodic, the others non-ergodic. Path-coalescence transition: at small values of χ the phase line is ω c χ /3, at large values of χ it is ω c const. Figure taken from [8]. 1.3 Formulation of the problem In a more realistic, 3-dimensional formulation of the problem, the particles suspended in the fluid flow satisfy the equations of motion ṙ = 1 m p (1.3) ṗ = mγ(u(r, t) ṙ) (1.4) where r = (r 1, r, r 3 ) denotes the position of a particle, p is its momentum, m is its mass, and u(r, t) denotes the velocity field. This model is appropriate for spherical particles when the Reynolds number of the flow referred to the particle diameter is small. Further conditions are required, but these can always be be satisfied if the radius of the particle and the molecular mean free path of the fluid are sufficiently small. Stokes s formula gives the relaxation rate γ γ = 6πaη (1.5) m where η is the viscosity of the fluid, a the radius of the particle, and m its mass. Effects due to the inertia of the displaced fluid are neglected, which is justified when the density of the suspended particles is large compared to that of the fluid. This is an excellent approximation for aerosol systems, and satisfactory for many examples of solid particles in water.

14 4 CHAPTER 1. INTRODUCTION 1.4 Possible applications A possible technological application suggested in [6] is the coagulation of small pollutant particles in an engine exhaust. An ultrasonic noise source could increase the size of the pollutant particles until they are large enough to be captured efficiently by a mechanical filter.

15 Chapter The Model.1 Introduction The model considered in this thesis is a one-dimensional simplification of the three-dimensional model discussed in the previous chapter. The main advantage of a one-dimensional model is that simulations can be run in far shorter time spans. It is also possible to make a twodimensional visualisation of the state space, i.e. momentum versus position, which helps in understanding the mechanics of the model. The velocity field of the surrounding fluid is modeled in terms of a random force correlated in both time and space.. Equations of motion Consider a system of i independent particles with positions x i (t) and momenta p i (t). The equations of motion for any particle are ẋ = p m (.1) ṗ = f(x, t) γp (.) where dots denote time derivatives, and γ characterises the strength of the viscous damping. f(x, t) is a random force fluctuating in both space and time, a detailed description follows..3 The random force The random force f(x, t) has the following statistical properties that are translationally invariant both in space and time: 5

16 6 CHAPTER. THE MODEL f(x, t) = (.3) f(x, t)f(x, t ) = c(x x, t t ) (.4) c(x x, t t ) = σ exp ( (x x ) ) ξ exp ( (t t ) ) τ (.5) c(x 1, x ) = Cov(x 1, x ) (.6) The correlation function c(x x, t t ) decays rapidly as x x and as t t. In this thesis, the form showed in eqn. (.5) is used. The random force f(x, t) can be viewed as the gradient of a scalar potential φ(x, t), f(x, t) = φ(x, t) (.7) which in the one dimensional case is just the derivative with respect to x. Two different cases of the random force is studied in this thesis. They are referred to as the unbounded case and the bounded case for reasons that will become evident after reading what follows..3.1 The unbounded case Consider a random variable r(x, t) of the following form r(x, t) = C x C t i= j= a i,j e (i x x) ξ e (i t t) τ. (.8) r(x, t) is a double sum of normally distributed random numbers with zero mean and unit variance, a i,j, weighted by two Gaussians with variance ξ and τ respectively, centred at (x, t). C x and C t are normalisation constants. The random numbers are separated by a distance x in x and t in t. The random variable r(x, t) is continuous in x and t while created from a sum of discrete random variables, thus making it suitable for computer simulation (where the sum is of course truncated). A random force satisfying the conditions stated in equations can then be created simply by putting f(x, t) = σr(x, t) (.9) ( ) 1 x 4 C x = (.1) ξ π ( ) 1 t 4 C t = (.11) τ π The constants C x and C t are calculated in appendix A.1.1 and A.1.. The corresponding potential φ(x, t) is the integral of r(x, t). That integral is unbounded, which is the reason for this case being called the unbounded case.

17 .4. PARAMETERS 7.3. The bounded case Another possible random force can be created by putting f(x, t) = σ r(x, t) (.1) ( ) 1 x 4 C x = ξ (.13) ξ π ( ) 1 t 4 C t = (.14) τ π The constants C x and C t are calculated in appendix A.1.1 and A.1.3. The corresponding potential φ(x, t) is equal to r(x, t), which is bounded. Therefore this is called the bounded case..4 Parameters The random force f(x, t) is characterised by its typical magnitude σ, the correlation length ξ and correlation time τ of the correlation function c(x, t), and by the size of x and t. The system of equations is characterised by two independent dimensionless parameters defined as χ = στ mξ, ω = γτ The parameter χ is a dimensionless measure of the strength of the force and ω is a dimensionless measure of the degree of damping. The simulations have the additional parameter h, which is the step size of the Euler algorithm used for time integration. In many calculations it is convenient to use p = mξ τ, D = π σ τ where p is a characteristic momentum in the system, and D is a characteristic diffusion rate..5 Correlations of the force The random force f(x, t) is correlated in both space and time. The correlation of the force a particle experiences depends on the momentum of the particle. If a particle travels very fast, several correlation lengths ξ in a time period shorter than the correlation time τ, the force the particle experience will de-correlate much faster than if the particle was kept at a fixed position in space.

18 8 CHAPTER. THE MODEL.5.1 A particle at a fixed position Since the force is normalised to unity multiplied by σ, the correlation for a fixed particle is the correlation of the time-dependent part of the force, calculated in appendix A.3.1 c(, t t ) = f(, t)f(, ) = σ e t τ (.15) Since this is independent of x it holds for both the unbounded and the bounded cases..5. The unbounded case The force correlation on the trajectory of a particle, f(x t, t)f(, ) can be calculated using the approximation x t p t t/m, discussed in appendix A., and then averaging f(p t t/m, t)f(, ) over the distribution of p. The result is an integral that can be evaluated numerically. f(x t, t)f(, ) = CP (p)f(pt/m, t)f(, )dp (.16) CP (p) exp [ t τ (1 + p p )] dp where P (p) is the distribution of the momentum, discussed in section.8.1, and C is a constant that normalises that distribution to unity..5.3 The bounded case Using the same approximation as in the unbounded case, i.e. x t p t t/m, the force correlation on a trajectory of a particle can be calculated in the bounded case. f(x t, t)f(, ) = CP (p)f(pt/m, t)f(, )dp (.17) CP (p) (p t m ξ ) m ξ exp [ t τ (1 + p p )] dp where P (p) is the distribution of the momentum, discussed in section.8., and C is a constant that normalises that distribution to unity..6 The diffusion model Consider a particle moving on a straight line, i.e. in one dimension (the results derived here can easily be generalised to higher dimensions). Assume that in a small time step δt the particle will jump a short distance δx with a certain probability w. The probability P (x, t) of a particle being at position x at a time t is then the probability of the particle being at x δx at time t δt times the probability w of jumping right plus the probability of the particle being at

19 .7. THE FOCKER-PLANCK EQUATION 9 x + δx at time t δt times the probability w + of jumping left. A formal expression is shown in eqn. (.18). P (x, t) = w P (x δx, t δt) + w + P (x + δx, t δt) (.18) Assume equal jump probabilities, w = w + = 1/ After Taylor expansion and some reordering of the terms (the details can be found in appendix A.4) eqn. (.18) can be rewritten as P (x, t) t = (δx) δt P (x, t) x + δt P (x, t) t +... (.19) By taking the limit δx, δt while keeping the ratio (δx) /δt = D constant, the following final result is reached: P (x, t) = D P (x, t) t x (.) which is a diffusion equation, where D is the diffusion constant. Assume any jump probabilities, w + w + = 1 Again, after Taylor expansion and reordering of the terms (the details can be found in appendix A.4), eqn. (.18) can be rewritten as P (x, t) t P (x, t) = (w + w )δx + (δx) P (x, t) x δt x + (w w + )δx P (x, t) t δt P (x, t) t (.1) By taking the limit δx, δt while keeping the ratio (δx) /δt = D constant, and (w w + ) δx δt = V constant, the following result is reached: P (x, t) = [ V + D ] P (x, t) (.) t x x.7 The Focker-Planck equation The Focker-Planck equation is also called Smoluchowski equation, second Kolmogorov equation or generalised diffusion equation. The last name might serve best to state the relation to the diffusion equation discussed previously. A general form of the Focker-Planck equation is P (p, t) t = p a 1(p)P + 1 p a (p)p (.3) where the coefficients a 1 (p) and a (p) may be any differentiable real functions with the only restriction a (p) >. It can be showed, see for instance [11], that for a solution P (p, t p, t o ) of eqn. (.3), the following is true when δt

20 1 CHAPTER. THE MODEL δp δt = a 1 (p ), When comparing to eqn. (.) it is apparent that (δp) δt = a (p ). (.4) a 1 = V, 1 a = D. (.5).8 Distribution of the momentum.8.1 The unbounded case The momentum p obeys a Langevin equation 1 of the form where dw = dp = γpdt + dw (.6) t+dt t dt f(x(t ), t ). (.7) The statistics of p described in eqn. (.4) can be calculated with aid of the approximation x t = x + p t t/m + x t, see appendix A.5 for the details, and the result is (δp) δt δp δt p D ( p + p ) 1/ = D p (.8) γp pp D ( p + p ) 3/ (.9) = γp + p D p. The Focker-Planck equation for the distribution P of the momentum p is thus P t = ( ) γp + D p P (.3) p p which has a stationary solution for ( ) γp + D p P =. (.31) p This is easily solved and the stationary distribution is [ P (p) = C exp γ ( p 3D p + p ) ] 3/ 1 see [11] for more details on the properties of such equations (.3)

21 .8. DISTRIBUTION OF THE MOMENTUM 11 where C is a normalisation constant chosen to normalise the distribution to unity. The distribution of the momentum is qualitatively different for small values of the ratio χ /ω compared to large values for the ratio. The distribution in eqn. (.3) can be rewritten as P (p) = C exp [ γp 3D ( ( ) 3/ 1 + p 1)]. (.33) p Assume χ /ω 1 be expanded,. Then the force is weak and p is typically small. The exponent can then and the distribution is approximately ) 3/ (1 + p 1 = 3p p p + O(p 3 ) (.34) ] P (p) C exp [ γp D which is Gaussian. The constant C can be calculated analytically, and the result is (.35) C = 1 γ. (.36) π D Assume χ /ω 1 approximately. Then the force is strong and p typically large. The distribution is then where the constant C is ] P (p) C exp [ γ p 3 3D p (.37) ( ) C = 3/3 γ 1/3. (.38) Γ(1/3) D p.8. The bounded case All calculations in the bounded case are analogous to those in the unbounded case. The distribution of the momentum is [ P (p) = C exp γ ( p 5D p 3 + p ) ] 5/ (.39) [ ( ( ) 5/ = C exp γp 1 + p 1)]. (.4) 5D p

22 1 CHAPTER. THE MODEL Assume χ /ω 1 be expanded,. Then the force is weak and p is typically small. The exponent can then and the distribution is approximately ) 5/ (1 + p 1 = 3p p p + O(p 3 ) (.41) ] P (p) C exp [ γp D which is identical to the Gaussian distribution in the unbounded case. (.4) Assume χ /ω 1 then approximately. Then the force is strong and p is typically large. The distribution is where the constant C is ] P (p) C exp [ γ p 5 5D p 3 (.43).9 Variance of the momentum ( ) C = 54/5 γ 1/5 Γ(1/5) D p 3. (.44) The variance of the momentum, V ar(p), can be calculated as follows: V ar(p) = E[p ] E[p] = E[p ] = p (.45) p = P (p)p dp (.46) where the fact that E[p] = has been used. As shown in the previous section, the distribution is qualitatively different for χ /ω 1 and χ /ω 1. In each case the distribution can be approximated and the calculations done analytically. The two cases are treated separately below..9.1 The unbounded case The distribution P of p is shown in eqn. (.3), repeated here for clarity where C is a normalisation constant. [ P (p) = C exp γ ( p 3D p + p ) ] 3/ (.47)

23 .9. VARIANCE OF THE MOMENTUM 13 Assume χ /ω 1. The distribution can then be approximated by P (p) 1 ] γ exp [ γp. (.48) π D D The expression shown in eqn. (.46) can be calculated (the details can be found in appendix A.6), and the variance becomes p = D π γ = σ τ 1 γ which can be rewritten on dimensionless form as (.49) Assume χ /ω 1 by p π p = χ ω when χ /ω 1. (.5). That means that p is typically large and eqn. (.3) can be approximated ( ) P (p) 3/3 γ 1/3 ] exp [ γ p 3. (.51) Γ(1/3) D p 3D p Once again, eqn. (.46) can be calculated, which yields p = = ( ) 3D p /3 1 γ Γ(1/3) ( ) 3 π /3 mξσ 1 γ Γ(1/3) (.5) where Γ is the gamma function (Γ(1/3).67894). On dimensionless form, the variance of p becomes ( ) p 3 π /3 p = χ4/3 1 ω Γ(1/3) when χ /ω 1. (.53).9. The bounded case The calculations of the variance of the momentum in the bounded case are analogous to those in the unbounded case. In fact, in the case of small χ /ω, the variance is identical. p p = π χ ω when χ /ω 1 (.54) p p = χ 4/5 ( 5 π ω ) /5 Γ(3/5) Γ(1/5) when χ /ω 1. (.55)

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25 Chapter 3 Implementation 3.1 Language considerations All programming for this thesis was done in C. A random number generator from Numeric recipes was the only part of the program not written by the author himself. 3. The random force The random force was implemented using a shifting window technique. Previously, in [13, 14], a different approach have been used, using an equivalent Fourier series representation of the force. The shifting window technique should perhaps be called the shifting square technique, since the window has both a time and a space dimension. The force at a given x and t is evaluated as a double sum of weighted random numbers. The random numbers are then shifted in time and space, and new random numbers created at the empty places The unbounded case The theoretical expression of the random force used is f(x, t) = σc x C t i= j= a i,j exp [ ] (i x x) ξ exp [ ] (i t t) τ. (3.1) All the variables are described in chapter. In practice, of course, the sums are cut off when the Gaussians are close enough to zero. The intervals summed over should be at least 5τ and 5ξ wide respectively. The matrix of normally distributed random numbers with zero mean and unit variance, a i,j, was created using an algorithm from [1]. 15

26 16 CHAPTER 3. IMPLEMENTATION 3.. The bounded case The bounded case is very similar to the unbounded. The force function is simply the spatial derivative of the unbounded force function, re-normalised. f(x, t) = σξc x C t 3.3 Source code i= j= a i,j (i x x) ξ Source code is available on request from the author. exp [ ] (i x x) ξ exp [ ] (i t t) τ (3.)

27 Chapter 4 Visualisation 4.1 Play A c program called play was written as an aid to visualise the data from the simulations. It is a simple open-gl program, that works on both mac and windows. It loads particle coordinates from one or two data files, and then displays the time evolution of the particles, much like a video player. A number of screen shots from such displays are shown in the following section. The program was written in a way that enables it to play data from different sources. It has also been used by Bernhard Mehlig to visualise simulated particles together with caustics on which they are predicted to cluster. An example is shown in fig

28 18 CHAPTER 4. VISUALISATION (a) (b) (c) (d) (e) (f) Figure 4.1: An example of how the output of play looks. The figure shows subsequent frames in the time evolution.

29 Chapter 5 Results of numerical simulation 5.1 Correlations of the force In section.5 the correlation of the force experienced by a particle at a fixed position was calculated analytically, the result is repeated here for clarity: c(, t t ) = f(, t)f(, ) = σ e t τ (5.1) which is plotted as a solid line in both fig. 5.1 and fig. 5.. The corresponding simulated force correlation is plotted as triangles, and shows excellent agreement. The correlation decays quickly, and is negligible at t = 5τ, confirming the statement made in section 3. saying that the sum of random variables can be truncated at a distance of 5τ, and 5ξ, respectively. A weak force means that particles will in general move slowly, and thus the correlation of the force they experience will be similar to that of a particle at a fixed position. If the strength of the force is increased, particles will typically move fast, and if the force is increased enough, making the particles move longer than a correlation length ξ in a single time step, they will experience no spatial correlation at all. Another approach would of course be to decrease the damping, which is discussed in section 5.4. The unbounded case The force correlation on a particle trajectory was calculated in section.5, repeated here for clarity: f(x t, t)f(, ) CP (p) exp [ (1 t τ + p p [ P (p) = C exp γ ( p 3D p + p ) ] 3/ )] dp (5.) (5.3) where P (p) is the distribution of the momentum, calculated in section.8. The integral can be evaluated numerically, and is plotted as a dashed line in fig. 5.1, and the corresponding simulated correlation is plotted as x s. For each subsequent subfigure, the force is increased, 19

30 CHAPTER 5. RESULTS OF NUMERICAL SIMULATION and as expected, the correlations decay more and more rapidly. The agreement between theory and simulation is good, with small fluctuations in the simulation which would most likely disappear if averaged over more runs. The bounded case The force correlation in the bounded case was calculated in section.5, repeated here for clarity: f(x t, t)f(, ) CP (p) (p t m ξ )] ) m ξ exp [ (1 t τ + p p dp (5.4) [ P (p) = C exp γ ( p 5D p 3 + p ) ] 5/ (5.5) where P (p) is the distribution of the momentum, calculated in section.8. The integral can be evaluated numerically, and is plotted as a dashed line in fig. 5., and the corresponding simulated correlation is plotted as x s. Subsequent subfigures show stronger forces, i.e. greater values of χ /ω. The simulated force correlation show some deviations from the theoretical curve, greater for stronger forces. The reason is probably that the requirement of the process to be ergodic is not met for the particular set of parameters, see [8] for details. 5. Distribution of the momentum The distribution of the momentum, calculated analytically in section.8, can be approximated by two qualitatively different distributions in the limiting cases of very weak forcing and very strong forcing respectively, which corresponds to χ /ω being very small or very large respectively. In both fig. 5.3 and fig. 5.4 subsequent subfigures show simulations ranging from weak forcing to strong forcing, and the transition between the approximate distributions can be seen clearly. In both the unbounded and bounded case, the momentum distribution is Gaussian in the limit of weak forcing, but in the case of strong forcing, the distributions are exponential of p 3 and p 5 respectively. The unbounded case in section.8, is The exact momentum distribution in the unbounded case, as calculated [ P (p) = C exp γ ( p 3D p + p ) 3/ ]. (5.6) The integral can be evaluated numerically, and is plotted as a solid line in fig The approximated distributions in the limiting cases are

31 5.3. VARIANCE OF THE MOMENTUM 1 ] P (p) C exp [ γp, for weak forcing (5.7) D ] P (p) C exp [ γ p 3, for strong forcing (5.8) 3D p which are plotted in fig. 5.3 as a dotted and a dash-dotted line, respectively. The simulated distributions agree very well with the theoretical, and the approximative distributions are clearly valid for the smallest and largest values of χ /ω respectively. As expected, only the exact theoretical distribution agrees with simulation in the case of intermediate forcing, i.e. when the distribution is neither Gaussian nor p 3 exponential. The bounded case In the bounded case, the exact momentum distribution is [ P (p) = C exp γ 5D p 3 ( p + p ) ] 5/ (5.9) as calculated in section.8. The integral can be evaluated numerically, and is plotted as a solid line in fig In the same way as in the unbounded case, this can be approximated for weak and strong forcing respectively, corresponding to χ /ω 1 and χ /ω 1, respectively, resulting in ] P (p) C exp [ γp, for weak forcing (5.1) D ] P (p) C exp [ γ p 5 5D p 3, for strong forcing (5.11) which are plotted in fig. 5.4 as a dotted and a dash-dotted line, respectively. Note that the approximate distribution for weak forcing, i.e. χ /ω 1, is Gaussian and identical to the distribution in the unbounded case. Theory and simulation agree well for weak forcing, but show some deviations for strong forcing, just as in the previous section. The reason is once again probably that the conditions for the system to be ergodic is not fully satisfied with the particular set of parameters, more details of this can be found in [8]. For χ /ω = 5 the agreement is excellent however, and the distribution is clearly non-gaussian. The approximate distributions are clearly valid in the limiting cases, but as expected not in the region of intermediate forcing. 5.3 Variance of the momentum The variance of the momentum, p, can be calculated analytically in the limiting cases of weak and strong forcing respectively, i.e. χ /ω 1 and χ /ω 1 respectively, as described in chapter.9. The behaviour of interest is how the variance of the momentum, p, depends on the strength of the forcing, ξ.

32 CHAPTER 5. RESULTS OF NUMERICAL SIMULATION The unbounded case In the limits of weak and strong forcing respectively, the variance of the momentum, rewritten on dimensionless form, are p p = π χ ω when χ /ω 1 and (5.1) ( ) p 3 π /3 p = χ 4/3 1 ω Γ(1/3) when χ /ω 1 (5.13) which are plotted in fig. 5.5 as dotted and dash-dotted lines respectively. The theoretical approximations agree well with simulations in the limiting cases of weak and strong forcing, but as expected there are deviations in the intermediate region. The bounded case In the bounded case, the analytically calculated approximations of the variance of the momentum, described in section.9, in the limiting cases of weak and strong forcing respectively, are p p = π χ ω when χ /ω 1 and (5.14) ( ) p 5 π /5 p = χ 4/5 Γ(3/5) ω Γ(1/5) when χ /ω 1 (5.15) which are plotted in fig. 5.6 as dotted and dash-dotted lines. In case of weak forcing, the variance of the momentum increases in exactly the same way as in the bounded case, which is not surprising since the distributions actually are the same, as shown in the previous section. In the case of strong forcing however, the variance increases slower than in the unbounded case. This can also be seen when comparing the widths of the distributions in fig. 5.3 and fig The simulations seem to agree well with theory in both the limiting cases. The deviations in the distribution of the momentum, discussed in the previous section does not effect the variance enough to be noticeable in fig. 5.6, although there must be some deviations there as well. 5.4 Anomalous diffusion In all previously described simulations, the system has reached a steady state due to the relation between the damping and the forcing. But what happens if the damping is very small, or even zero? As stated in section 1., the variance of the positions of particles in diffusion increases with time as V ar(x) t α (5.16) where α = 1 in usual diffusion, and α = is called ballistic diffusion, characterised by the fact that all particles move away at a constant rate.

33 5.4. ANOMALOUS DIFFUSION 3 Simulations in the under-damped region where γ = show that anomalous diffusion arise, both for the particle momentum p and position x. Initially, the diffusion is ballistic, i.e. the parameter α p corresponding to the slope is α p =, as shown in both fig. 5.8 and fig Then after a time t the diffusion changes and becomes anomalous. The transition occurs when the particles typically move longer than a correlation length ξ in each time step, and thus experience no force correlation, which can be formulated as t = mξ p (5.17) which agrees well with fig. 5.7 and fig. 5.9 if the variance of the momentum is used as an approximation the actual momentum The unbounded case The parameter α can be fitted to values of α p = /3 and α x = + /3, obtained in [8, 9], for the momentum and position respectively as shown in fig. 5.7 and fig The bounded case In the bounded case, the transition region between ballistic and anomalous diffusion is longer, but after some time, the results of simulations can be fitted to α p = /5 and α x = + /5, obtained in [8, 9, 1], as shown in fig. 5.9 and fig. 5.1.

34 4 CHAPTER 5. RESULTS OF NUMERICAL SIMULATION 1. 1 fixed x simulated actual trajectory simulated fixed x theoretical actual trajectory theoretical 1. 1 fixed x simulated actual trajectory simulated fixed x theoretical actual trajectory theoretical.8.8 <f(x t,t),f(x,)>.6.4 <f(x t,t),f(x,)> t t (a) χ =.5, ω =.1, χ /ω =.5 (b) χ =.158, ω =.1, χ /ω = fixed x simulated actual trajectory simulated fixed x theoretical actual trajectory theoretical 1. 1 fixed x simulated actual trajectory simulated fixed x theoretical actual trajectory theoretical.8.8 <f(x t,t),f(x,)>.6.4 <f(x t,t),f(x,)> t t (c) χ =.5, ω =.1, χ /ω =.5 (d) χ =.158, ω =.1, χ /ω = fixed x simulated actual trajectory simulated fixed x theoretical actual trajectory theoretical 1. 1 fixed x simulated actual trajectory simulated fixed x theoretical actual trajectory theoretical.8.8 <f(x t,t),f(x,)>.6.4 <f(x t,t),f(x,)> t t (e) χ =.5, ω =.1, χ /ω = 5 (f) χ = 1.58, ω =.1, χ /ω = 5 Figure 5.1: Correlations of the force in the unbounded case, for different values of χ /ω. All runs were made using the same parameter settings, save for σ. They were: t =.1, x =.1, ξ =.1, τ =.1, and γ =.1.

35 5.4. ANOMALOUS DIFFUSION fixed x simulated actual trajectory simulated fixed x theoretical actual trajectory theoretical 1. 1 fixed x simulated actual trajectory simulated fixed x theoretical actual trajectory theoretical.8.8 <f(x t,t),f(x,)>.6.4 <f(x t,t),f(x,)> t t (a) χ =.158, ω =.1, χ /ω =.5 (b) χ =.5, ω =.1, χ /ω = fixed x simulated actual trajectory simulated fixed x theoretical actual trajectory theoretical 1. 1 fixed x simulated actual trajectory simulated fixed x theoretical actual trajectory theoretical.8.8 <f(x t,t),f(x,)>.6.4 <f(x t,t),f(x,)> t t (c) χ =.158, ω =.1, χ /ω =.5 (d) χ =.5, ω =.1, χ /ω = fixed x simulated actual trajectory simulated fixed x theoretical actual trajectory theoretical 1. 1 fixed x simulated actual trajectory simulated fixed x theoretical actual trajectory theoretical.8.8 <f(x t,t),f(x,)>.6.4 <f(x t,t),f(x,)> t t (e) χ = 1.58, ω =.1, χ /ω = 5 (f) χ = 5., ω =.1, χ /ω = 5 Figure 5.: Correlations of the force in the bounded case, for different values of χ /ω. All runs were made using the same parameter settings, save for σ. They were: t =.1, x =.1, ξ =.1, τ =.1, and γ =.1.

36 6 CHAPTER 5. RESULTS OF NUMERICAL SIMULATION 8 7 simulated analytical p approximation p 3 approximation.5 simulated analytical p approximation p 3 approximation P(p) 4 P(p) p p (a) χ =.5, ω =.1, χ /ω =.5 (b) χ =.158, ω =.1, χ /ω = simulated analytical p approximation p 3 approximation.35.3 simulated analytical p approximation p 3 approximation P(p).4 P(p) p p (c) χ =.5, ω =.1, χ /ω =.5 (d) χ =.158, ω =.1, χ /ω = simulated analytical p approximation p 3 approximation.6.5 simulated analytical p approximation p 3 approximation P(p) P(p) p p (e) χ =.5, ω =.1, χ /ω = 5 (f) χ = 1.58, ω =.1, χ /ω = 5 Figure 5.3: Distribution of the momentum in the unbounded case, for different values of χ /ω. The black line show the exact theoretical curve, numerically evaluated, the green line shows the gaussian approximation, and the red line shows the p 3 approximation. All runs were made using the same parameter settings, save for σ. They were: t =.1, x =.1, ξ =.1, τ =.1, and γ =.1.

37 5.4. ANOMALOUS DIFFUSION simulated analytical p approximation p 5 approximation.9.8 simulated analytical p approximation p 5 approximation P(p) 1.5 P(p) p p (a) χ =.158, ω =.1, χ /ω =.5 (b) χ =.5, ω =.1, χ /ω = simulated analytical p approximation p 5 approximation.5 simulated analytical p approximation p 5 approximation P(p) P(p) p p (c) χ =.158, ω =.1, χ /ω =.5 (d) χ =.5, ω =.1, χ /ω = simulated analytical p approximation p 5 approximation.9.8 simulated analytical p approximation p 5 approximation P(p) P(p) p p (e) χ = 1.58, ω =.1, χ /ω = 5 (f) χ = 5., ω =.1, χ /ω = 5 Figure 5.4: Distribution of the momentum in the bounded case, for different values of χ /ω. The black line show the exact theoretical curve, numerically evaluated, the green line shows the p 4 approximation, and the red line shows the p 5 approximation. All runs were made using the same parameter settings, save for σ. They were: t =.1, x =.1, ξ =.1, τ =.1, and γ =.1.

38 8 CHAPTER 5. RESULTS OF NUMERICAL SIMULATION 6 4 log Var(p) 4 simulated χ /ω << 1 χ /ω >> log χ Figure 5.5: The variance of the momentum, p versus dimensionless parameter χ, for the unbounded case. All runs were made using the same parameter settings, save for σ. They were: t =.1, x =.1, ξ =.1, τ =.1, and γ =.1. The values of σ were.5,.158,.5, 1.58, 5., and log Var(p) 4 simulated χ /ω << 1 χ /ω >> log χ Figure 5.6: The variance of the momentum, p, versus dimensionless parameter χ, for the bounded case. All runs were made using the same parameter settings, save for σ. They were: t =.1, x =.1, ξ =.1, τ =.1, and γ =.1. The values of σ were.5,.158,.5, 1.58, 5., 15.8, and 5.

39 5.4. ANOMALOUS DIFFUSION log Var(p) simulated fit with slope / log t Figure 5.7: log V ar(p) versus log t for the unbounded case, with parameters τ =.1, ξ =.1, σ = log Var(x) simulated fit with slope: log t Figure 5.8: log V ar(x) versus log t for the unbounded case, with parameters τ =.1, ξ =.1, σ = 5

40 3 CHAPTER 5. RESULTS OF NUMERICAL SIMULATION log Var(p).5 simulated with χ = 5 fit with slope /5 simulated with χ = 5 fit with slope / log t Figure 5.9: log V ar(p) versus log t for the bounded case, with parameters τ =.1, ξ =.1, σ = 5 or σ = log Var(x) simulated with χ = 5 fit with slope + /5 simulated with χ = 5 fit with slope + / log t Figure 5.1: log V ar(x) versus log t for the bounded case, with parameters τ =.1, ξ =.1, σ = 5 or σ = 5

41 Chapter 6 Discussion and future work 6.1 A simple model explains a lot It is quite remarkable how such a simple model as the one described in this thesis can account for such a range of varying phenomena. Particles exposed to a random force field that is all Gaussian can be shown to exhibit non-gaussian momentum distributions, which might sound surprising, but analytical calculations agree with the simulations. The anomalous diffusion was a surprise outcome of the simulations at first, but has later been calculated analytically, in agreement with simulations. During the course of the creation of this thesis, the area understood and explained in the phase diagram shown in fig. 1. has grown from a small patch to almost the whole area. 6. A new system with a ladder spectra There are only a few physically significant systems with ladder spectra (exactly evenly spaced energy levels). Examples are the harmonic oscillator and the Zeeman-splitting Hamiltonian. A simplified version of the model used in this thesis, ṗ = γp + f(t) (6.1) which is the same as eqn. (.) except that the force is spatially independent, describes a standard Ornstein-Uhlenbeck process, discussed in [11]. A physically significant extension of this (basically a Fokker-Plank equation with no damping) can then be formulated as exactly solvable eigenvalue problems that have ladder spectra. Details of this can be found in [8, 9]. 6.3 What to do next In reality the force field is not as random as in the simplified model used in this thesis. An interesting next step might be to incorporate models of fluid mechanics and turbulence in the force field. The model considered in this thesis is one-dimensional, it would of course be interesting to extend it to higher dimensions. 31

42

43 Appendix A Calculations A.1 Normalization constants of the random force The normalization constants of eqn. (.8), repeated here fore clarity, can be calculated as follows. r(x, t) = C x C t i= j= a i,j e (i x x) ξ e (i t t) τ (A.1) A.1.1 Time dependent part The time dependent part should be normalised by C t and the space dependent part by C x. Let s first consider only the time dependent part. f(t) = σr(t) = σc t i= ) (i t t) a i exp ( τ (A.) where C t is a constant, a i are normally distributed random numbers with zero mean and unit variance, and τ is the correlation time. The statement that r(t) should be normalised to one can then be formulated as r(t), r(t) = 1. (A.3) The task is then to calculate the constant C t. The correlation function of r can be calculated as follows. r(t), r(t) = C t i= j= ( a i, a j exp (i t t) + (j t t) ) ) τ (A.4) Since a i and a j are independent random numbers it follows that 33

44 34 APPENDIX A. CALCULATIONS a i, a j = and eqn. (A.4) can we rewritten as follows r(t), r(t) = C t i= { 1 ; i = j ; i j This sum can be approximated by an integral, which yields (A.5) ) (i t t) exp ( τ. (A.6) r(t), r(t) C t Thus, by setting the constant C t to = C t C t = τ π t. t dx exp ( τ ( ) 1 4 π ) (x t t) τ eqn. (A.3) is satisfied, and r(t) has the desired statistical properties. (A.7) (A.8) A.1. Space dependent part, unbounded case Assume that the spatially dependent part f(x) has the following form: f(x) = r(x) = C x i= ) (i x x) a i exp ( ξ (A.9) where C x is a constant, a i are normally distributed random numbers with zero mean and unit variance, and ξ is the correlation length. In order to normalise f(x) to one, the constant C x is calculated completely analogous to C t as described in appendix A.1.1, thus yielding A.1.3 C x = x ξ Space dependent part, bounded case Assume that the spatially dependent part f(x) has the following form: f(x) = r(x) x = C x i= ( ) 1 4. (A.1) π ) (i x x) (i x x) a i exp ( ξ ξ (A.11) where C x is a constant, a i are normally distributed random numbers with zero mean and unit variance, and ξ is the correlation length.

45 A.. LINEAR APPROXIMATION OF A PARTICLE TRAJECTORY 35 f(x) should be normalised to one, which can be formulated as The task is then to calculate the constant C x. f(x), f(x) = 1. (A.1) f(x), f(x) = C x i= j= exp ( a i, a j exp ( (j x x) ξ Since a i and a j are independent random numbers it follows that ) ) (i x x) ξ (A.13) 4 i x x j x x ξ ξ and eqn. (A.13) can be rewritten as a i, a j = { 1 ; i = j ; i j (A.14) f(x), f(x) = C x i= ) (i x x) (i x x) exp ( ξ 4 ξ 4. (A.15) This sum can be approximated by an integral, and calculated as follows f(x), f(x) C x = C x π dz exp ( x ξ 1 x x = 1. ) (z x x) (z x x) ξ 4 ξ 4 (A.16) The desired normalization can thus be accomplished by putting ( ) 1 x 4 C x = ξ. (A.17) ξ π Note that this only differs by a factor ξ from the unbounded case. A. Linear approximation of a particle trajectory A..1 The momentum of a particle The momentum of a particle at a time t is determined by the equation ṗ = γp + f(x t, t). (A.18)

46 36 APPENDIX A. CALCULATIONS The momentum p t is thus p t = t dt e γ(t t ) f(x t, t ). Since this might not be obvious, the above relation is shown below: (A.19) p t = t dt e γ(t t ) f(x t, t ) = d dt p t = F (t, t) dt F (t, ) + dt = e γ(t t) f(x t, t) + = f(x t, t) γ = f(x t, t) γp t t where the following known expression was used: A.. d dx v(x) u(x) t t t F (t, t )dt t F (t, t )dt t e γ(t t ) f(x t, t )dt dt e γ(t t ) f(x t, t ) F (x, t)dt = F (x, v) dv v(x) F (x, u)du dx dx + F (x, t)dt. u(x) x The position of a particle The position x t of a particle at time t is where x t = p tt m + x t (A.) x t = t m [ p t t dt e γ(t t ) f( p ] t t m, t ). (A.1) In many calculations, it is sufficient to only consider the first linear term, and neglect the higher order corrections. A.3 Force correlations When the particles move fast, they can travel a distance longer than the correlation length ξ in a time interval shorter than the correlation time τ, and thus the correlation of the force they experience decay faster. The force correlation of a particle on a fixed position in space, and of a particle on an actual trajectory are calculated below.

47 A.3. FORCE CORRELATIONS 37 A.3.1 Fixed position in space The space dependent part of the force correlation for a particle at a fixed position in space is always one, as calculated in section A.1. and A.1.3. The time dependent force correlation is f(t), f() = σ C t i= j= i= j= a i, a j exp ( (i t t) + (j t) τ ). (A.) Since a i and a j are independent random numbers it follows that a i, a j = { 1 ; i = j ; i j (A.3) and thus f(t), f() = σ C t i= i= exp ( (i t t) + (i t) τ ). (A.4) This sum can be approximated by an integral, which yields f(t), f() σ C t = σ exp dz exp ( (z t t) + (z t) ) ) ( t τ. τ (A.5) A.3. On a trajectory in the unbounded case The exact correlation of a force experienced by a particle moving along the actual trajectory is very complicated to calculated. Simulations show that first order approximation of the trajectory gives a sufficiently accurate theoretical prediction. The strategy is then to calculate correlation function of the force, and then replace x t by p t t/m and take the average over the distribution of the momentum p, which yields a function that can be computed numerically. In the unbounded case, the spatial correlation is completely analogous to the correlation in time, and thus ] ] f(x, t)f(, ) = σ exp [ x ξ exp [ t τ (A.6) is the general expression of the force correlation in the unbounded case. Using the approximation x t = p t t/m and neglecting higher order corrections, this becomes

48 38 APPENDIX A. CALCULATIONS f( p tt m, t)f(, ) = σ exp [ p t ] ] m ξ exp [ t τ (A.7) = σ exp [ (1 t τ + p τ )] m ξ = σ exp [ t τ (1 + p where the final expression should be averaged over the distribution of p, which is calculated in section A.5. A.4 The diffusion model The one-dimensional diffusion equation can be derived as follows. Consider a particle moving in one dimension. In a small time step δt the particle will jump a short distance δx to the left with a certain probability w + or to the right with probability w. The probability of a particle being at position x at a time t is thus p )] P (x, t) = w P (x δx, t δt) + w + P (x + δx, t + δt). (A.8) A.4.1 Assuming equal jump probabilities Lets first consider the case where w = w + = 1. The two terms in the right hand side of eqn. (A.8) can be Taylor expanded in the following way P (x, t) P (x δx, t δt) = P (x, t) δx x + δx P (x, t) x + δxδt P (x, t) x t + δt + P (x, t) P (x + δx, t δt) = P (x, t) + δx x + δx P (x, t) x δxδt P (x, t) x t + δt + P (x, t) δt t P (x, t) t P (x, t) δt t P (x, t) t (A.9) (A.3)

49 A.5. DISTRIBUTION OF THE MOMENTUM 39 After some reordering of the terms, the resulting expression is P (x, t) t = δx P (x, t) δt x + δt P (x, t) t + (A.31) By taking the limit δx, δt while keeping the ratio (δx) /δt = D constant, the following final result is reached: P (x, t) t = D P (x, t) x. (A.3) A.4. Assuming different jump probabilities Assume w + w + = 1. After Taylor expansion, eqn. (A.8) becomes P (x, t) P (x, t) = (w + w + )P (x, t) + (w + w )δx x P (x, t) (w + w + )δt t + (w + w + ) δt After some reordering of the terms, this becomes + (w + w + ) δx P (x, t) x P (x, t) t (w w + )δxδt P (x, t) + x t (A.33) P (x, t) t = (w + w ) δx P (x, t) + δx P (x, t) δt x δt x + δt P (x, t) t + (w w + )δx P (x, t). x t (A.34) By taking the limit δx, δt while keeping the ratio (δx) /δt = D constant and (w w + )δx/δt = V, the following final result is reached: P (x, t) t = [ V + D ] P (x, t). x x (A.35) A.5 Distribution of the momentum The momentum p obeys a Langevin equation of the form dp = γpdt + dw (A.36)