Investigation of the Equilibrium Speed Distribution of Relativistic Particles

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1 I. Introduction Investigation of the Equilibrium Speed Distribution of Relativistic Pavlos Apostolidis Supervisor: Prof. Ian Ford Condensed Matter & Material Physics Dept. of Physics & Astronomy, UCL June-July 14 The equilibrium velocity distribution of ultra relativistic particles has puzzled scientists for at least a century, mainly due to the vague understanding of fundamental concepts such as the temperature of a moving body in relativistic regimes, as well as due to the mathematical difficulty of constructing a fully relativistic and covariant mathematical framework. In the relativistic limit, the Maxwell-Boltzmann velocity distribution of particles is believed to no longer hold, as it gives non-zero probability for particles to have speeds greater than the speed of light and the distribution function itself is not Lorentz invariant. At high relativistic speeds, the Maxwell-Boltzmann distribution is replaced by the Maxwell- Jüttner distribution, which is believed to be a generalisation of the former and to reduce to it at classical speeds [1]. Due to the lack of understanding of this area in relativistic kinetic theory, mainly because of the inability to perform experiments for high relativistic particles, this project is aimed to investigate, using simulations, how the equilibrium velocity distribution of relativistic particles differs from the standard Maxwell-Boltzmann distribution function and examine which probability function describes the equilibrium distribution better in 3-dimensions, for both neutral and charged particles. II. Procedures and Theory For this project, simulations for particles were implemented in Java (Eclipse IDE). The particles were used as an implementation of a low density gas. Two types of interactions were investigated: interactions via particle (elastic) collisions and electromagnetic interactions. For the latter case, particles were randomly given a +q or -q charge, where q is times the elementary electric charge. For the colliding particles, each particle was given a random initial position in a cubic box of side length 1 cm and assigned a random speed between.9c and.9999c, where c is the speed of light. The speed was broken down into Cartesian components upon initiation using spherical polar coordinates with random values for the polar and azimuthal angles within [, π] and [, π), respectively. For the polar angle, a Gaussian function was used so that the probability distribution of the random angle values was peaked at π/. The box was centred at the origin and periodic boundary conditions applied for the walls of the box (i.e. a particle that hits a wall is absorbed and emitted from the opposite side of the box) so that the particles could move around with no constraints. The particles were all given the same mass (1-6 kg) and the system was left to evolve with time for 1 4 time steps, of 1 ns each. For the electromagnetic interactions, the speed of the particles was reduced between.c and.5c in order to ignore retardation effects in the simulation, which become important for high relativistic interactive charged particles and possess a major mathematical difficulty if 1

2 taken into account. In this case, the periodic boundary conditions of the box were no longer used and the box was made to have adiabatic walls, so the corresponding momentum component of a particle hitting a wall was to be reversed. For the classical speed electromagnetic interactions, the distribution was expected to follow a Maxwell-Boltzmann distribution upon reaching equilibrium. For the interaction of particles via elastic collisions, the particles were assumed to be point-like and with their momenta changing instantaneously when colliding. The collision criterion was chosen such that a collision occurs then the distance between the centres of two particles is equal to or less than a critical range, which was orders of magnitude smaller than the size of the box. It was assumed that when a collision occurs, the force is exerted along the line that connects the centres of the particles, that is r ij = r i - r j, where r denotes a position vector relative to the centre of the box (origin). This implies that momentum components parallel to that line change with a collision, while those perpendicular to it do not change. The parallel momentum components p // for the i th particle change as in the one-dimensional case indicated below: [] ' p (v ) [v E (1 v ) p ] (1) i,// cm cm i cm i,// where v cm is the relativistic centre of mass velocity (collision invariant), given by pi,// pj,// vcm, with E i being the energy of the i th particle. E E i j The collection of particles was left to evolve as described above until the speed distribution was no longer time-dependent. This was found to emerge after about 1 4 time steps that correspond to about 14-1 collisions per particle. The equilibrium speed distribution was plotted in units of the speed of light and it was compared with the theoretical equilibrium speed distribution for high relativistic particles, the Maxwell-Jüttner distribution. The Maxwell-Jüttner distribution function for a particle with velocity v in 3-dimensions is given below: [3] 3 5 m f ( v) exp J mc (3) Z v 1 where 1, J, m is the mass of the particle, k B is the Boltzmann constant, Z is c kt B a normalisation constant and T is the temperature of the system, related to the mean energy per particle and used as a fitting parameter, obtained to be 1.3x1 14 K in this case. The same procedure was followed for a moving frame of reference (S ), moving with v=.c with respect to the stationary (lab S) frame. The x-components of the velocities of the particles were transformed using relativistic velocity addition and the system was left to equilibrate, accounting for time dilation effects and length contraction along the x-direction. The x-component of the velocities was then transformed back and the speed distribution was obtained. The equation for relativistic velocity addition is given by: [4] u ' u v uv 1 c (4)

3 where u is the speed of the particle as measured by S, v is the speed of S as seen by S and u is the speed of the particle as measured by S. III. Results and Analysis A. Elastic Collisions To compare the equilibrium speed distribution with the classical expectations (i.e. Maxwell-Boltzmann distribution), all distributions were plotted on the same graph for comparison as shown in Figure 1. The speed distributions obtained for frames S and S were found to be in a good agreement, although not identical as one would expect. The small discrepancies between the two distributions were concluded to arise because of minor approximations and assumptions in the calculations. In general, it was shown that for at high relativistic speeds the equilibrium speed distribution of the particles is best described by the Maxwell-Jüttner distribution, while there are major discrepancies when compared to the Maxwell-Boltzmann distribution (see Figures 1a,b). Small discrepancies between expected and obtained values suggest the probability that the gas had not yet fully equilibrate, although a number of factors could have had an impact on the distribution, including small semi-classical approximations in the equations for collisions. In Figure 1c, the distributions obtained for the fames S and S were shown to almost overlap, suggesting that a Lorentz invariant approach has been implemented. 4 of 3 3 Distributions for Stationary Frame Distribu tion MB Figure 1a: shows the equilibrium speed distribution for the stationary frame S for particles, including the theoretical Maxwell-Jüttner and Maxwell-Boltzmann distributions. 3

4 4 of 3 3 Distributions for Moving Frame Distribu tion MB Figure 1b: shows the equilibrium speed distribution for the frame S for particles, including the theoretical Maxwell-Jüttner and Maxwell-Boltzmann distributions. 4 of Distributions for Stationary and Moving frames S S' Figure 1c: shows the equilibrium speed distribution for the frames S and S for particles. The individual velocity components for the stationary and moving distributions were also investigated. After equilibrium was reached, all three velocity components for the stationary frame were obtained and shown in Figure a, while for the moving frame the v x velocity component (the one that underwent transformation with Eq. 4) is shown in Figure b, before being transformed back. For the former case, the distributions were found to have some significant discrepancies when compared to the expected velocity distribution based on the Maxwell-Jüttner function, as shown. Some further investigation was suggested for future simulations to find out why that is so. For the latter case, more particles were found to lie on the right-hand side of the distribution as expected, since S is moving along the positive x-axis with respect to S. 4

5 Velocity Components for Stationary Frame of vx vy vz v i /c Figure a: shows the velocity components at equilibrium for the frame S for particles, along with the theoretical distribution based on the Maxwell-Jüttner function. Vx Distribution for Moving Frame of Vx/c Figure b: shows the v x velocity component at equilibrium for the frame S for particles. B. Electromagnetic Interactions Similarly, the same collection of particles was evolved using electromagnetic interactions instead of collisions, after assigning a charge to each particle, as already described. As mentioned earlier, retarded potentials become important for high speeds and they have to be 5

6 taken into consideration for obtaining the correct distribution. However, this requires a complex mathematical approach, so non-relativistic speeds were considered. The force acting on each particle due to the electric and magnetic field of the other particles is the linear sum of the Lorentz forces each particle exerts on the particle of interest. The total force on a particle with charge q and velocity v is given below: [5] F qe qv B (4) where E and B is the electric and magnetic field at the position of the particle of interest that emerge from the charge and movement of all the other particles in the box. The electric and magnetic fields emerging from an individual particle of charge Q and velocity Q u are given by ˆ E r and B u rˆ, respectively, where r is the distance of the 4 r 4 r particle from the point of interest, and ε ο and μ o are the permittivity and permeability of free space, respectively. The particles were left to equilibrate for the same amount of steps and the speed distributions were extracted as before. The results are shown in Figure 3. It was shown that the speed distribution is best described by a Maxwell-Boltzmann rather than a Maxwell-Jüttner distribution, as initially expected. For the examination of the validity of the Maxwell-Jüttner distribution at high relativistic speeds for charged particles, a rigorous mathematical model must be implemented, that includes retarded electric and magnetic potentials which depend on the path followed by the particle and how that affects the force felt by the neighbouring particles. of Distributions for Electromagnetic Interactions Distribu tion MB Figure 3: shows the equilibrium speed distribution for charged particles interacting electromagnetically, including the theoretical Maxwell-Jüttner and Maxwell-Boltzmann distributions. 6

7 IV. Conclusions The equilibrium speed distributions were investigated using particle simulations for high relativistic particles interacting via elastic collisions (stationary and moving frames) and nonrelativistic particles interacting via electromagnetic interactions. Comparing the distributions obtained from the simulations with theory, it was verified that the Maxwell-Boltzmann distribution breaks down at high relativistic speeds for colliding particles, while the Maxwell- Jüttner distribution describes the data in a more realistic manner. For electromagnetic interactions, the retardation effects are a significant obstacle on dealing with high relativistic charged particles and obtaining the correct speed distribution, so those interactions were investigated for classical speeds, where retardation effects are negligible and the distribution was found to agree with the classical Maxwell-Boltzmann distribution. For the colliding particles, simulation data and theoretical data for the Maxwell-Jüttner distribution were not found to be in a complete agreement, although the general behaviour of the simulated particles was shown to follow the Maxwell-Jüttner trend, while having major defects from the classical Maxwell-Boltzmann distribution. Moreover, distributions emerging from both a stationary and a moving frame of reference were compared and found to agree with each other, as well as with theory. Discrepancies between expected and obtained values were assumed to arise because of approximations and assumptions in calculations, for example the use of classical collision equations (although with relativistic masses), which was taken as a necessary step due to the inability of the use of a fully relativistic 4-momentum approach in 3- dimensions. There is also the probability that the gas was not completely in equilibrium and more time steps were required for it to equilibrate fully, although increasing the number of time steps by a small factor did not result in any major changes of the distributions. Major discrepancies between the individual velocity component distributions and their corresponding Maxwell-Jüttner distribution indicate that a further investigation is necessary for the understanding of this aspect. For electromagnetically interacting particles at lower speeds, the speed distribution was shown to be in agreement with the classical Maxwell-Boltzmann distribution, while having major differences with the Maxwell-Jüttner distribution, as expected. For a full understanding of the Maxwell-Jüttner distribution at high relativistic speeds for electromagnetically interacting particles, retarded potentials need to be considered in calculations. V. References [1] F. Jüttner, Ann. Phys. Lpz. 34, (1911). [] D. Cubero, J. Casado-Pascual, J. Dunkel, P. Talkner, and P. Hanggi, Phys. Rev. Lett. 99, 1761 (7). [3] A. Montakhab, M. Ghodrat, and M. Barati, Phys. Rev. E 79, 3114 (9). [4] R. Mould, The Physical Arguments, in Basic Relativity, Springer, 1st Ed., ch., sec. 3, p. 35. [5] D. Griffiths, The Lorentz Force Law, in Introduction to Electrodynamics, Prentice Hall, 3rd Ed., 1999, ch.5, sec.1, p. 4. 7