On billiards and time irreversibility… The legacy of Ludwig Boltzmann Irene M. Gamba Mathematics Department Institute for Computational Engineering and Sciences (ICES) The University of Texas at Austin SMM lecture - UT Austin, November 4 th, 2006

1- Mathematics is a Language Outline calculus 2- Some basic concepts of calculus stochastic processes 3- An experiment with stochastic processes 4- The Boltzmann Problem: collisions and billiards Timeirreversibility = 5- Time irreversibility = stability & memory loss 100 years after L. Boltzmann years after L. Boltzmann: billiards simulations in aerospace and nano-electronics design

Derivative Derivative of a function f(x) at x=x 0 It is the slope of the tangent line to the graph curve It is the slope of the tangent line to the graph curve that passes through the point ( x 0, f(x 0 ) ) It is defined by x0=x0= a0 If f(x) = a constant then df (x) = 0 Notation: means the rate of change f x Notation: d f = f’ means the rate of change of f at the point x dx calculus 2- Some basic concepts of calculus

How to compute the area under a curve? given by the plot of (x, f(x)) for x between a and b: a b f(x) The area can be approximated by

a area If the function f(x)= a constant the area is just a rectangle. a The general conclusion here is that the integral of a constant a is just that atimesx a times the variable of integration x. a For a function f(x) = a x, the area is a triangle:

random Galton Board : an example of a random walk equally separated rods 1.Suppose we drop balls through a fix chute between equally separated rods, having the same probability to go to the right or left. 2.We record the location where they land. 3.After dropping many, many balls, what is the number of balls per unit length we would observed in a given landing location? measured from the distance to the chute location it approximates : G # balls/per unit length = G(x) is the Gaussian “bell curve” as the rods get more densely packed long time behavior This “bell curve” represents the long time behavior of the event repeated infinitely many times. stochastic processes 3 - An experiment with stochastic processes

4- The Boltzmann Problem: collisions and billiards… … in 3 dimensional space

Astronaut Catch: Imagine that you are hovering next to the space shuttle in earth-orbit and your buddy of equal mass who is moving 4 m/s (with respect to the ship) bumps into you. In any instance in which two objects collide with pre-collision velocities v i1 = 4 m/s and v i2 = 0, can be considered isolated from all other net forces Collisions between objects are governed by laws of momentum and energy If she holds onto you, then how fast do the two of you move after the collision? the conservation of momentum principle can be utilized to determine the post-collision velocities v f1 and v f2 of the two objects : 1. Conservation of momentum principle

When a collision occurs in an isolated system the total momentum of the system is conserved: before the collision the combined momentum of the two astronauts before the collision = after the collision the combined momentum of the two astronauts after the collision The mathematics of this problem: a momentum analysis shows that all the momentum was concentrated in the moving astronaut before the collisionafter the collision before the collision. And after the collision, all the momentum was single object moving at one-half the velocity the result of a single object moving at one-half the velocity. Simple equations for the four “interacting velocities” p i = m 1 v i1 + m 2 v i2 = m 1 v f1 + m 2 v f2 = p f Thus, the two astronauts move together with after the collision velocities v f1 = 2 m/s and v f2 = 2 m/s after the collision.

Energy can be defined as the capacity for doing work: It may exist in many forms and may be transformed from one type of energy to another. These energy transformations are constrained by the fundamental principle of Conservation of Energy: this means before the collision after the collision all of the kinetic energy of the objects before the collision is still in the form of kinetic energy after the collision. ‘the total energy of an isolated system remains constant ’ before collision ‘Work’ before collision equals E i = m 1 |v| 2 i1 + m 2 |v| 2 i2 = m 1 |v| 2 f1 + m 2 |v| 2 f2 = E f after collision ‘Work’ after collision For our astronauts in space 2. Conservation of Energy principle: "Energy can neither be created nor destroyed".

An elastic collision is defined as one in which one observes both conservation of momentum and conservation of kinetic energy. Elastic Collisions Collisions between hard steel balls are nearly elastic. elastic“Collisions” in which the objects do not touch each other (due to repulsive forces like in atomic or nuclear scattering) are elastic: Coulomb forces keeps the particles out of contact with each other. Collisions in ideal gases are very nearly elastic: “billiard” Here is the ground for Boltzmann’s amazing ideas !!

Elastic collisions Before collision: p i = m 1 v i1 + m 2 v i2 momentum E i = m 1 |v| 2 i1 + m 2 |v| 2 i2 energy After collision: p f = m 1 v f1 + m 2 v f2 E f = m 1 |v| 2 f1 + m 2 |v| 2 f2 Conservation means Conservation means : equate the total momentum and energy before and after the collision momentum m 1 v i1 + m 2 v i2 = m 1 v f1 + m 2 v f2 momentum energy m 1 |v| 2 i1 + m 2 |v| 2 i2 = m 1 |v| 2 f1 + m 2 |v| 2 f2 energy

Entropy Boltzmann new concept: Entropy as time's arrow Timeirreversibility = 5- Time irreversibility = stability & memory loss

Can you find an equation that expresses these concepts, perhaps … in a probabilistic sense? Yes xv Yes : Think of f(x,v,t) to be the probability of finding a particle x, v, at a point x, moving with velocity v, at a time t. conservation of momentum Then df = conservation of energy dt vvvdv f(x,v,t) {1, v, |v| 2 } dv = 0

Boltzmann H-theorem for elastic interactions : Entropy decay the Boltzmann H-theorem for elastic interactions : Entropy decay xxy log x =log x – log y y Is monotone increasing, Is monotone increasing, that is y xxy (y - x) ( log x – log y) <= 0 xvt <= 0 Then the rate of change of f(x,v,t) with respect to time t is always <= 0 And it is zero (no more changes in time!) for xv v v f(x,v) = a 0 (x) e - |v-u 0 (x)|/ T 0 (x) = G( x, v) “bell curve” No matter the shape of f(x,v,0) was initially !!! No matter the shape of f(x,v,0) was initially !!! - memory loss!! This means … Time irreversibility

Evolution of an initial velocity distribution to a Gaussian state Different initial states converges to the same probability distribution function Deterministic Numerical simulation of the Boltzmann Transport equation Time irreversibility = Time irreversibility = only remembers vvvdv f 0 (x,v) {1, v, |v|2} dv = {a 0 (x), u 0 (x), T 0 (x)}

Direct Simulation Monte Carlo DSMC developed by Graeme Bird (late 60’s) Popular in aerospace engineering (70’s) Variants & improvements (early 80’s) Applications in physics & chemistry (late 80’s) Used for micro/nano-scale flows (early 90’s) Extended to dense gases & liquids (late 90’s) Used for granular gas simulations (early 00’s) billiards models DSMC is a numerical method for molecular simulations of dilute gases (billiards models) Development of DSMC 100 years after L. Boltzmann 100 years after L. Boltzmann: billiards simulations in aerospace and nano-electronics design

Molecular Simulations of Gases Exact molecular dynamics inefficient for simulating a dilute gas. Interesting time scale is mean free time. Computational time step limited by time of collision. Mean Free Path Collision

DSMC Algorithm Initialize system with particles Loop over time steps –Create particles at open boundaries –Move all the particles –Process any interactions of particle & boundaries –Sort particles into cells –Select and execute random collisions –Sample statistical values Example: Flow past a sphere

DSMC Collisions Sort particles into spatial collision cells Loop over collision cells –Compute collision frequency in a cell –Select random collision partners within cell –Process each collision Probability that a pair collides only depends on their relative velocity. Two collisions during  t

Collisions (cont.) Post-collision velocities (6 variables) given by: Conservation of momentum (3 constraints) Conservation of energy (1 constraint) Random collision solid angle (2 choices) v1v1 v2v2 v2’v2’ v1’v1’ V cm vrvr vr’vr’ Direction of v r ’ is uniformly distributed in the unit sphere

DSMC in Aerospace Engineering International Space Station experienced an unexpected degree roll about its X-axis during usage of the U.S. Lab vent relief valve. Analysis using DSMC provided detailed insight into the anomaly and revealed that the “zero thrust” T-vent imparts significant torques on the ISS when it is used. NASA DAC (DSMC Analysis Code) Mean free path ~ 1 10  1 Pa

Computer Disk Drives Mechanical system resembles phonograph, with the read/write head located on an air bearing slider positioned at the end of an arm. Platter Air flow

DSMC Simulation of Air Slider  30 nm Spinning platter R/W Head Pressure Inflow Outflow F. Alexander, A. Garcia and B. Alder, Phys. Fluids (1994). Navier-Stokes 1 st order slip Cercignani/ Boltzmann DSMC Position Pressure Flow between platter and read/write head of a computer disk drive

DS MC Direct Simulation Monte Carlo

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