# GasLab Circular Benchmark

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### VERSION

$Id: GasLab Circular Benchmark.nlogo 38506 2008-03-05 23:59:14Z tisue $

### WHAT IS IT?

This model is one in a series of GasLab models. They use the same basic rules for simulating the behavior of gases. Each model integrates different features in order to highlight different aspects of gas behavior. This model is different from the other GasLab models in that the collision calculations take the circular shape and size of the particles into account, instead of modeling the particles as dimensionless points.

### HOW IT WORKS

The model determines the resulting motion of particles that collide, with no loss in their total momentum or total kinetic energy (an elastic collision).

To calculate the outcome of collision, it is necessary to calculate the exact time at which the edge of one particle (represented as a circle), would touch the edge of another particle (or the walls of a container) if the particles were allowed to continue with their current headings and speeds.

By performing such a calculation, one can determine when the next collision anywhere in the system would occur in time. From this determination, the model then advances the motion of all the particles using their current headings and speeds that far in time until this next collision point is reached. Exchange of kinetic energy and momentum between the two particles, according to conservation of kinetic energy and conservation of momentum along the collision axis (a line drawn between the centers of the two particles), is then calculated, and the particles are given new headings and speeds based on this outcome.

### HOW TO USE IT

NUMBER determines the number of gas particles used with SETUP. If the world is too small or the particles are too large, the SETUP procedure of the particles will stop so as to prevent overlapping particles.

SMALLEST-PARTICLE-SIZE and LARGEST-PARTICLE-SIZE determines the range of particle sizes that will be created when SETUP is pressed. (Particles are also assigned a mass proportional to the area of the particle that is created.)

SHOW-SPEED-AS-COLOR? allows you to visualize particle speed using a color palette.

The "blue-green-red" setting shows the lower half of the speeds of the starting population as blue, and the upper half as red.

The "blue shades" setting shows a gradient of dark blue to light blue for slow to fast particle speed.

The "all green" setting does not show a different color for each particle based on its speed.

### THINGS TO NOTICE

Particles never overlap or penetrate into each other or the wall as they move about.

SET the SPEED-AS-COLOR? chooser to red-green-blue and run the model with RANDOMIZE-MASS-SIZES? set turned on. With many different mass particles colliding over time, different sized particles start to move at different speed ranges (in general). The smallest mass particles will be usually moving faster (red) than the average particle speed and the largest mass particles will be usually slower (blue) than the average particle speed. This emergent result is what happens in a gas that is a mixture of particles of different masses. At any given temperature, the higher mass particles are moving slower (such as Nitrogen gas: N2) then the lower mass particles (such as water vapor: H2O).

### THINGS TO TRY

Setting all the particles to have a very slow speed (e.g. 0.001) and one particle to have a very fast speed helps show how kinetic energy is eventually transferred to all the particles through a series of collisions and would serve as a good model for energy exchange through conduction between hot and cold gases.

To see what the approximate mass of each particle is, type this in the command center:

| ask particles [set label precision mass 0]

### EXTENDING THE MODEL

Collisions between boxes and circles could also be explored. Variations in size between particles could investigated or variations in the mass of some of the particle could be made to explore other factors that affect the outcome of collisions.

### NETLOGO FEATURES

MANAGE-VIEW-UPDATES? uses the VIEW-UPDATE-RATE slider to enforce a requirement that a minimum amount of ticks (simulation time) must pass before the view is updated. VIEW-UPDATE-RATE sets what this minimum value is. This helps smooth out the motion of the particles that is seen in the WORLD & VIEW. The particles should appear to have a linear rate of change in their motion when they are not colliding. Without a minimum VIEW-UPDATE-RATE, however, this would not appear to be the case, since without a the model would update the graphics every time after every tick and ticks vary in length, depending on how close the next projected collision is. As the tick length moved up and down, the simulation time the particles were allowed to move forward before they were redisplayed would also move up and down. This would cause the motion of the particles to look jerky and non-linear.

### RELATED MODELS

Look at the other GasLab models to see collisions of "point" particles, that is, the particles are assumed to have an area or volume of zero.

### CREDITS AND REFERENCES

## Comments and Questions

globals [ result ;; for benchmarking tick-length ;; how much simulation will pass in this step box-edge ;; distance of box edge from axes colliding-particles sorted-colliding-particles colliding-particle-1 colliding-particle-2 original-tick-length last-view-update manage-view-updates? view-update-rate ;; specifies the minimum amount of simulation time that must ;; pass before the view is updated ] breed [ particles particle ] particles-own [ speed mass ] ;;;;;;;;;;;;;;;; to benchmark random-seed 12345 reset-timer setup set manage-view-updates? false repeat 3000 [ go ] set result timer end to setup ca set-default-shape particles "circle" set manage-view-updates? true set view-update-rate 0.2 set box-edge (max-pxcor - 1) make-box make-particles ;; set variable tick length based on fastest particle. If the fastest particle has a speed of 1, ;; then tick-length is 1. If the fastest particles has a speed of 10, then tick-length is 1/10. set tick-length (1 / (ceiling max [speed] of particles)) set original-tick-length tick-length set colliding-particle-1 nobody set colliding-particle-2 nobody rebuild-collision-list end to rebuild-collision-list set colliding-particles [] ask particles [check-for-wall-collision] ask particles [check-for-particle-collision ] end to go ;;Since only collisions involving the particles that collided most recently could be affected, ;;we filter those out of colliding-particles. Then we recalculate all possible collisions for the ;;particles that collided last. The ifelse statement is necessary because colliding-particle-2 ;;can be either a particle or a string representing a wall. If it is a wall, we don't want to ;;invalidate all collisions involving that wall (because the wall's position wasn't affected, those ;;collisions are still valid. ifelse is-turtle? colliding-particle-2 [ set colliding-particles filter [item 1 ? != colliding-particle-1 and item 2 ? != colliding-particle-1 and item 1 ? != colliding-particle-2 and item 2 ? != colliding-particle-2] colliding-particles ask colliding-particle-2 [check-for-wall-collision] ask colliding-particle-2 [check-for-particle-collision] ] [ set colliding-particles filter [item 1 ? != colliding-particle-1 and item 2 ? != colliding-particle-1] colliding-particles ] if colliding-particle-1 != nobody [ask colliding-particle-1 [check-for-wall-collision]] if colliding-particle-1 != nobody [ask colliding-particle-1 [check-for-particle-collision]] sort-collisions ask particles [ jump speed * tick-length ] collide-winners tick-advance tick-length ;; flag that updates display only after enough simulation time has passed. ;; the display-update-rate sets the minimum simulation time that must pass ;; before updating the display. This avoids many redisplays of the view for ;; a series of small time steps in the simulation (which would make the view show ;; what looks like particles slowing down as they get near multiple collision points) if manage-view-updates? [ if (ticks - last-view-update) > view-update-rate [ display set last-view-update ticks ] ] end to-report convert-heading-x [heading-angle] report sin heading-angle end to-report convert-heading-y [heading-angle] report cos heading-angle end to check-for-particle-collision ;; check-for-particle-collision is a particle procedure that determines the time it takes to the collision between ;; two particles (if one exists). It solves for the time by representing the equations of motion for ;; distance, velocity, and time in a quadratic equation of the vector components of the relative velocities ;; and changes in position between the two particles and solves for the time until the next collision let my-x xcor let my-y ycor let my-particle-size size let my-x-speed (speed * convert-heading-x heading) let my-y-speed (speed * convert-heading-y heading) ask other particles [ let dpx (xcor - my-x) ;; relative distance between particles in the x direction let dpy (ycor - my-y) ;; relative distance between particles in the y direction let x-speed (speed * convert-heading-x heading) ;; speed of other particle in the x direction let y-speed (speed * convert-heading-y heading) ;; speed of other particle in the x direction let dvx (x-speed - my-x-speed) ;; relative speed difference between particles in the x direction let dvy (y-speed - my-y-speed) ;; relative speed difference between particles in the y direction let sum-r (((my-particle-size) / 2 ) + (([size] of self) / 2 )) ;; sum of both particle radii ;; To figure out what the difference in position (P1) between two particles at a future time (t) would be, ;; one would need to know the current difference in position (P0) between the two particles ;; and the current difference in the velocity (V0) between of the two particles. ;; The equation that represents the relationship would be: P1 = P0 + t * V0 ;; we want find when in time (t), P1 would be equal to the sum of both the particle's radii (sum-r). ;; When P1 is equal to is equal to sum-r, the particles will just be touching each other at ;; their edges (a single point of contact). ;; Therefore we are looking for when: sum-r = P0 + t * V0 ;; This equation is not a simple linear equation, since P0 and V0 should both have x and y components ;; in their two dimensional vector representation (calculated as dpx, dpy, and dvx, dvy). ;; By squaring both sides of the equation, we get: (sum-r) * (sum-r) = (P0 + t * V0) * (P0 + t * V0) ;; When expanded gives: (sum-r ^ 2) = (P0 ^ 2) + (t * PO * V0) + (t * PO * V0) + (t ^ 2 * VO ^ 2) ;; Which can be simplified to: 0 = (P0 ^ 2) - (sum-r ^ 2) + (2 * PO * V0) * t + (VO ^ 2) * t ^ 2 ;; Below, we will let p-squared represent: (P0 ^ 2) - (sum-r ^ 2) ;; and pv represent: (2 * PO * V0) ;; and v-squared represent: (VO ^ 2) ;; then the equation will simplify to: 0 = p-squared + pv * t + v-squared * t^2 let p-squared ((dpx * dpx) + (dpy * dpy)) - (sum-r ^ 2) ;; p-squared represents difference of the ;; square of the radii and the square ;; of the initial positions let pv (2 * ((dpx * dvx) + (dpy * dvy))) ;;the vector product of the position times the velocity let v-squared ((dvx * dvx) + (dvy * dvy)) ;; the square of the difference in speeds ;; represented as the sum of the squares of the x-component ;; and y-component of relative speeds between the two particles ;; p-squared, pv, and v-squared are coefficients in the quadratic equation shown above that ;; represents how distance between the particles and relative velocity are related to the time, ;; t, at which they will next collide (or when their edges will just be touching) ;; Any quadratic equation that is the function of time (t), can represented in a general form as: ;; a*t*t + b*t + c = 0, ;; where a, b, and c are the coefficients of the three different terms, and has solutions for t ;; that can be found by using the quadratic formula. The quadratic formula states that if a is not 0, ;; then there are two solutions for t, either real or complex. ;; t is equal to (b +/- sqrt (b^2 - 4*a*c)) / 2*a ;; the portion of this equation that is under a square root is referred to here ;; as the determinant, D1. D1 is equal to (b^2 - 4*a*c) ;; and: a = v-squared, b = pv, and c = p-squared. let D1 pv ^ 2 - (4 * v-squared * p-squared) ;; the next line next line tells us that a collision will happen in the future if ;; the determinant, D1 is >= 0, since a positive determinant tells us that there is a ;; real solution for the quadratic equation. Quadratic equations can have solutions ;; that are not real (they are square roots of negative numbers). These are referred ;; to as imaginary numbers and for many real world systems that the equations represent ;; are not real world states the system can actually end up in. ;; Once we determine that a real solution exists, we want to take only one of the two ;; possible solutions to the quadratic equation, namely the smaller of the two the solutions: ;; (b - sqrt (b^2 - 4*a*c)) / 2*a ;; which is a solution that represents when the particles first touching on their edges. ;; instead of (b + sqrt (b^2 - 4*a*c)) / 2*a ;; which is a solution that represents a time after the particles have penetrated ;; and are coming back out of each other and when they are just touching on their edges. let time-to-collision -1 if D1 >= 0 [set time-to-collision (- pv - sqrt D1) / (2 * v-squared) ] ;;solution for time step ;; if time-to-collision is still -1 there is no collision in the future - no valid solution ;; note: negative values for time-to-collision represent where particles would collide ;; if allowed to move backward in time. ;; if time-to-collision is greater than 1, then we continue to advance the motion ;; of the particles along their current trajectories. They do not collide yet. if time-to-collision > 0 [ ;; time-to-collision is relative (ie, a collision will occur one second from now) ;; We need to store the absolute time (ie, a collision will occur at time 48.5 seconds. ;; So, we add clock to time-to-collision when we store it. let colliding-pair (list (time-to-collision + ticks) self myself) ;; sets a three element list of ;; time to collision and the colliding pair set colliding-particles fput colliding-pair colliding-particles ;; adds above list to collection ;; of colliding pairs and time ;; steps ] ] end to check-for-wall-collision ;; particle procedure for determining if a particle will hit one of the ;; four walls of the box let x-speed (speed * convert-heading-x heading) let y-speed (speed * convert-heading-y heading) let xpos-plane (box-edge - 0.5) ;;inside boundary of right side of the box let xneg-plane (- box-edge + 0.5) ;;inside boundary of left side of the box let ypos-plane (box-edge - 0.5) ;;inside boundary of top side of the box let yneg-plane (- box-edge + 0.5) ;;inside boundary of bottom side of the box ;; find point of contact on edge of circle ;; points of contact located at 1 radius above, below, to the left, and to the right ;; of the center of the particle let contact-point-xpos (xcor + (size / 2)) let contact-point-xneg (xcor - (size / 2)) let contact-point-ypos (ycor + (size / 2)) let contact-point-yneg (ycor - (size / 2)) ;; find difference in position between plane location and edge of circle let dpxpos (xpos-plane - contact-point-xpos) let dpxneg (xneg-plane - contact-point-xneg) let dpypos (ypos-plane - contact-point-ypos) let dpyneg (yneg-plane - contact-point-yneg) let t-plane-xpos 0 ;; solve for the time it will take the particle to reach the wall by taking ;; the distance to the wall and dividing it by the speed in the direction to the wall ifelse x-speed != 0 [set t-plane-xpos (dpxpos / x-speed)] [set t-plane-xpos 0] if t-plane-xpos > 0 [ assign-colliding-wall t-plane-xpos "plane-xpos" ] let t-plane-xneg 0 ifelse x-speed != 0 [set t-plane-xneg (dpxneg / x-speed)] [set t-plane-xneg 0] if t-plane-xneg > 0 [ assign-colliding-wall t-plane-xneg "plane-xneg" ] let t-plane-ypos 0 ifelse y-speed != 0 [set t-plane-ypos (dpypos / y-speed)] [set t-plane-ypos 0] if t-plane-ypos > 0 [ assign-colliding-wall t-plane-ypos "plane-ypos" ] let t-plane-yneg 0 ifelse y-speed != 0 [set t-plane-yneg (dpyneg / y-speed)] [set t-plane-yneg 0] if t-plane-yneg > 0 [ assign-colliding-wall t-plane-yneg "plane-yneg" ] end to assign-colliding-wall [time-to-collision wall] ;; this procedure is used by the check-for-wall-collision procedure ;; to assemble the correct particle-wall pair ;; time-to-collision is relative (ie, a collision will occur one second from now) ;; We need to store the absolute time (ie, a collision will occur at time 48.5 seconds. ;; So, we add clock to time-to-collision when we store it. let colliding-pair (list (time-to-collision + ticks) self wall) set colliding-particles fput colliding-pair colliding-particles end to sort-collisions ;; Slight errors in floating point math can cause a collision that just ;; happened to be calculated as happening again a very tiny amount of ;; time into the future, so we remove any collisions that involves ;; the same two particles (or particle and wall) as last time. set colliding-particles filter [item 1 ? != colliding-particle-1 or item 2 ? != colliding-particle-2] colliding-particles set colliding-particle-1 nobody set colliding-particle-2 nobody set tick-length original-tick-length if colliding-particles = [] [ stop ] ;; Sort the list of projected collisions between all the particles into an ordered list. ;; Take the smallest time-step from the list (which represents the next collision that will ;; happen in time). Use this time step as the tick-length for all the particles to move through let winner first colliding-particles foreach colliding-particles [if first ? < first winner [set winner ?]] ;;winner is now the collision that will occur next let dt item 0 winner if dt > 0 [ ;;If the next collision is more than 1 in the future, ;;clear the winners and advance the simulation one tick. ;;This helps smooth the model on smaller particle counts. ifelse dt - ticks <= 1 ;;We have to subtract clock back out because now we want the relative time until collision, ;;not the absolute time the collision will occur. [set tick-length dt - ticks set colliding-particle-1 item 1 winner set colliding-particle-2 item 2 winner] ;;Since there are no collisions in the next second, we will set winners to [] to keep from ;;mistakenly colliding any particles that shouldn't collide yet. [set tick-length 1] ] end to collide-winners ;; deal with 3 possible cases of collisions: ;; particle and one wall, particle and two walls, and two particles if colliding-particle-1 = nobody [ stop ] ;; deal with a case where the next collision in time is between a particle and a wall if colliding-particle-2 = "plane-xpos" or colliding-particle-2 = "plane-xneg" [ask colliding-particle-1 [set heading (- heading)] stop] if colliding-particle-2 = "plane-ypos" or colliding-particle-2 = "plane-yneg" [ask colliding-particle-1 [set heading (180 - heading)] stop] ;; deal with the remaining case of the next collision in time being between two particles. ask colliding-particle-1 [collide-with colliding-particle-2] end to collide-with [ other-particle ] ;; particle procedure ;;; PHASE 1: initial setup ;; for convenience, grab some quantities from other-particle let mass2 [mass] of other-particle let speed2 [speed] of other-particle let heading2 [heading] of other-particle ;;modified so that theta is heading toward other particle let theta towards other-particle ;;; PHASE 2: convert velocities to theta-based vector representation ;; now convert my velocity from speed/heading representation to components ;; along theta and perpendicular to theta let v1t (speed * cos (theta - heading)) let v1l (speed * sin (theta - heading)) ;; do the same for other-particle let v2t (speed2 * cos (theta - heading2)) let v2l (speed2 * sin (theta - heading2)) ;;; PHASE 3: manipulate vectors to implement collision ;; compute the velocity of the system's center of mass along theta let vcm (((mass * v1t) + (mass2 * v2t)) / (mass + mass2) ) ;; now compute the new velocity for each particle along direction theta. ;; velocity perpendicular to theta is unaffected by a collision along theta, ;; so the next two lines actually implement the collision itself, in the ;; sense that the effects of the collision are exactly the following changes ;; in particle velocity. set v1t (2 * vcm - v1t) set v2t (2 * vcm - v2t) ;;; PHASE 4: convert back to normal speed/heading ;; now convert my velocity vector into my new speed and heading set speed sqrt ((v1t * v1t) + (v1l * v1l)) ;; if the magnitude of the velocity vector is 0, atan is undefined. but ;; speed will be 0, so heading is irrelevant anyway. therefore, in that ;; case we'll just leave it unmodified. if v1l != 0 or v1t != 0 [ set heading (theta - (atan v1l v1t)) ] ;; and do the same for other-particle ask other-particle [ set speed sqrt (((v2t * v2t) + (v2l * v2l))) ] if v2l != 0 or v2t != 0 [ ask other-particle [ set heading (theta - (atan v2l v2t)) ] ] ;; PHASE 5: final updates ;; now recolor, since color is based on quantities that may have changed recolor ask other-particle [ recolor ] end to recolor if color-scheme = "red-green-blue" [ recolor-banded ] if color-scheme = "blue shades" [ recolor-shaded ] if color-scheme = "one color" [ recolor-none ] end to recolor-banded ;; particle procedure let avg-speed 1 ;; avg-speed is assumed to be 0.5, since particles are assigned a random speed between 0 and 1 ;; particle coloring procedures for visualizing speed with a color palette, ;; red are fast particles, blue slow, and green in between. ifelse speed < (0.5 * avg-speed) ;; at lower than 50% the average speed [ set color blue ;; slow particles colored blue ] [ ifelse speed > (1.5 * avg-speed) ;; above 50% higher the average speed [ set color red ] ;; fast particles colored blue [ set color green ] ;; medium speed particles colored green ] end to recolor-shaded let avg-speed 1 ;; avg-speed is assumed to be 0.5, since particles are assigned a random speed between 0 and 1 ;; a particle shading gradient is applied to all particles less than speed 1.5, ;; the uppermost threshold speed to apply the shading gradient to. ifelse speed < (3 * avg-speed) [ set color (sky - 3.001) + (8 * speed / (3 * avg-speed)) ] [ set color (sky + 4.999)] end to recolor-none set color green - 1 end ;;; ;;; drawing procedures ;;; to make-box ask patches with [ ((abs pxcor = box-edge) and (abs pycor <= box-edge)) or ((abs pycor = box-edge) and (abs pxcor <= box-edge)) ] [ set pcolor yellow ] end ;; creates some particles to make-particles create-ordered-particles number [ set speed 1 set size smallest-particle-size + random-float (largest-particle-size - smallest-particle-size) set mass (size * size) ;; set the mass proportional to the area of the particle recolor set heading random-float 360 ] arrange particles end ;; If the number of particles requested by the user won't fit in the box, ;; this code will go into an infinite loop. To work around a NetLogo bug ;; where some kinds of infinite loops cannot be halted by Tools->Halt, ;; we put this code in a separate procedure and write it a certain way. ;; (It's necessary for the loop not to be within an ASK. The reason ;; for this is very obscure. We plan to fix the problem in a future ;; NetLogo version.) to arrange [particle-set] if not any? particle-set [ stop ] ask particle-set [ random-position ] arrange particle-set with [overlapping?] end to-report overlapping? ;; particle procedure ;; here, we use in-radius just for improved speed report any? other particles in-radius ((size + largest-particle-size) / 2) with [distance myself < (size + [size] of myself) / 2] end ;; place particle at random location inside the box. to random-position ;; particle procedure setxy one-of [1 -1] * random-float (box-edge - 0.5 - size / 2) one-of [1 -1] * random-float (box-edge - 0.5 - size / 2) end ;;; ;;; procedure for reversing time ;;; to reverse-time ask particles [ rt 180 ] rebuild-collision-list ;; the last collision that happened before the model was paused ;; (if the model was paused immediately after a collision) ;; won't happen again after time is reversed because of the ;; "don't do the same collision twice in a row" rule. We could ;; try to fool that rule by setting colliding-particle-1 and ;; colliding-particle-2 to nobody, but that might not always work, ;; because the vagaries of floating point math means that the ;; collision might be calculated to be slightly in the past ;; (the past that used to be the future!) and be skipped. ;; So to be sure, we force the collision to happen: collide-winners end ;; Here's a procedure that demonstrates time-reversing the model. ;; You can run it from the command center. When it finishes, ;; the final particle positions may be slightly different because ;; the amount of time that passes after the reversal might not ;; be exactly the same as the amount that passed before; this ;; doesn't indicate a bug in the model. ;; For larger values of n, you will start to notice larger ;; discrepancies, eventually causing the behavior of the system ;; to diverge totally. Unless the model has some bug we don't know ;; about, this is due to accumulating tiny inaccuracies in the ;; floating point calculations. Once these inaccuracies accumulate ;; to the point that a collision is missed or an extra collision ;; happens, after that the reversed model will diverge rapidly. to test-time-reversal [n] setup ask particles [ stamp ] while [ticks < n] [ go ] let old-clock ticks reverse-time while [ticks < 2 * old-clock] [ go ] ask particles [ set color white ] end

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