3f:null 40:["$","title",null,{"children":"billard, work and heat | xdze2 | Observable"}] 46:T583,md`# billard, work and heat A hard sphere model for a 2d gas, i.e. a friction-less billard. It would be possible to obtain the relation between the temperature, the pressure and the volume (i.e. the ideal gas law). Original method proposed in 1960 by B. J. Alder and T. E. Wainwright : [Studies in Molecular Dynamics. I. General Method](http://pbx-brasil.com/outrasDisciplinas/DinMol/Notas/IIarea/aula201/papers/JChemPhys_31_459.pdf). > The calculation was first set up for the _Univac computer_. The small capacity of the high-speed memory of this machine limited the system which could be efficiently handled to about 100 rigid spheres. With this number of molecules, roughly 100 collisions (that is, two mean collision times) could be calculated in one hour. The _IBM 704 calculator_ handles about 2000 collisions per hour for 100 molecules and about 500 collisions per hour for 500 molecules. The time of the next collision is computed by a geometric reasoning. The evolution of the system between two collisions is straightforward, because the velocities are constants between each collision. No attrative force is used here. Therefore only low density gas can be modeled. In the original work, a square potential is used, increasing the number of computation, but could model liquid state. Nowadays, molecular dynamic simulation uses continuous attractive and repulsive forces between particles. `47:T556,Particle = { // Constructor function Particle(position, vitesse, radius, mass) { this.x = new Vector( position ); this.v = new Vector( vitesse ); this.radius = radius; this.mass = mass; } /* Compute the time until collision with another particle * return `Infinity` if no collision */ Particle.prototype.timeToCollide = function( that ){ const V = this.v.minus( that.v ) const X = this.x.minus( that.x ) const vx = V.dot( X ) let t = Infinity if( vx < 0 ){ // particles are getting closer const v2 = V.dot(V) const x2 = X.dot(X) const r2 = Math.pow( this.radius + that.radius, 2 ) const vx2 = vx*vx; if( x2-r2>0 ){ // non contact const discriminant = vx2 - v2*(x2-r2) if(discriminant > 0){ t = ( -vx - Math.sqrt( discriminant ) )/v2 } } } return t } /* Compute the time until a collision with a Wall */ Particle.prototype.timeToCollideTheWall = function( wall ){ const nx = wall.n.dot( this.x ) let t = Infinity if( nx - wall.nx > this.radius ){ const nv = wall.n.dot( this.v ) if( nv < 0 ){ t = (this.radius - nx + wall.nx)/nv } } return t } /* Move the particle for the time dt */ Particle.prototype.moveTo = function( dt ){ this.x = this.x.plus( this.v.time( dt ) ) } return Particle }48:T45b,vizu = { loop; const height = 300 const margin = 10 const scale = Math.min( (height-2*margin)/Ly, (width-2*margin)/Lx ) const velocityScale = 60 const svg = d3.select(DOM.svg(width, height)); svg.append('rect') .attr('x', margin) .attr('y', margin) .attr('width', Lx*scale) .attr('height', Ly*scale) .attr('fill', 'none' ) .attr('stroke', 'black') svg.selectAll('circle') .data(mutable particles).enter() .append("circle") .attr("r", p => scale * p.radius) .attr("fill", 'red') .attr("stroke", 'none') .attr('cx', p => margin + scale*p.x.coords[0]) .attr('cy', p => margin + scale*p.x.coords[1]) svg.selectAll('line') .data(mutable particles).enter() .append("line") .attr("stroke", 'black') .attr('x1', p => margin + scale*p.x.coords[0]) .attr('y1', p => margin + scale*p.x.coords[1]) .attr('x2', p => margin + scale*p.x.coords[0] + velocityScale*p.v.coords[0]) .attr('y2', p => margin + scale*p.x.coords[1] + velocityScale*p.v.coords[1]) return svg.node() }49:Tebb,loop = { /* Main loop of the computation * * The 2 steps in the loop are: * 1. Compute all the collision time, * or update the table only for particles which just collide * * 2. Perform the first collision to happen by moving all the particles * and computing the new velocities for the two collided particles */ /* -- Initialization -- */ // -debug- //let particles = [ new Particle( [2, 2], [1, 0], .5, 1 ), // new Particle( [6, 2.4], [0, 0], .5, 1 )] let particles = mutable particles let timeTableForWalls = computeAllTimeForWalls( walls, particles ) let timeTableParticles = computeAllTime( particles ) let nextWallCollision = findNextCollision( timeTableForWalls ) let nextParticleCollision = findNextCollision( timeTableParticles ) let update = [] /* -- loop -- */ //for (let k=0; k<1200; k++) { while(run){ mutable numberOfCollision ++ // Compute the next collision time by updating the timeTable: if(update.length>0){ for(const i of update){ // Collision between particles: for(let j=0; j60){ copy_lnwc.shift() } mutable lastNwallcollisions = copy_lnwc } else { // ... between particles // 1. move all the particles for (const p of particles) { p.moveTo( nextParticleCollision.time ) } // 2. change the velocities collideWithParticle( particles[ nextParticleCollision.indices[0]+1 ], //<- offset semi matrix particles[ nextParticleCollision.indices[1] ] ) time = nextParticleCollision.time update = nextWallCollision.indices } const timescale = 100 mutable particles = particles yield Promises.delay(timescale*time, true) } }4a:[] 44:{"id":"5cb593818a082d03","slug":"billard-work-and-heat","title":"billard, work and 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*/\nVector.prototype.dot = function( that ){\n\treturn this.x*that.x + this.y*that.y\n}\n/* Scalar product with a scalar a */\nVector.prototype.time = function( scalar ){\n\treturn new Vector( [scalar*this.x, scalar*this.y] )\n}\n/* Addition with a vector v */\nVector.prototype.plus = function( that ){\n\treturn new Vector( [ this.x+that.x, this.y+that.y ] )\n}\n/* Soustraction: this - that */\nVector.prototype.minus = function( that ){\n\treturn new Vector( [ this.x-that.x, this.y-that.y ] )\n}\n/* return the unit vector */\nVector.prototype.normed = function(){\n \tconst n = Math.sqrt( this.dot( this ) ) \n\treturn this.time( 1/n ) \n}\n\nreturn Vector\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2,"value":"$47","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":628,"value":"/* Perform a collision with a wall\n * change the velocity of the particle\n */\ncollideWithWall = function( particule, wall ){\n\tconst nv = wall.n.dot( particule.v )\n const dirv = wall.dir.dot( particule.v )\n \n particule.v = wall.n.time( -nv ).plus( wall.dir.time(dirv) )\n \n \tconst momentum = -2*particule.mass*nv\n \treturn momentum\n} ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":634,"value":"/* Perform a collision between particles A and B\n* change the velocity of each particle\n*/\ncollideWithParticle = function( particleA, particleB ){\n const reducedmass = 2*particleB.mass*particleA.mass/( particleA.mass + particleB.mass )\n const k = particleB.x.minus( particleA.x ).normed()\n const u = particleB.v.minus( particleA.v )\n \n const alpha = reducedmass*k.dot(u)\n \n particleA.v = particleA.v.plus( k.time( alpha/particleA.mass ) )\n particleB.v = particleB.v.minus( k.time( alpha/particleB.mass ) )\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":162,"value":"Wall = {\n\n/* Constructor for a Wall object \n orientation convention : x1 to x2 inside on the left (anti-clockwise rotation)\n \n x is th position of the wall\n u is the direction of the wall\n */\nfunction Wall( x, u ) {\n this.x = new Vector( x )\n this.u = new Vector( u )\n \n \tconst normU = Math.sqrt( this.u.dot( this.u ) )\n \tthis.n = new Vector( [-u[1]/normU, u[0]/normU] ) // [2d] normed normal vector\n\tthis.dir = new Vector( [u[0]/normU, u[1]/normU] ) // [2d] tangent vector\n \n\tthis.nx = this.n.dot( this.x ) // used to compute the collision with a particle\n}\n\nreturn Wall\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":168,"value":"/* initiate particules on a regular lattice */\nmutable particles = {\n const n = 6 // n*n particles\n const a = .6 // spacing\n const radius = 0.2\n const mass = 1\n const v0 = .4323\n \n let particles = []\n for(let i=0; i ({index:j, time:x }) ) , x=>x.time )\n }\n\tconst minOnRow = timeTable.map( argminRow )\n \treturn _.minBy( minOnRow.map( (jt, u) => ({indices:[u, jt.index], time:jt.time }) ) , x=>x.time )\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":208,"value":"computeAllTime = function( particles ){\n /* Compute for each pair of particles the time to the next collision\n * return a half matrix (without the diagonal, size n-1)\n */\n\tlet timeTable = []\n \n // lower half matrix, index i is offset by one, i.e. no diagonal\n for(let u=0; u 0.5*p.mass*p.v.dot( p.v ) ) )","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":789,"value":"viewof temperature = html`${_.mean( particles.map( p=> 0.5*p.mass*p.v.dot( p.v ) ) )} (average kinetic energy of all the particles)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":859,"value":"viewof run = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":180,"value":"$48","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":812,"value":"mutable numberOfCollision = 0","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":960,"value":"mutable lastNwallcollisions = []","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":974,"value":"pressure = _.sum( lastNwallcollisions.map( c => c[1] ) )/_.sum( lastNwallcollisions.map( c => c[0] ) )/2/(Lx+Ly)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":286,"value":"$49","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":185,"value":"md`## Require`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":16,"value":"_ = require('lodash')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":183,"value":"d3 = require(\"https://d3js.org/d3.v5.min.js\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":871,"value":"import {button} from \"@jashkenas/inputs\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W4a","stdlib":"1"}} 45:"$44:body"