{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 16 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier " 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot " -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 64 "Classical kinetic theory \+ for evaporative cooling in an atom trap" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1979 "Classical statistical mechanics al lows one to describe an ensemble of atoms which move in an external po tential, and which interact with each other by means of elastic collis ions. During these binary collisions energy is exchanged between the a toms. The common motion of many non-interacting particles in an extern al field is described by the classical Liouville equation, which repre sents a reformulation of the many Newton equations in terms of a singl e phase-space distribution functions. When an elastic collision integr al is added to this equation it becomes the Boltzmann equation. The Li ouville or Boltzmann equations are formulated for continuous distribut ion functions, but practical realizations are often performed by retur ning to discrete testparticle equations. These are Newton (Hamilton) e quations for particles moving in a common potential - which may includ e a long-distance interaction between the particles - supplemented by \+ a collision term which represents the elastic exchange of energy among particle pairs. The Boltzmann equation is interesting insofar as it d isplays irreversible behaviour caused by the collision term: the Liouv ille dynamics is based on Newton's equations which are symmetric under a reversal of time; the collision integral, however, introduces chang es to the particle motions such that different initial distributions e volve towards Maxwell-Boltzmann type equilibrium distributions. Obviou sly, it cannot work in reverse, i.e., it cannot undo the relaxations a nd turn an equilibrium distribution (valid at large times) back into t he original distribution by going backwards in time (the non-uniquenes s alone would prohibit that). In that sense it contains the arrow of t ime as observed in nature (2nd law of thermodynamics), but at the same time it remains a heuristic equation. For a discussion of this and ot her features it is worthwhile to consult texts on statistical mechanic s, a good starting point would be: Gregory H. Wannier: " }{TEXT 258 19 "Statistical Physics" }{TEXT -1 58 " (paperback reprint by Dover fr om the 1966 Wiley edition)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 552 "This kinetic motion approach allows one to und erstand how a sample acquires an equilibrium state through energy exch ange by elastic collisions. We can start the atoms all with equal velo city, or we can equidistribute the energies initially over all possibl e states: when we wait long enough an equilibrium state will develop w hich is characterized by the average kinetic energy available to the e nsemble. The steady-state or equilibrium distribution produced by the \+ elastic collisions between the atoms is the well-known Maxwell-Boltzma nn distribution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1038 "Collisions have the tendency to turn a pair of two part icles of similar-energy into a pair in which one partner is slow, and \+ one is faster. In the present worksheet this is demonstrated by simula ting evaporative cooling. The idea is to remove the fast particles (in evaporative cooling in the coffee or tea cup the hottest molecules le ave the fluid in the form of steam, thereby leaving an ensemble which \+ is colder on average). In an atom trap one can have a finite barrier h eight which separates the trap from the environment. The idea is that \+ once an atom has acquired enough energy to go over the barrier it is g one for good. If we keep the barrier height fixed, the cooling will wo rk up to some point: once the whole ensemble has cooled off such that \+ the average kinetic energy is well below the barrier height, it is unl ikely that further eleastic collisions will produce many more atoms of sufficient energy to make it over the barrier. Therefore, efficient c ooling requires a lowering of the trap height as the ensemble cools do wn." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 640 "O f course we cannot integrate the full Boltzmann equation in Maple for \+ a three-dimensional system. In fact, a substantial simplification is p ossible when the system is ergodic. This means that the system visits \+ all energetically accessible points in phase space if given sufficient time, and that different sequences of the time history selected at ra ndom look similar to each other. In such a system it is not important \+ to know (in order to characterize the state) how the convection of the particle flow in phase space looks in detail. One can characterize th e system by its energy dependence alone. The phase-space distribution \+ function " }{TEXT 19 8 "f(r,p,t)" }{TEXT -1 51 " is replaced by the pr oduct of a density of states " }{TEXT 19 6 "rho(E)" }{TEXT -1 28 " tim es the level population " }{TEXT 19 4 "f(E)" }{TEXT -1 24 ". The densi ty of states " }{TEXT 19 6 "rho(E)" }{TEXT -1 242 " is usually a simpl e function of the potential. Commonly one employes the technique of co unting the allowed quantum energy levels (taking into account their de generacy in a multi-dimensional setting such as the spherical harmonic oscillator)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 265 "The details of reducing the six-dimensional phase-space \+ Boltzmann equation to a one-dimensional energy-space equation are give n, e.g., in the paper by J. Walraven et al., Phys. Rev. A 53, 381(1996 ). The idea is to discretize the energy range into bins according to \+ " }{TEXT 19 12 "E=(i-1/2)*dE" }{TEXT -1 6 " with " }{TEXT 19 6 "i=1..N " }{TEXT -1 131 " covering predominantly the energy range from the tra p bottom up to the barrier height. The time evolution of the level pop ulation " }{TEXT 19 10 "f_i=f(E_i)" }{TEXT -1 113 " is determined by a collision term which involves the product of level populations for tw o arbitrary states with " }{TEXT 19 3 "E_k" }{TEXT -1 5 " and " } {TEXT 19 3 "E_l" }{TEXT -1 26 " to connect to the states " }{TEXT 19 3 "E_i" }{TEXT -1 15 " combined with " }{TEXT 19 3 "E_j" }{TEXT -1 37 ". While determining the evolution of " }{TEXT 19 6 "f(E_i)" }{TEXT -1 50 " one sums over all possible k,l while restricting " }{TEXT 19 15 "E_j=E_k+E_l-E_i" }{TEXT -1 275 ", thus ensuring energy conservatio n. The collision term is weighted with the probability for collisions \+ to take place (the cross section which is assumed to be energy-indepen dent at low energies enters). The equation is also weighted by the den sity of states: on the left by " }{TEXT 19 5 "rho_i" }{TEXT -1 22 ", a nd on the right by " }{TEXT 19 5 "rho_h" }{TEXT -1 25 ", where the ene rgy index " }{TEXT 19 14 "h=min(i,j,k,l)" }{TEXT -1 241 ". The selecti on of the energy index on the right follows from the reduction of the \+ collision integral when the phase-space probability is replaced by the energy-level population. Schematically we arrive at a system of diffe rential equations:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 19 72 "rho_i diff(f_i,t) = c*add(rho_h*(f_k*f_l - f_i*f_j) , k=1 ..N ,l=1..N) " }{TEXT 2 5 "where" }{TEXT 19 25 " h=min(i,j,k,l), j= k+l-i" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 " Here the level population " }{TEXT 19 3 "f_i" }{TEXT -1 74 " is dimens ionless and the constant c contains the dimensionful quantities " } {TEXT 19 29 "c=m*sigma*dE^2/(Pi^2*hbar^3) " }{TEXT -1 10 "such that " }{TEXT 19 4 "c*dt" }{TEXT -1 83 " represents a measure of the total nu mber of collisions that occured. Furthermore, " }{TEXT 19 16 "rho_i = \+ rho(E_i)" }{TEXT -1 109 " is usually a power-law density of states, i .e., one can show that for the isotropic 3D harmonic oscillator " } {TEXT 19 5 "rho_i" }{TEXT -1 11 " goes like " }{TEXT 19 5 "E_i^2" } {TEXT -1 36 ", since more generally it goes like " }{TEXT 19 11 "E_i^( 1/2+d)" }{TEXT -1 24 " for potentials of form " }{TEXT 19 12 "V(r)=a*r ^3/d" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 58 "This system of differential equations can be solved usi ng " }{TEXT 19 15 "dsolve[numeric]" }{TEXT -1 289 ", but it is more st raightforward to apply an extrapolation (Euler) method, i.e., to discr etize time. To illustrate the method we assume that the energy scale i s set by the trap depth (in our scaled dimensionless variables we dist ribute intially all particles equally over the energy range " }{TEXT 19 6 "E=0..1" }{TEXT -1 77 "). At first we will not remove particles, \+ i.e., we start the simulation with " }{TEXT 19 3 "f_i" }{TEXT -1 41 " \+ representing a step function cut off at " }{TEXT 19 3 "E=1" }{TEXT -1 60 ", and just watch the evolution of this distribution in time." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }} {PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r edefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "We introduce parame ters for the dimensionless trap height, the total number of energy bin s " }{TEXT 19 1 "N" }{TEXT -1 78 ", and the number of energy bins devo ted to the range above the barrier height " }{TEXT 19 2 "N0" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "E_t:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$E_tG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "N:=40;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "N0:=10;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#N0G\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dE:=E_t/(N-N0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dEG#\"\"\"\" #I" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 162 "The bin energies and the d ensity of states are now specified. We begin by choosing the density o f states as appropriate for a harmonic oscillator in 3 dimensions." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "E:=i->(i-0.5)*dE;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EGf*6#%\"iG6\"6$%)operatorG%&arrowGF(*&, &9$\"\"\"$\"\"&!\"\"F2F/%#dEGF/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "rho:=i->E(i)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $rhoGf*6#%\"iG6\"6$%)operatorG%&arrowGF(*$)-%\"EG6#9$\"\"#\"\"\"F(F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot([seq([E(i),rho(i)] ,i=1..N)]);" }}{PARA 13 "" 1 "" {GLPLOT2D 791 299 299 {PLOTDATA 2 "6%- %'CURVESG6$7J7$$\"3-+++nmmm;!#>$\"3)*******yxxxF!#@7$$\"3G+++++++]F*$ \"31+++++++D!#?7$$\"3^*****HLLLL)F*$\"35+++WWWWpF37$$\"3/+++nmmm6!#=$ \"3++++766h8F*7$$\"3%**************\\\"F<$\"3#*************\\AF*7$$\"3 7+++LLLL=F<$\"3/+++566hLF*7$$\"35+++nmmm@F<$\"3%)*****fWWWp%F*7$$\"3++ ++++++DF<$\"3+++++++]iF*7$$\"3!******HLLL$GF<$\"3?+++wxxF!)F*7$$\"3))* ****pmmm;$F<$\"3&******zxxF+\"F<7$$\"3w*************\\$F<$\"3)******** ****\\A\"F<7$$\"3B+++LLLLQF<$\"37+++WWWp9F<7$$\"3A+++nmmmTF<$\"3$***** *466ht\"F<7$$\"35+++++++XF<$\"39++++++D?F<7$$\"3,+++LLLL[F<$\"3\"***** *466hL#F<7$$\"3a+++nmmm^F<$\"3))*****\\WW%pEF<7$$\"3U+++++++bF<$\"3\"* ***********\\-$F<7$$\"3M+++LLLLeF<$\"31+++xxx-MF<7$$\"3K+++nmmmhF<$\"3 %******zxxF!QF<7$$\"3A+++++++lF<$\"3()***********\\A%F<7$$\"37+++LLLLo F<$\"3#******RWW%pYF<7$$\"35+++nmmmrF<$\"3D+++766O^F<7$$\"3++++++++vF< $\"3+++++++DcF<7$$\"3!******HLLL$yF<$\"3&******466h8'F<7$$\"3))*****pm mm;)F<$\"37+++XWWpmF<7$$\"3w*************\\)F<$\"3J++++++DsF<7$$\"3o** ***HLLL$))F<$\"3`*****pxxF!yF<7$$\"3m*****pmmm;*F<$\"39+++yxx-%)F<7$$ \"3a*************\\*F<$\"3p***********\\-*F<7$$\"3X*****HLLL$)*F<$\"3[ +++WWWp'*F<7$$\"3(******pmmm,\"!#<$\"3++++76hL5F^u7$$\"3/++++++]5F^u$ \"3/+++++]-6F^u7$$\"3!******HLLL3\"F^u$\"3'*******46ht6F^u7$$\"31+++nm m;6F^u$\"3-+++XW%pC\"F^u7$$\"3#*************\\6F^u$\"3,+++++]A8F^u7$$ \"3)******HLLL=\"F^u$\"3*)*****pxx-S\"F^u7$$\"3#******pmmm@\"F^u$\"3!* ******yxF![\"F^u7$$\"3+++++++]7F^u$\"3++++++]i:F^u7$$\"33+++LLL$G\"F^u $\"31+++WW%pk\"F^u7$$\"3,+++nmm;8F^u$\"3'******>66Ot\"F^u-%'COLOURG6&% $RGBG$\"#5!\"\"$\"\"!FfxFex-%+AXESLABELSG6$Q!6\"Fjx-%%VIEWG6$%(DEFAULT GF_y" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "This is the level density of a potential that continues harmonically beyond " }{TEXT 19 3 "E=1" } {TEXT -1 60 ". We choose all energy levels to be equally populated bel ow " }{TEXT 19 3 "E=1" }{TEXT -1 24 ", and to be empty above." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "fv:=array([seq(1,i=1..N-N0), seq(0,i=N-N0+1..N)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fvG-%'vect orG6#7J\"\"\"F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F )\"\"!F*F*F*F*F*F*F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 173 "The \+ distribution over the energies has to take the density of states into \+ account. Note that we should really be displaying a histogram - theref ore the step is not sharp at " }{TEXT 19 3 "E=1" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "PL0:=plot([seq([E(i),(rho(i) *fv[i])],i=1..N)]): display(PL0);" }}{PARA 13 "" 1 "" {GLPLOT2D 840 299 299 {PLOTDATA 2 "6%-%'CURVESG6$7J7$$\"3-+++nmmm;!#>$\"3)*******yxx xF!#@7$$\"3G+++++++]F*$\"31+++++++D!#?7$$\"3^*****HLLLL)F*$\"35+++WWWW pF37$$\"3/+++nmmm6!#=$\"3++++766h8F*7$$\"3%**************\\\"F<$\"3#** ***********\\AF*7$$\"37+++LLLL=F<$\"3/+++566hLF*7$$\"35+++nmmm@F<$\"3% )*****fWWWp%F*7$$\"3++++++++DF<$\"3+++++++]iF*7$$\"3!******HLLL$GF<$\" 3?+++wxxF!)F*7$$\"3))*****pmmm;$F<$\"3&******zxxF+\"F<7$$\"3w********* ****\\$F<$\"3)************\\A\"F<7$$\"3B+++LLLLQF<$\"37+++WWWp9F<7$$\" 3A+++nmmmTF<$\"3$******466ht\"F<7$$\"35+++++++XF<$\"39++++++D?F<7$$\"3 ,+++LLLL[F<$\"3\"******466hL#F<7$$\"3a+++nmmm^F<$\"3))*****\\WW%pEF<7$ $\"3U+++++++bF<$\"3\"************\\-$F<7$$\"3M+++LLLLeF<$\"31+++xxx-MF <7$$\"3K+++nmmmhF<$\"3%******zxxF!QF<7$$\"3A+++++++lF<$\"3()********** *\\A%F<7$$\"37+++LLLLoF<$\"3#******RWW%pYF<7$$\"35+++nmmmrF<$\"3D+++76 6O^F<7$$\"3++++++++vF<$\"3+++++++DcF<7$$\"3!******HLLL$yF<$\"3&******4 66h8'F<7$$\"3))*****pmmm;)F<$\"37+++XWWpmF<7$$\"3w*************\\)F<$ \"3J++++++DsF<7$$\"3o*****HLLL$))F<$\"3`*****pxxF!yF<7$$\"3m*****pmmm; *F<$\"39+++yxx-%)F<7$$\"3a*************\\*F<$\"3p***********\\-*F<7$$ \"3X*****HLLL$)*F<$\"3[+++WWWp'*F<7$$\"3(******pmmm,\"!#<$\"\"!F`u7$$ \"3/++++++]5F^uF_u7$$\"3!******HLLL3\"F^uF_u7$$\"31+++nmm;6F^uF_u7$$\" 3#*************\\6F^uF_u7$$\"3)******HLLL=\"F^uF_u7$$\"3#******pmmm@\" F^uF_u7$$\"3+++++++]7F^uF_u7$$\"33+++LLL$G\"F^uF_u7$$\"3,+++nmm;8F^uF_ u-%'COLOURG6&%$RGBG$\"#5!\"\"F_uF_u-%+AXESLABELSG6$Q!6\"Ffw-%%VIEWG6$% (DEFAULTGF[x" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "The product of " } {TEXT 19 4 "c*dt" }{TEXT -1 116 " has to be small for Euler's extrapol ation method to work. The algorithm updates the level population accor ding to: " }{TEXT 19 33 "f_i(t+dt) = f_i(t) + dt*(df_i/dt)" }{TEXT -1 67 ". Implementation details: the sums on the right are truncated when " }{TEXT 19 1 "j" }{TEXT -1 176 " becomes negative or exceeds the ene rgy range. The update loop at the end is easily modified to set the di stribution to zero for energies above the half-way point, i.e., above \+ " }{TEXT 19 3 "E=1" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "cdt:=1/1000;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$cd tG#\"\"\"\"%+5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "N_col:=10 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&N_colG\"$+\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "for icol from 1 to N_col do:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for i from 1 to N do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "RHS:=0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " for k from 1 to N do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for l from 1 to N do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "j:=k+l-i:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "if j>0 and j " 0 "" {MPLTEXT 1 0 66 "h:=min(i,j,k,l): RHS:=RHS+rho(h)/rho(i)*(fv[k]*fv[l]- fv[i]*fv[j]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 7 "od: od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "i f i<=N then fv[i]:=fv[i]+cdt*RHS: else fv[i]:=0: fi: od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "if trunc(icol/10)*10=icol then PL[trunc(icol /10)]:=plot([seq([E(i),(rho(i)*fv[i])],i=1..N)],title=cat(\"Collision \+ number : \",(N/2)*icol*cdt/dE)): fi: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "display([PL0,seq(PL[i],i=1..N_col/10)],insequence=tru e);" }}{PARA 13 "" 1 "" {GLPLOT2D 666 298 298 {PLOTDATA 2 "6#-%(ANIMAT EG6-7%-%'CURVESG6$7J7$$\"3-+++nmmm;!#>$\"3)*******yxxxF!#@7$$\"3G+++++ ++]F.$\"31+++++++D!#?7$$\"3^*****HLLLL)F.$\"35+++WWWWpF77$$\"3/+++nmmm 6!#=$\"3++++766h8F.7$$\"3%**************\\\"F@$\"3#*************\\AF.7 $$\"37+++LLLL=F@$\"3/+++566hLF.7$$\"35+++nmmm@F@$\"3%)*****fWWWp%F.7$$ \"3++++++++DF@$\"3+++++++]iF.7$$\"3!******HLLL$GF@$\"3?+++wxxF!)F.7$$ \"3))*****pmmm;$F@$\"3&******zxxF+\"F@7$$\"3w*************\\$F@$\"3)** **********\\A\"F@7$$\"3B+++LLLLQF@$\"37+++WWWp9F@7$$\"3A+++nmmmTF@$\"3 $******466ht\"F@7$$\"35+++++++XF@$\"39++++++D?F@7$$\"3,+++LLLL[F@$\"3 \"******466hL#F@7$$\"3a+++nmmm^F@$\"3))*****\\WW%pEF@7$$\"3U+++++++bF@ $\"3\"************\\-$F@7$$\"3M+++LLLLeF@$\"31+++xxx-MF@7$$\"3K+++nmmm hF@$\"3%******zxxF!QF@7$$\"3A+++++++lF@$\"3()***********\\A%F@7$$\"37+ ++LLLLoF@$\"3#******RWW%pYF@7$$\"35+++nmmmrF@$\"3D+++766O^F@7$$\"3++++ ++++vF@$\"3+++++++DcF@7$$\"3!******HLLL$yF@$\"3&******466h8'F@7$$\"3)) *****pmmm;)F@$\"37+++XWWpmF@7$$\"3w*************\\)F@$\"3J++++++DsF@7$ $\"3o*****HLLL$))F@$\"3`*****pxxF!yF@7$$\"3m*****pmmm;*F@$\"39+++yxx-% )F@7$$\"3a*************\\*F@$\"3p***********\\-*F@7$$\"3X*****HLLL$)*F @$\"3[+++WWWp'*F@7$$\"3(******pmmm,\"!#<$\"\"!Fdu7$$\"3/++++++]5FbuFcu 7$$\"3!******HLLL3\"FbuFcu7$$\"31+++nmm;6FbuFcu7$$\"3#*************\\6 FbuFcu7$$\"3)******HLLL=\"FbuFcu7$$\"3#******pmmm@\"FbuFcu7$$\"3++++++ +]7FbuFcu7$$\"33+++LLL$G\"FbuFcu7$$\"3,+++nmm;8FbuFcu-%'COLOURG6&%$RGB G$\"#5!\"\"FcuFcu-%+AXESLABELSG6$Q!6\"Fjw-%%VIEWG6$%(DEFAULTGF_x7&-F(6 $7J7$F,$\"3#******H(GOV5F77$F3$\"39*****z'QT(y)F77$F9$\"31+++Mmt'G#F.7 $F>$\"3-+++1^+&>%F.7$FD$\"3m******HzM$['F.7$FI$\"3R*****>yMX/*F.7$FN$ \"3'******>5&Ry6F@7$FS$\"35+++?p!>Y\"F@7$FX$\"3!******fZyyu\"F@7$Fgn$ \"36+++3bEI?F@7$F\\o$\"3-+++]x+/BF@7$Fao$\"3-+++C!\\\\c#F@7$Ffo$\"3!)* *****3)z(4GF@7$F[p$\"37+++L([f.$F@7$F`p$\"3/+++W+dTKF@7$Fep$\"3v***** \\^M`U$F@7$Fjp$\"3')*****HtCke$F@7$F_q$\"3u******z^VCPF@7$Fdq$\"33+++% =(HRQF@7$Fiq$\"3y*****RP37$RF@7$F^r$\"3%)*****zAm0+%F@7$Fcr$\"3++++?m) )\\SF@7$Fhr$\"3C+++>^x!3%F@7$F]s$\"3E+++D?1&4%F@7$Fbs$\"3=+++5cy%4%F@7 $Fgs$\"3A+++'***=#3%F@7$F\\t$\"37+++ritfSF@7$Fat$\"38+++]i7ISF@7$Fft$ \"37+++W\"Rj*RF@7$F[u$\"3C+++E>ohRF@7$F`u$\"31+++>u#*eEF@7$Ffu$\"33+++ `tNiCF@7$Fiu$\"37+++RpuqAF@7$F\\v$\"33+++Jf;&3#F@7$F_v$\"3()*****p6Mk! 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In fact, the truncation of the distri bution at the tail is the key idea behind evaporative cooling. We will introduce now a removal of atoms at " }{TEXT 19 6 "E(i)>1" }{TEXT -1 82 ", and look for long-term behaviour. We start with the same distrib ution as before:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "fv:=arr ay([seq(1,i=1..N-N0),seq(0,i=N-N0+1..N)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fvG-%'vectorG6#7J\"\"\"F)F)F)F)F)F)F)F)F)F)F)F)F)F)F )F)F)F)F)F)F)F)F)F)F)F)F)F)F)\"\"!F*F*F*F*F*F*F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "PL0:=plot([seq([E(i),(rho(i)*fv[i])],i=1. .N)]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "We increase the length \+ of the loop and in the update of the energy population " }{TEXT 19 5 " fv[i]" }{TEXT -1 88 " we set the distribution function to zero for ato ms above the barrier height, i.e., for " }{TEXT 19 3 "E>1" }{TEXT -1 213 ". This corresponds to removal of atoms which have acquired such e nergies. This idea is valid on physical grounds provided the time betw een collision is long enough for evaporating atoms to disappear from t he trap." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "N_col:=100;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&N_colG\"$+\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 28 "for icol from 1 to N_col do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for i from 1 to N do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "RHS:=0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for k fr om 1 to N do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for l from 1 to N \+ do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "j:=k+l-i:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "if j>0 and j " 0 "" {MPLTEXT 1 0 66 "h:=min(i,j,k,l): RHS:=RHS+rho(h)/rho(i)*(fv[k]*fv[l]- fv[i]*fv[j]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 7 "od: od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "i f i<=N-N0 then fv[i]:=fv[i]+cdt*RHS: else fv[i]:=0: fi: od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "if trunc(icol/10)*10=icol then PL[trunc( icol/10)]:=plot([seq([E(i),(rho(i)*fv[i])],i=1..N)],title=cat(\"Collis ion number : \",(N/2)*icol*cdt/dE)): fi: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "display([PL0,seq(PL[i],i=1..N_col/10)],insequenc e=true,view=[0..1,0..0.8]);" }}{PARA 13 "" 1 "" {GLPLOT2D 666 298 298 {PLOTDATA 2 "6$-%(ANIMATEG6-7%-%'CURVESG6$7J7$$\"3-+++nmmm;!#>$\"3)*** ****yxxxF!#@7$$\"3G+++++++]F.$\"31+++++++D!#?7$$\"3^*****HLLLL)F.$\"35 +++WWWWpF77$$\"3/+++nmmm6!#=$\"3++++766h8F.7$$\"3%**************\\\"F@ $\"3#*************\\AF.7$$\"37+++LLLL=F@$\"3/+++566hLF.7$$\"35+++nmmm@ F@$\"3%)*****fWWWp%F.7$$\"3++++++++DF@$\"3+++++++]iF.7$$\"3!******HLLL $GF@$\"3?+++wxxF!)F.7$$\"3))*****pmmm;$F@$\"3&******zxxF+\"F@7$$\"3w** ***********\\$F@$\"3)************\\A\"F@7$$\"3B+++LLLLQF@$\"37+++WWWp9 F@7$$\"3A+++nmmmTF@$\"3$******466ht\"F@7$$\"35+++++++XF@$\"39++++++D?F @7$$\"3,+++LLLL[F@$\"3\"******466hL#F@7$$\"3a+++nmmm^F@$\"3))*****\\WW %pEF@7$$\"3U+++++++bF@$\"3\"************\\-$F@7$$\"3M+++LLLLeF@$\"31++ +xxx-MF@7$$\"3K+++nmmmhF@$\"3%******zxxF!QF@7$$\"3A+++++++lF@$\"3()*** ********\\A%F@7$$\"37+++LLLLoF@$\"3#******RWW%pYF@7$$\"35+++nmmmrF@$\" 3D+++766O^F@7$$\"3++++++++vF@$\"3+++++++DcF@7$$\"3!******HLLL$yF@$\"3& ******466h8'F@7$$\"3))*****pmmm;)F@$\"37+++XWWpmF@7$$\"3w************* \\)F@$\"3J++++++DsF@7$$\"3o*****HLLL$))F@$\"3`*****pxxF!yF@7$$\"3m**** *pmmm;*F@$\"39+++yxx-%)F@7$$\"3a*************\\*F@$\"3p***********\\-* F@7$$\"3X*****HLLL$)*F@$\"3[+++WWWp'*F@7$$\"3(******pmmm,\"!#<$\"\"!Fd u7$$\"3/++++++]5FbuFcu7$$\"3!******HLLL3\"FbuFcu7$$\"31+++nmm;6FbuFcu7 $$\"3#*************\\6FbuFcu7$$\"3)******HLLL=\"FbuFcu7$$\"3#******pmm m@\"FbuFcu7$$\"3+++++++]7FbuFcu7$$\"33+++LLL$G\"FbuFcu7$$\"3,+++nmm;8F buFcu-%'COLOURG6&%$RGBG$\"#5!\"\"FcuFcu-%+AXESLABELSG6$Q!6\"Fjw-%%VIEW G6$%(DEFAULTGF_x7&-F(6$7J7$F,$\"3!******flSi?\"F77$F3$\"3(*******p'H(3 5F.7$F9$\"3&******4sAvg#F.7$F>$\"33+++2$R+v%F.7$FD$\"3'*******f%*z'G(F .7$FI$\"3/+++e$y&35F@7$FN$\"3*)******RP?.8F@7$FS$\"3.+++adp-;F@7$FX$\" 33+++*3L()*=F@7$Fgn$\"3#******f**>W=#F@7$F\\o$\"3-+++%4iTX#F@7$Fao$\"3 A+++`(\\Nq#F@7$Ffo$\"33+++$Q$HHHF@7$F[p$\"36+++%G!4HJF@7$F`p$\"3B+++9< [,LF@7$Fep$\"3;+++njrXMF@7$Fjp$\"3u*****H:A;c$F@7$F_q$\"3/+++tr[\\OF@7 $Fdq$\"3%******p**\\*4PF@7$Fiq$\"3*******f'))*Qu$F@7$F^r$\"3#*******p) yBv$F@7$Fcr$\"3%******p>\"QTPF@7$Fhr$\"3%******4!\\e8PF@7$F]s$\"3%**** **4k&yrOF@7$Fbs$\"3z*****>&*z)=OF@7$Fgs$\"3%)*******3pyb$F@7$F\\t$\"3G +++F%y=\\$F@7$Fat$\"3z*****>?!>CMF@7$Fft$\"3!******zf)GeLF@7$F[u$\"3A+ ++)=I-BF77$F3$\"35+++#\\%GT=F.7$F9 $\"3)******f/&y[XF.7$F>$\"3h+++e)*[>zF.7$FD$\"3-+++eSGh6F@7$FI$\"32+++ `txO:F@7$FN$\"3!*******p$=!**=F@7$FS$\"3(*******)GPTB#F@7$FX$\"3/+++$R pF`#F@7$Fgn$\"3/+++*4[\"*y#F@7$F\\o$\"3@+++(QE.+$F@7$Fao$\"3!******4uF b;$F@7$Ffo$\"3++++a0k&G$F@7$F[p$\"3++++!ekFO$F@7$F`p$\"3#******zxN)*R$ F@7$Fep$\"3$)*****f.E.S$F@7$Fjp$\"31+++-j+oLF@7$F_q$\"3x*****pbinI$F@7 $Fdq$\"3<+++r]X?KF@7$Fiq$\"37+++gC\"G6$F@7$F^r$\"31+++eKN()HF@7$Fcr$\" 3#*******e>][GF@7$Fhr$\"3%******z4)\\*p#F@7$F]s$\"3-+++/uFVDF@7$Fbs$\" 3%******\\)p[#Q#F@7$Fgs$\"3!******HP6&>AF@7$F\\t$\"3%******>x1l0#F@7$F at$\"3++++BHW&*=F@7$Fft$\"3'******\\W`\"Q$\"3%*******yfhj5F@7$FD$\"35+++$RfM^\"F@7$FI$\"3/+++x,$R%>F@7$FN$\" 3))*****>*>-KBF@7$FS$\"3?+++^C0kEF@7$FX$\"3++++8jDLHF@7$Fgn$\"36+++u+' y8$F@7$F\\o$\"3!)*****\\dp&zKF@7$Fao$\"3'******Hg2CO$F@7$Ffo$\"3u***** fF`=R$F@7$F[p$\"3!)*****>*z.GF@7$Fiq$\"3.+++tsmKEF@7$F^r$\"3%******\\cRVX#F@7$Fcr$\"33+++6n3s AF@7$Fhr$\"3'******pzW*)3#F@7$F]s$\"37+++>uU2>F@7$Fbs$\"3))******>(z&H $\"3++++j37(H\"F@7$FD$\"3)*******[[!\\!=F@7$FI$\"3'******Ry$QnAF@7 $FN$\"3u*****>:%ygEF@7$FS$\"3)******\\6iQ(HF@7$FX$\"3#******Rr-S?$F@7$ Fgn$\"39+++zJ$F@7$F`p$\"3'********yBm=$F@7$Fep$\"3***** **RU%=LIF@7$Fjp$\"3w********o=fGF@7$F_q$\"3A+++$\\n7n#F@7$Fdq$\"35+++7 .6vCF@7$Fiq$\"3#******HD$[vAF@7$F^r$\"3/+++**4Iw?F@7$Fcr$\"3%******zi; 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With increasing co llision number the maximum shifts towards lower energies, but it becom es a rather slow process in the long term, because it becomes less lik ely that particles be evaporated over the high barrier. A solution to \+ this problem is to reduce the barrier height as the ensemble cools dow n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 225 "Th e calculated daverage energy is now a more realistic measure given tha t the distribution is bounded over the interval. Before we proceed wit h the idea of lowering the barrier height we improve the algorithm in \+ two respects." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 11 "Exercise 1:" }}{PARA 0 "" 0 "" {TEXT -1 174 "Increase th e number of collisions by running the loop over longer times and obser ve the modest change in the final distribution. Also perform small var iations of the product " }{TEXT 19 4 "c*dt" }{TEXT -1 75 " and verify \+ whether your results are consistent for fixed collision number." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "A good question to ask i s: How much would we gain from an " }{TEXT 19 7 "O(dt^2)" }{TEXT -1 107 " method? Twice the computational effort per dt-step might be comp ensated by 4-fold improvement in accuracy." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plo ts):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords ha s been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "E_t:=1 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$E_tG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "N:=40;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"NG\"#S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "N0:=10;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#N0G\"#5" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "dE:=E_t/(N-N0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#dEG#\"\"\"\"#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "E:=i-> (i-0.5)*dE;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EGf*6#%\"iG6\"6$%)o peratorG%&arrowGF(*&,&9$\"\"\"$\"\"&!\"\"F2F/%#dEGF/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "rho:=i->E(i)^2;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$rhoGf*6#%\"iG6\"6$%)operatorG%&arrowGF(*$)-%\"EG6# 9$\"\"#\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "We need to farm out the RHS-calculation to a procedure:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "RHSpr:=proc(fv,i) local j,k,l,h,RHS; global N,cd t,rho;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "RHS:=0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for k from 1 to N do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for l from 1 to N do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "j:=k+l-i:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "if j>0 and j " 0 "" {MPLTEXT 1 0 66 "h:=min(i,j,k,l): RHS:=RHS +rho(h)/rho(i)*(fv[k]*fv[l]-fv[i]*fv[j]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "od: od: RHS; end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "N_col:=100;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&N_colG\"$+\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 11 "cdt:=1/400;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$cdtG#\"\"\"\"$+%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "fv: =array([seq(1,i=1..N-N0),seq(0,i=N-N0+1..N)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fvG-%'vectorG6#7J\"\"\"F)F)F)F)F)F)F)F)F)F)F)F)F)F)F )F)F)F)F)F)F)F)F)F)F)F)F)F)F)\"\"!F*F*F*F*F*F*F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "PL0:=plot([seq([E(i),(rho(i)*fv[i])],i=1. .N)]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 168 "The collision loop is \+ modified by introducing a simple predictor-corrector method: extrapola tion is used as before in the first step to predict the level populati on at " }{TEXT 19 4 "t+dt" }{TEXT -1 33 ", and then the rate of change in " }{TEXT 19 5 "fv[i]" }{TEXT -1 134 " is recomputed from this pred icted population. The real time-step is carried out using an average o f both rate of change calculations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "for icol from 1 to N_col do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "for i from 1 to N do: RHSp[i]:=RHSpr(fv,i):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "if i<=N-N0 then gv[i]:=fv[i]+cdt*RHSp[i]: else gv[i]:=0: fi: od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "for i fr om 1 to N do: RHS:=0.5*(RHSpr(gv,i)+RHSp[i]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "if i<=N-N0 then fv[i]:=fv[i]+cdt*RHS: else fv[i]:=0: \+ fi: od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "if trunc(icol/10)*10=ic ol then PL[trunc(icol/10)]:=plot([seq([E(i),(rho(i)*fv[i])],i=1..N)],t itle=cat(\"Collision number : \",trunc((N/2)*icol*cdt/dE))): fi: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "display([PL0,seq(PL[i],i= 1..N_col/10)],insequence=true,view=[0..1,0..0.5]);" }}{PARA 13 "" 1 " " {GLPLOT2D 666 298 298 {PLOTDATA 2 "6$-%(ANIMATEG6-7%-%'CURVESG6$7J7$ $\"3-+++nmmm;!#>$\"3)*******yxxxF!#@7$$\"3G+++++++]F.$\"31+++++++D!#?7 $$\"3^*****HLLLL)F.$\"35+++WWWWpF77$$\"3/+++nmmm6!#=$\"3++++766h8F.7$$ \"3%**************\\\"F@$\"3#*************\\AF.7$$\"37+++LLLL=F@$\"3/+ ++566hLF.7$$\"35+++nmmm@F@$\"3%)*****fWWWp%F.7$$\"3++++++++DF@$\"3++++ +++]iF.7$$\"3!******HLLL$GF@$\"3?+++wxxF!)F.7$$\"3))*****pmmm;$F@$\"3& ******zxxF+\"F@7$$\"3w*************\\$F@$\"3)************\\A\"F@7$$\"3 B+++LLLLQF@$\"37+++WWWp9F@7$$\"3A+++nmmmTF@$\"3$******466ht\"F@7$$\"35 +++++++XF@$\"39++++++D?F@7$$\"3,+++LLLL[F@$\"3\"******466hL#F@7$$\"3a+ ++nmmm^F@$\"3))*****\\WW%pEF@7$$\"3U+++++++bF@$\"3\"************\\-$F@ 7$$\"3M+++LLLLeF@$\"31+++xxx-MF@7$$\"3K+++nmmmhF@$\"3%******zxxF!QF@7$ $\"3A+++++++lF@$\"3()***********\\A%F@7$$\"37+++LLLLoF@$\"3#******RWW% pYF@7$$\"35+++nmmmrF@$\"3D+++766O^F@7$$\"3++++++++vF@$\"3+++++++DcF@7$ $\"3!******HLLL$yF@$\"3&******466h8'F@7$$\"3))*****pmmm;)F@$\"37+++XWW pmF@7$$\"3w*************\\)F@$\"3J++++++DsF@7$$\"3o*****HLLL$))F@$\"3` *****pxxF!yF@7$$\"3m*****pmmm;*F@$\"39+++yxx-%)F@7$$\"3a************* \\*F@$\"3p***********\\-*F@7$$\"3X*****HLLL$)*F@$\"3[+++WWWp'*F@7$$\"3 (******pmmm,\"!#<$\"\"!Fdu7$$\"3/++++++]5FbuFcu7$$\"3!******HLLL3\"Fbu Fcu7$$\"31+++nmm;6FbuFcu7$$\"3#*************\\6FbuFcu7$$\"3)******HLLL =\"FbuFcu7$$\"3#******pmmm@\"FbuFcu7$$\"3+++++++]7FbuFcu7$$\"33+++LLL$ G\"FbuFcu7$$\"3,+++nmm;8FbuFcu-%'COLOURG6&%$RGBG$\"#5!\"\"FcuFcu-%+AXE SLABELSG6$Q!6\"Fjw-%%VIEWG6$%(DEFAULTGF_x7&-F(6$7J7$F,$\"3/+++()[%pM#F 77$F3$\"3))*****f8-#f=F.7$F9$\"3I+++$[pSa%F.7$F>$\"3=+++?U0MyF.7$FD$\" 3*******pk1(Q6F@7$FI$\"3'*******y+9&\\\"F@7$FN$\"3)******pJ1Z$=F@7$FS$ \"35+++J00X@F@7$FX$\"36+++5*=#=CF@7$Fgn$\"3w*****zq\\'\\EF@7$F\\o$\"3' ******ROet$GF@7$Fao$\"31+++o'*G\")HF@7$Ffo$\"37+++(QIG3$F@7$F[p$\"3,++ +NvQWJF@7$F`p$\"3-+++O\"[!pJF@7$Fep$\"3A+++OAJgJF@7$Fjp$\"38+++ce)=7$F @7$F_q$\"3#)*****>nAv0$F@7$Fdq$\"3A+++=\\!4(HF@7$Fiq$\"3=+++]bblGF@7$F ^r$\"38+++!*\\xWFF@7$Fcr$\"3()******)*Ha7EF@7$Fhr$\"3++++ei#=Z#F@7$F]s $\"3(******R[,`K#F@7$Fbs$\"35+++8/Ov@F@7$Fgs$\"3#******4;CT-#F@7$F\\t$ \"3'******pjlM(=F@7$Fat$\"3-+++a$R]s\"F@7$Fft$\"3)******zy=.e\"F@7$F[u $\"3'******>XU1W\"F@F_uFeuFhuF[vF^vFavFdvFgvFjvF]wF`w-%&TITLEG6#Q6Coll ision~number~:~15F[xFgwF\\x7&-F(6$7J7$F,$\"35+++uk)fE%F77$F3$\"3\"**** ***>&yt@$F.7$F9$\"3k+++[:=([(F.7$F>$\"3'*******oB4H7F@7$FD$\"3,+++a*=7 q\"F@7$FI$\"3\"******pU(=F@F@7$FN$\"3'******4mNf[#F@7$FS$\"3#)*****Rn/ \"oFF@7$FX$\"3A+++QuAsHF@7$Fgn$\"3++++Vn$>5$F@7$F\\o$\"3y*****\\:ZR;$F @7$Fao$\"3#******4(RimJF@7$Ffo$\"3>+++D1+>JF@7$F[p$\"31+++7=5IIF@7$F`p $\"31+++`H]3HF@7$Fep$\"3++++$yj?w#F@7$Fjp$\"3)******>J'y(f#F@7$F_q$\"3 !******R\")f\"F@7$Fgs$\"3++++GK6]5F@7$F\\t$\"3j*****4Hm p8*F.7$Fat$\"3i+++'3l\\(yF.7$Fft$\"3D+++&)309nF.7$F[u$\"3G+++JPV^cF.F_ uFeuFhuF[vF^vFavFdvFgvFjvF]wF`w-F_^l6#Q6Collision~number~:~30F[xFgwF\\ x7&-F(6$7J7$F,$\"3L+++0t6;fF77$F3$\"34+++=\"=sK%F.7$F9$\"3y******f8`m( *F.7$F>$\"3)*******4Q/b:F@7$FD$\"3!*******Q#Rx3#F@7$FI$\"3%******R=GA` #F@7$FN$\"3A+++M,tqGF@7$FS$\"3))******e55,JF@7$FX$\"3%)*****RnB0B$F@7$ Fgn$\"37+++p@@rKF@7$F\\o$\"3\"*******Q3cPKF@7$Fao$\"3v*****pn 9F@7$Fcr$\"3)*******)R(3X7F@7$Fhr$\"3-+++C'>U3\"F@7$F]s$\"3'******f[d. 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In order to make a quantitative comparison track the maximum of the \+ distribution as a function of collision number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 254 "To accomplish co oling to low temperatures (i.e., energies in the trap) the method beco mes inefficient. The obvious strategy is to lower the trap height, i.e ., to allow particles of lower energy to evaporate as time goes on. We lower the trap height from " }{TEXT 19 7 "E(N-10)" }{TEXT -1 4 " to \+ " }{TEXT 19 6 "E(N/4)" }{TEXT -1 69 " over the time of the run. This i s accomplished by defining an index " }{TEXT 19 6 "N_trap" }{TEXT -1 150 " which varies with time (collision number), and which is used in \+ the update algorithm as the cut-off parameter beyond which the distrib ution function " }{TEXT 19 5 "fv[i]" }{TEXT -1 49 " is set to zero at \+ the end of a propagation step." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 199 "First we improve the algorithm by avoiding bookkeeping for the over-the-barrier states. We discretize t he energy range for the (initally) trapped states and propagate only t hose. The double sum over " }{TEXT 19 3 "k,l" }{TEXT -1 64 " in the ca lculation of the right hand side is extended to cover " }{TEXT 19 2 "N 0" }{TEXT -1 26 " additional energy levels." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "E_t:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$E_tG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "N:=40;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "dE:=E_t/(N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dEG#\"\"\"\"#S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "E:=i->(i-0.5)*dE;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"EGf*6#%\"iG6\"6$%)operatorG%&arrowGF(*&,&9$\"\"\"$\"\"&!\"\"F2F/%# dEGF/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "rho:=i->E(i) ^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoGf*6#%\"iG6\"6$%)operator G%&arrowGF(*$)-%\"EG6#9$\"\"#\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "cdt:=1/400;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c dtG#\"\"\"\"$+%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "N_col:=5 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&N_colG\"#]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "RHSpr:=proc(fv,i) local j,k,l,h,RHS,fvk,f vl,fvj,N0; global N,cdt,rho;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "RHS :=0: N0:=15;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "for k from 1 to N+N 0 do: if k<=N then fvk:=fv[k]: else fvk:=0: fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "for l from 1 to N+N0 do: if l<=N then fvl:=fv[l]: els e fvl:=0: fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "j:=k+l-i:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "if j>0 and j<=N+N0 then if j<=N then fvj: =fv[j]: else fvj:=0: fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "h:=min( i,j,k,l): RHS:=RHS+rho(h)/rho(i)*(fvk*fvl-fv[i]*fvj):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "od: od: RHS; end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "fv:=array([se q(1,i=1..N)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fvG-%'vectorG6#7J \"\"\"F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F) F)F)F)F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "PL0:=plot ([seq([E(i),(rho(i)*fv[i])],i=1..N)]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "for icol from 1 to N_col do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "for i from 1 to N do: RHSp[i]:=RHSpr(fv,i):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "gv[i]:=fv[i]+cdt*RHSp[i]: od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "for i from 1 to N do: RHS:=0.5*(RHSpr(gv,i) +RHSp[i]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "fv[i]:=fv[i]+cdt*RHS: od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "if trunc(icol/10)*10=icol \+ then PL[trunc(icol/10)]:=plot([seq([E(i),(rho(i)*fv[i])],i=1..N)],titl e=cat(\"Collision number : \",trunc((N/2)*icol*cdt/dE))): fi: od:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "display([PL0,seq(PL[i],i=1.. 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How sensitive are the results to the resolution \+ " }{TEXT 19 2 "dE" }{TEXT -1 70 "? How do the results differ for other powers in the density of states?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 238 "Now we come to the main point: we will l ower the barrier height for the trap as a function of time (actually a s a function of the number of collisions between the atoms). Over the \+ total number of collisions we reduce the trap height from " }{TEXT 19 6 "E(N)=1" }{TEXT -1 4 " to " }{TEXT 19 5 "E(N1)" }{TEXT -1 61 ". The \+ index which is used as a criterion as to how to update " }{TEXT 19 5 " fv[i]" }{TEXT -1 11 " is called " }{TEXT 19 6 "N_trap" }{TEXT -1 131 " and is calculated in the collision loop to change gradually. 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rho(i),i=1..N)/add(fv[i]*rho(i),i=1..N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&E_avgG$\"+1!GZ3#!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "N_TP:=add(fv[i]*rho(i),i=1..N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%N_TPG$\"+\\(G)yU!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 146 "Ev idently the distribution peaks now at a lower energy value, but for th e same number of collisions a smaller fraction of bound particles rema ins." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 11 " Exercise 5:" }}{PARA 0 "" 0 "" {TEXT -1 177 "Carry out Boltzmann simul ations in which the reduction in barrier height happens faster (or mor e slowly) by changing the total number of collisions (controlled by th e parameter " }{TEXT 19 5 "N_col" }{TEXT -1 226 ", which is proportion al to the collision number) in the above loop. Is the final result sim ply a function of collision number, or are there differences when the \+ the barrier is lowered at a different rate with collision number?" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 11 "Exercise 6:" }}{PARA 0 "" 0 "" {TEXT -1 244 "Change the power law for the dens ity of states from the harmonic oscillator case and observe the effici ency of evaporative cooling with collision number. How do the results \+ depend on the power when it is changed in steps of 1/2 up or down from " }{TEXT 19 3 "n=2" }{TEXT -1 1 "?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "68 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }