Walker: Integrating the number-fraction beta SDE
This example runs Walker to integrate the number-fraction beta SDE (see DiffEq/NumberFractionBeta.h) using constant coefficients. The number-fraction beta SDE is based on the beta SDE, additionally computing two more stochastic variables that are functions of the beta variables integrated. For more detail on the beta SDE, see https:/
Control file
title "Test Ray's closure ideas for <v^2> and <v^3>" walker term 25.0 # Max time dt 0.002 # Time step size npar 10000 # Number of particles ttyi 100 # TTY output interval rngs mkl_r250 end end numfracbeta # Select the number-fraction beta SDE depvar x # Dependent variable: X, denoting mole or number fractions ncomp 15 # = 3x5 = 5 systems each computing 3 variables: # X - number fraction, # R - density, # V - specific volume init zero coeff const # alpha = Sb/kappa, beta = (1-S)b/kappa kappa 2.0 0.76923 0.5 0.15873 0.1 end b 0.4 1.0 1.0 1.0 100.0 end S 0.5 0.53846 0.5 0.39683 0.5 end rng mkl_r250 rho2 1.0 1.0 1.0 1.0 1.0 end # low-A rcomma 1.0e-2 1.0e-2 1.0e-2 1.0e-2 1.0e-2 end # high-A #rcomma 0.9 0.9 0.9 0.9 0.9 end end statistics # column numbers in output file # <X>, mole fraction means <X1> # 3 <X2> # 4 <X3> # 5 <X4> # 6 <X5> # 7 # <rho>, mean densities <X6> # 8 <X7> # 9 <X8> # 10 <X9> # 11 <X10> # 12 # <V>, mean specific volumes <X11> # 13 <X12> # 14 <X13> # 15 <X14> # 16 <X15> # 17 # <x^2>, mole fraction variances <x1x1> # 18 <x2x2> # 19 <x3x3> # 20 <x4x4> # 21 <x5x5> # 22 # <rho v>, density-specific-volume covariances <x6x11> # 25 <x7x12> # 30 <x8x13> # 25 <x9x14> # 40 <x10x15> # 45 # <rho v^2> <x6x11x11> # 26 <x7x12x12> # 31 <x8x13x13> # 36 <x9x14x14> # 41 <x10x15x15> # 46 # <rho^2 v> <x6x6x11> # 23 <x7x7x12> # 28 <x8x8x13> # 33 <x9x9x14> # 38 <x10x10x15> # 43 # <rho^2v^2> <x6x6x11x11> # 24 <x7x7x12x12> # 29 <x8x8x13x13> # 34 <x9x9x14x14> # 39 <x10x10x15x15> # 43 # <rho v^3> <x6x11x11x11> # 27 <x7x12x12x12> # 32 <x8x13x13x13> # 37 <x9x14x14x14> # 42 <x10x15x15x15> # 47 # <v^2>, specific volume variances <x11x11> # 48 <x12x12> # 50 <x13x13> # 52 <x14x14> # 54 <x15x15> # 56 # <v^3>, specific volume third central moments <x11x11x11> # 49 <x12x12x12> # 51 <x13x13x13> # 53 <x14x14x14> # 55 <x15x15x15> # 57 end pdfs interval 100 filetype txt policy overwrite centering elem format scientific precision 4 p1( X1 : 1.0e-2 ) p2( X2 : 1.0e-2 ) p3( X3 : 1.0e-2 ) p4( X4 : 1.0e-2 ) p5( X5 : 1.0e-2 ) end end
Example run on 8 CPUs
./charmrun +p8 Main/walker -v -c ../../tmp/numfracbeta.q -u 0.9
Results
The rationale for these runs is to integrate the system in time, extract the time evolution of various statistics, and then test several different closure hypotheses among the statistics. Example gnuplot commands to plot and test some closure ideas are:
# vim: filetype=gnuplot: # Nomenclature: # ------------- # V - instantaneous specific volume # R - instantaneous density # v = V - <V>, fluctuating specific volume # r = R - <R>, fluctuating density # b = -<rv>, density-specific-volume covariance # B - as postfix means the Bousinessq approximation # <v^2> # IA: <v^2>B = b/<R>^2 plot "stat.txt" u 2:(-$25/$8/$8), "stat.txt" u 2:48 w l lt 2 # no-mix plot "stat.txt" u 2:(-$30/$9/$9), "stat.txt" u 2:50 w l lt 2 plot "stat.txt" u 2:(-$35/$10/$10), "stat.txt" u 2:52 w l lt 2 # uniform plot "stat.txt" u 2:(-$40/$11/$11), "stat.txt" u 2:54 w l lt 2 plot "stat.txt" u 2:(-$45/$12/$12), "stat.txt" u 2:56 w l lt 2 # almost mixed #################### COLUMNS NOT UPDATED TO LATEST COLUMN NUMBER UNTIL HASHES # IB: <v^2>(B+1) = b/<R>^2 [ 1 + b(1+b)/(1+(1+b)^2) ] plot "stat.txt" u 2:(-$23/$8/$8*(1.0+(-$23*(1.0-$23)/(1.0+(1.0-$23)*(1.0-$23))))), "stat.txt" u 2:33 w l lt 2 # no-mix plot "stat.txt" u 2:(-$25/$9/$9*(1.0+(-$25*(1.0-$25)/(1.0+(1.0-$25)*(1.0-$25))))), "stat.txt" u 2:35 w l lt 2 plot "stat.txt" u 2:(-$27/$10/$10*(1.0+(-$27*(1.0-$27)/(1.0+(1.0-$27)*(1.0-$27))))), "stat.txt" u 2:37 w l lt 2 # uniform plot "stat.txt" u 2:(-$29/$11/$11*(1.0+(-$29*(1.0-$29)/(1.0+(1.0-$29)*(1.0-$29))))), "stat.txt" u 2:39 w l lt 2 plot "stat.txt" u 2:(-$31/$12/$12*(1.0+(-$31*(1.0-$31)/(1.0+(1.0-$31)*(1.0-$31))))), "stat.txt" u 2:41 w l lt 2 # almost mixed # <v^3> # IIA: <v^3>B = -<rv^2>/<R>^2 plot "stat.txt" u 2:(-$24/$8/$8), "stat.txt" u 2:34 w l lt 2 # no-mix plot "stat.txt" u 2:(-$26/$9/$9), "stat.txt" u 2:36 w l lt 2 plot "stat.txt" u 2:(-$28/$10/$10), "stat.txt" u 2:38 w l lt 2 # uniform plot "stat.txt" u 2:(-$30/$11/$11), "stat.txt" u 2:40 w l lt 2 plot "stat.txt" u 2:(-$32/$12/$12), "stat.txt" u 2:42 w l lt 2 # almost mixed # IIA: <v^3>B = (b<V> + <R><v^2>) / <R>^2 plot "stat.txt" u 2:(($23*$13+$8*$33)/$8/$8), "stat.txt" u 2:34 w l lt 2 # no-mix plot "stat.txt" u 2:(($25*$14+$9*$35)/$9/$9), "stat.txt" u 2:36 w l lt 2 plot "stat.txt" u 2:(($27*$15+$10*$37)/$10/$10), "stat.txt" u 2:38 w l lt 2 # uniform plot "stat.txt" u 2:(($29*$16+$11*$39)/$11/$11), "stat.txt" u 2:40 w l lt 2 plot "stat.txt" u 2:(($31*$17+$12*$41)/$12/$12), "stat.txt" u 2:42 w l lt 2 # almost mixed # IIB: <v^3>B = r'/rho2 <v^2>B (1-2<X>) plot "stat.txt" u 2:(1.0e-2/1.0*(-$23)/$8/$8*(1.0-2.0*$3)), "stat.txt" u 2:34 w l lt 2 # no-mix plot "stat.txt" u 2:(1.0e-2/1.0*(-$25)/$9/$9*(1.0-2.0*$4)), "stat.txt" u 2:36 w l lt 2 plot "stat.txt" u 2:(1.0e-2/1.0*(-$27)/$10/$10*(1.0-2.0*$5)), "stat.txt" u 2:38 w l lt 2 # uniform plot "stat.txt" u 2:(1.0e-2/1.0*(-$29)/$11/$11*(1.0-2.0*$6)), "stat.txt" u 2:40 w l lt 2 plot "stat.txt" u 2:(1.0e-2/1.0*(-$31)/$12/$12*(1.0-2.0*$7)), "stat.txt" u 2:42 w l lt 2 # almost mixed # IIB2: <v^3>(B+1) = r'/rho2 <v^2>(B+1) (1-2<X>) plot "stat.txt" u 2:(1.0e-2/1.0*(-$23/$8/$8*(1.0+(-$23*(1.0-$23)/(1.0+(1.0-$23)*(1.0-$23)))))*(1.0-2.0*$3)), "stat.txt" u 2:34 w l lt 2 # no-mix plot "stat.txt" u 2:(1.0e-2/1.0*(-$25/$9/$9*(1.0+(-$25*(1.0-$25)/(1.0+(1.0-$25)*(1.0-$25)))))*(1.0-2.0*$4)), "stat.txt" u 2:36 w l lt 2 plot "stat.txt" u 2:(1.0e-2/1.0*(-$27/$10/$10*(1.0+(-$27*(1.0-$27)/(1.0+(1.0-$27)*(1.0-$27)))))*(1.0-2.0*$5)), "stat.txt" u 2:38 w l lt 2 # uniform plot "stat.txt" u 2:(1.0e-2/1.0*(-$29/$11/$11*(1.0+(-$29*(1.0-$29)/(1.0+(1.0-$29)*(1.0-$29)))))*(1.0-2.0*$6)), "stat.txt" u 2:40 w l lt 2 plot "stat.txt" u 2:(1.0e-2/1.0*(-$31/$12/$12*(1.0+(-$31*(1.0-$31)/(1.0+(1.0-$31)*(1.0-$31)))))*(1.0-2.0*$7)), "stat.txt" u 2:42 w l lt 2 # almost mixed # IIC: <v^3>B = 2(1-2<V>)<v^2>B (1-theta)/(2-theta) plot "stat.txt" u 2:(2.0*(1.0-2.0*$13)*(-$23/$8/$8)*(1.0-(1.0+$23/(-$23+1.0*1.0e-2*$3*(1.0-$3))))/(2.0-(1.0+$23/(-$23+1.0*1.0e-2*$3*(1.0-$3))))), "stat.txt" u 2:34 w l lt 2 # no-mix plot "stat.txt" u 2:(2.0*(1.0-2.0*$14)*(-$25/$9/$9)*(1.0-(1.0+$25/(-$25+1.0*1.0e-2*$4*(1.0-$4))))/(2.0-(1.0+$25/(-$25+1.0*1.0e-2*$4*(1.0-$4))))), "stat.txt" u 2:36 w l lt 2 plot "stat.txt" u 2:(2.0*(1.0-2.0*$15)*(-$27/$10/$10)*(1.0-(1.0+$27/(-$27+1.0*1.0e-2*$5*(1.0-$5))))/(2.0-(1.0+$27/(-$27+1.0*1.0e-2*$5*(1.0-$5))))), "stat.txt" u 2:38 w l lt 2 # uniform plot "stat.txt" u 2:(2.0*(1.0-2.0*$16)*(-$29/$11/$11)*(1.0-(1.0+$29/(-$29+1.0*1.0e-2*$6*(1.0-$6))))/(2.0-(1.0+$29/(-$29+1.0*1.0e-2*$6*(1.0-$6))))), "stat.txt" u 2:40 w l lt 2 plot "stat.txt" u 2:(2.0*(1.0-2.0*$17)*(-$31/$12/$12)*(1.0-(1.0+$31/(-$31+1.0*1.0e-2*$7*(1.0-$7))))/(2.0-(1.0+$31/(-$31+1.0*1.0e-2*$7*(1.0-$7))))), "stat.txt" u 2:42 w l lt 2 # almost mixed # IIC: <v^3>B = 2(1-2<V>)<v^2>(B+1) (1-theta)/(2-theta) plot "stat.txt" u 2:(2.0*(1.0-2.0*$13)*(-$23/$8/$8*(1.0+(-$23*(1.0-$23)/(1.0+(1.0-$23)*(1.0-$23)))))*(1.0-(1.0+$23/(-$23+1.0*1.0e-2*$3*(1.0-$3))))/(2.0-(1.0+$23/(-$23+1.0*1.0e-2*$3*(1.0-$3))))), "stat.txt" u 2:34 w l lt 2 # no-mix plot "stat.txt" u 2:(2.0*(1.0-2.0*$14)*(-$25/$9/$9*(1.0+(-$25*(1.0-$25)/(1.0+(1.0-$25)*(1.0-$25)))))*(1.0-(1.0+$25/(-$25+1.0*1.0e-2*$4*(1.0-$4))))/(2.0-(1.0+$25/(-$25+1.0*1.0e-2*$4*(1.0-$4))))), "stat.txt" u 2:36 w l lt 2 plot "stat.txt" u 2:(2.0*(1.0-2.0*$15)*(-$27/$10/$10*(1.0+(-$27*(1.0-$27)/(1.0+(1.0-$27)*(1.0-$27)))))*(1.0-(1.0+$27/(-$27+1.0*1.0e-2*$5*(1.0-$5))))/(2.0-(1.0+$27/(-$27+1.0*1.0e-2*$5*(1.0-$5))))), "stat.txt" u 2:38 w l lt 2 # uniform plot "stat.txt" u 2:(2.0*(1.0-2.0*$16)*(-$29/$11/$11*(1.0+(-$29*(1.0-$29)/(1.0+(1.0-$29)*(1.0-$29)))))*(1.0-(1.0+$29/(-$29+1.0*1.0e-2*$6*(1.0-$6))))/(2.0-(1.0+$29/(-$29+1.0*1.0e-2*$6*(1.0-$6))))), "stat.txt" u 2:40 w l lt 2 plot "stat.txt" u 2:(2.0*(1.0-2.0*$17)*(-$31/$12/$12*(1.0+(-$31*(1.0-$31)/(1.0+(1.0-$31)*(1.0-$31)))))*(1.0-(1.0+$31/(-$31+1.0*1.0e-2*$7*(1.0-$7))))/(2.0-(1.0+$31/(-$31+1.0*1.0e-2*$7*(1.0-$7))))), "stat.txt" u 2:42 w l lt 2 # almost mixed # IIC: <v^3>B = 2(1-2<V>)<v^2>B (1-thetav)/(2-thetav) plot "stat.txt" u 2:(2.0*(1.0-2.0*$13)*(-$23/$8/$8)*(1.0-(1.0-$33/($33+1.0*1.0e-2/1.0/0.99*$3*(1.0-$3))))/(2.0-(1.0-$33/($33+1.0*1.0e-2/1.0/0.99*$3*(1.0-$3))))), "stat.txt" u 2:34 w l lt 2 # no-mix plot "stat.txt" u 2:(2.0*(1.0-2.0*$14)*(-$25/$9/$9)*(1.0-(1.0-$35/($35+1.0*1.0e-2/1.0/0.99*$4*(1.0-$4))))/(2.0-(1.0-$35/($35+1.0*1.0e-2/1.0/0.99*$4*(1.0-$4))))), "stat.txt" u 2:36 w l lt 2 plot "stat.txt" u 2:(2.0*(1.0-2.0*$15)*(-$27/$10/$10)*(1.0-(1.0-$37/($37+1.0*1.0e-2/1.0/0.99*$5*(1.0-$5))))/(2.0-(1.0-$37/($37+1.0*1.0e-2/1.0/0.99*$5*(1.0-$5))))), "stat.txt" u 2:38 w l lt 2 # uniform plot "stat.txt" u 2:(2.0*(1.0-2.0*$16)*(-$29/$11/$11)*(1.0-(1.0-$39/($39+1.0*1.0e-2/1.0/0.99*$6*(1.0-$6))))/(2.0-(1.0-$39/($39+1.0*1.0e-2/1.0/0.99*$6*(1.0-$6))))), "stat.txt" u 2:40 w l lt 2 plot "stat.txt" u 2:(2.0*(1.0-2.0*$17)*(-$31/$12/$12)*(1.0-(1.0-$41/($41+1.0*1.0e-2/1.0/0.99*$7*(1.0-$7))))/(2.0-(1.0-$41/($41+1.0*1.0e-2/1.0/0.99*$7*(1.0-$7))))), "stat.txt" u 2:42 w l lt 2 # almost mixed # IIC: <v^3>B = 2(1-2<V>)<v^2>(B+1) (1-thetav)/(2-thetav) plot "stat.txt" u 2:(2.0*(1.0-2.0*$13)*(-$23/$8/$8*(1.0+(-$23*(1.0-$23)/(1.0+(1.0-$23)*(1.0-$23)))))*(1.0-(1.0-$33/($33+1.0*1.0e-2/1.0/0.99*$3*(1.0-$3))))/(2.0-(1.0-$33/($33+1.0*1.0e-2/1.0/0.99*$3*(1.0-$3))))), "stat.txt" u 2:34 w l lt 2 # no-mix plot "stat.txt" u 2:(2.0*(1.0-2.0*$14)*(-$25/$9/$9*(1.0+(-$25*(1.0-$25)/(1.0+(1.0-$25)*(1.0-$25)))))*(1.0-(1.0-$35/($35+1.0*1.0e-2/1.0/0.99*$4*(1.0-$4))))/(2.0-(1.0-$35/($35+1.0*1.0e-2/1.0/0.99*$4*(1.0-$4))))), "stat.txt" u 2:36 w l lt 2 plot "stat.txt" u 2:(2.0*(1.0-2.0*$15)*(-$27/$10/$10*(1.0+(-$27*(1.0-$27)/(1.0+(1.0-$27)*(1.0-$27)))))*(1.0-(1.0-$37/($37+1.0*1.0e-2/1.0/0.99*$5*(1.0-$5))))/(2.0-(1.0-$37/($37+1.0*1.0e-2/1.0/0.99*$5*(1.0-$5))))), "stat.txt" u 2:38 w l lt 2 # uniform plot "stat.txt" u 2:(2.0*(1.0-2.0*$16)*(-$29/$11/$11*(1.0+(-$29*(1.0-$29)/(1.0+(1.0-$29)*(1.0-$29)))))*(1.0-(1.0-$39/($39+1.0*1.0e-2/1.0/0.99*$6*(1.0-$6))))/(2.0-(1.0-$39/($39+1.0*1.0e-2/1.0/0.99*$6*(1.0-$6))))), "stat.txt" u 2:40 w l lt 2 plot "stat.txt" u 2:(2.0*(1.0-2.0*$17)*(-$31/$12/$12*(1.0+(-$31*(1.0-$31)/(1.0+(1.0-$31)*(1.0-$31)))))*(1.0-(1.0-$41/($41+1.0*1.0e-2/1.0/0.99*$7*(1.0-$7))))/(2.0-(1.0-$41/($41+1.0*1.0e-2/1.0/0.99*$7*(1.0-$7))))), "stat.txt" u 2:42 w l lt 2 # almost mixed #################### # <rv^3> = <rv><v^2> plot "stat.txt" u 2:($25*$48), "stat.txt" u 2:27 w l lt 2 # no-mix plot "stat.txt" u 2:($30*$50), "stat.txt" u 2:32 w l lt 2 plot "stat.txt" u 2:($35*$52), "stat.txt" u 2:37 w l lt 2 # uniform plot "stat.txt" u 2:($40*$54), "stat.txt" u 2:42 w l lt 2 plot "stat.txt" u 2:($45*$56), "stat.txt" u 2:47 w l lt 2 # almost mixed # <r^2v^2> = <rv>^2 plot "stat.txt" u 2:($25*$25), "stat.txt" u 2:24 w l lt 2 # no-mix plot "stat.txt" u 2:($30*$30), "stat.txt" u 2:29 w l lt 2 plot "stat.txt" u 2:($35*$35), "stat.txt" u 2:34 w l lt 2 # uniform plot "stat.txt" u 2:($40*$40), "stat.txt" u 2:39 w l lt 2 plot "stat.txt" u 2:($45*$45), "stat.txt" u 2:43 w l lt 2 # almost mixed # <v^2> = b/<R>^2 ( 1-b^2-b^3 ) plot "stat.txt" u 2:(-$25/$8/$8*(1.0-$25*$25-$25*$25*$25)), "stat.txt" u 2:48 w l lt 2 # no-mix plot "stat.txt" u 2:(-$30/$9/$9*(1.0-$30*$30-$30*$30*$30)), "stat.txt" u 2:50 w l lt 2 plot "stat.txt" u 2:(-$35/$10/$10*(1.0-$35*$35-$35*$35*$35)), "stat.txt" u 2:52 w l lt 2 # uniform plot "stat.txt" u 2:(-$40/$11/$11*(1.0-$40*$40-$40*$40*$40)), "stat.txt" u 2:54 w l lt 2 plot "stat.txt" u 2:(-$45/$12/$12*(1.0-$45*$45-$45*$45*$45)), "stat.txt" u 2:56 w l lt 2 # almost mixed # <rv^2>/<R>/<V>^2 = b^2(2+b)/(1+b)^2 plot "stat.txt" u 2:($25*$25*(2.0-$25)/(1.0-$25)/(1.0-$25)), "stat.txt" u 2:($26/$8/$13/$13) w l lt 2 # no-mix plot "stat.txt" u 2:($30*$30*(2.0-$30)/(1.0-$30)/(1.0-$30)), "stat.txt" u 2:($31/$9/$14/$14) w l lt 2 plot "stat.txt" u 2:($35*$35*(2.0-$35)/(1.0-$35)/(1.0-$35)), "stat.txt" u 2:($36/$10/$15/$15) w l lt 2 # uniform plot "stat.txt" u 2:($40*$40*(2.0-$40)/(1.0-$40)/(1.0-$40)), "stat.txt" u 2:($41/$11/$16/$16) w l lt 2 plot "stat.txt" u 2:($45*$45*(2.0-$45)/(1.0-$45)/(1.0-$45)), "stat.txt" u 2:($46/$12/$17/$17) w l lt 2 # almost mixed # <r^2v> = -<R>^2 <rv^2> plot "stat.txt" u 2:(-$8*$8*$26), "stat.txt" u 2:23 w l lt 2 # no-mix plot "stat.txt" u 2:(-$9*$9*$31), "stat.txt" u 2:28 w l lt 2 plot "stat.txt" u 2:(-$10*$10*$36), "stat.txt" u 2:33 w l lt 2 # uniform plot "stat.txt" u 2:(-$11*$11*$41), "stat.txt" u 2:38 w l lt 2 plot "stat.txt" u 2:(-$12*$12*$46), "stat.txt" u 2:43 w l lt 2 # almost mixed # <Rv^3>/<R>/<V>^3 = -b^2 (3+b)/(1+b)^3 plot "stat.txt" u 2:(-$25*$25*(3.0-$25)/(1.0-$25)**3.0), "stat.txt" u 2:(($8*$49+$27)/$8/$13/$13/$13) w l lt 2 # no-mix plot "stat.txt" u 2:(-$30*$30*(3.0-$30)/(1.0-$30)**3.0), "stat.txt" u 2:(($9*$51+$32)/$9/$14/$14/$14) w l lt 2 plot "stat.txt" u 2:(-$35*$35*(3.0-$35)/(1.0-$35)**3.0), "stat.txt" u 2:(($10*$53+$37)/$10/$15/$15/$15) w l lt 2 # uniform plot "stat.txt" u 2:(-$40*$40*(3.0-$40)/(1.0-$40)**3.0), "stat.txt" u 2:(($11*$55+$42)/$11/$16/$16/$16) w l lt 2 plot "stat.txt" u 2:(-$45*$45*(3.0-$45)/(1.0-$45)**3.0), "stat.txt" u 2:(($12*$57+$47)/$12/$17/$17/$17) w l lt 2 # almost mixed # <v^2> vs. rho2 * <v^3> plot "stat.txt" u 2:48, "stat.txt" u 2:49 w l lt 2 # no-mix plot "stat.txt" u 2:50, "stat.txt" u 2:51 w l lt 2 plot "stat.txt" u 2:52, "stat.txt" u 2:53 w l lt 2 # uniform plot "stat.txt" u 2:54, "stat.txt" u 2:55 w l lt 2 plot "stat.txt" u 2:56, "stat.txt" u 2:57 w l lt 2 # almost mixed # <v^2> = (1+r') b /<R>^2 plot "stat.txt" u 2:((1.0+9)*(-$25)/$8/$8), "stat.txt" u 2:48 w l lt 2 # no-mix plot "stat.txt" u 2:((1.0+9)*(-$30)/$9/$9), "stat.txt" u 2:50 w l lt 2 plot "stat.txt" u 2:((1.0+9)*(-$35)/$10/$10), "stat.txt" u 2:52 w l lt 2 # uniform plot "stat.txt" u 2:((1.0+9)*(-$40)/$11/$11), "stat.txt" u 2:54 w l lt 2 plot "stat.txt" u 2:((1.0+9)*(-$45)/$12/$12), "stat.txt" u 2:56 w l lt 2 # almost mixed # <v^2> = b(1+b)[(r-2)/r]/<R>^2 + b/(r <R> rho1) plot "stat.txt" u 2:(-$25*(1.0-$25)*(0.0101-2.0)/0.0101/$8/$8-$25/0.0101/$8/0.99), "stat.txt" u 2:48 w l lt 2 # no-mix plot "stat.txt" u 2:(-$30*(1.0-$30)*(0.0101-2.0)/0.0101/$9/$9-$30/0.0101/$9/0.99), "stat.txt" u 2:50 w l lt 2 plot "stat.txt" u 2:(-$35*(1.0-$35)*(0.0101-2.0)/0.0101/$10/$10-$35/0.0101/$10/0.99), "stat.txt" u 2:52 w l lt 2 # uniform plot "stat.txt" u 2:(-$40*(1.0-$30)*(0.0101-2.0)/0.0101/$11/$11-$40/0.0101/$11/0.99), "stat.txt" u 2:54 w l lt 2 plot "stat.txt" u 2:(-$45*(1.0-$45)*(0.0101-2.0)/0.0101/$12/$12-$45/0.0101/$12/0.99), "stat.txt" u 2:56 w l lt 2 # almost mixed # <v^2> = [(2+r)/(1+r)]b/(<R> rho1) - b(1+b)/<R>^2 plot "stat.txt" u 2:((2.0+9.0)/(1.0+9.0)*(-$25)/$8/0.1+$25*(1.0-$25)/$8/$8), "stat.txt" u 2:48 w l lt 2 # no-mix plot "stat.txt" u 2:((2.0+9.0)/(1.0+9.0)*(-$30)/$9/0.1+$30*(1.0-$30)/$9/$9), "stat.txt" u 2:50 w l lt 2 plot "stat.txt" u 2:((2.0+9.0)/(1.0+9.0)*(-$35)/$10/0.1+$35*(1.0-$35)/$10/$12), "stat.txt" u 2:52 w l lt 2 # uniform plot "stat.txt" u 2:((2.0+9.0)/(1.0+9.0)*(-$40)/$11/0.1+$40*(1.0-$40)/$11/$12), "stat.txt" u 2:54 w l lt 2 plot "stat.txt" u 2:((2.0+9.0)/(1.0+9.0)*(-$45)/$12/0.1+$45*(1.0-$45)/$12/$12), "stat.txt" u 2:56 w l lt 2 # almost mixed # <v^2> = [b/<R>^2][1 + (b/b_{nm})[<R>^2/(rho_1rho_2) - 1]] with b_{nm} = r^2/(1+r) [X(1-X)] plot "stat.txt" u 2:((-$25/$8/$8)*(1.0-$25/(9.0*9.0/(1.0+9.0)*$3*(1.0-$3))*($8*$8/0.1-1.0))), "stat.txt" u 2:48 w l lt 2 # no-mix plot "stat.txt" u 2:((-$30/$9/$9)*(1.0-$30/(9.0*9.0/(1.0+9.0)*$4*(1.0-$4))*($9*$9/0.1-1.0))), "stat.txt" u 2:50 w l lt 2 plot "stat.txt" u 2:((-$35/$10/$10)*(1.0-$35/(9.0*9.0/(1.0+9.0)*$5*(1.0-$5))*($10*$10/0.1-1.0))), "stat.txt" u 2:52 w l lt 2 # uniform plot "stat.txt" u 2:((-$40/$11/$11)*(1.0-$40/(9.0*9.0/(1.0+9.0)*$6*(1.0-$6))*($11*$11/0.1-1.0))), "stat.txt" u 2:54 w l lt 2 plot "stat.txt" u 2:((-$45/$12/$12)*(1.0-$45/(9.0*9.0/(1.0+9.0)*$7*(1.0-$7))*($12*$12/0.1-1.0))), "stat.txt" u 2:56 w l lt 2 # almost mixed # (b/b_{nm}) plot "stat.txt" u 2:(-$25/(9.0*9.0/(1.0+9.0)*$3*(1.0-$3))) # <v^2> = b(1+b)/<R>^2 + (b/<R>) [ (2+r)/rho_2 - 2(1+b)/<R> ] (b/b_{nm})^{1/2} where b_{nm} = r^2/(1+r) [X(1-X)], high At plot "stat.txt" u 2:((-$25*(1.0-$25)/$8/$8 - $25/$8 * ((2.0+9.0)-2.0*(1.0-$25)/$8)) * (-$25/(9.0*9.0/(1.0+9.0)*($3*(1.0-$3))))**(1.0/2.0)), "stat.txt" u 2:48 w l lt 2 # no-mix plot "stat.txt" u 2:((-$30*(1.0-$30)/$9/$9 - $30/$9 * ((2.0+9.0)-2.0*(1.0-$30)/$9)) * (-$30/(9.0*9.0/(1.0+9.0)*($4*(1.0-$4))))**(1.0/2.0)), "stat.txt" u 2:50 w l lt 2 plot "stat.txt" u 2:((-$35*(1.0-$35)/$10/$10 - $35/$10 * ((2.0+9.0)-2.0*(1.0-$35)/$10)) * (-$35/(9.0*9.0/(1.0+9.0)*($5*(1.0-$5))))**(1.0/2.0)), "stat.txt" u 2:52 w l lt 2 # uniform plot "stat.txt" u 2:((-$40*(1.0-$40)/$11/$11 - $40/$11 * ((2.0+9.0)-2.0*(1.0-$40)/$11)) * (-$40/(9.0*9.0/(1.0+9.0)*($6*(1.0-$6))))**(1.0/2.0)), "stat.txt" u 2:54 w l lt 2 plot "stat.txt" u 2:((-$45*(1.0-$45)/$12/$12 - $45/$12 * ((2.0+9.0)-2.0*(1.0-$45)/$12)) * (-$45/(9.0*9.0/(1.0+9.0)*($7*(1.0-$7))))**(1.0/2.0)), "stat.txt" u 2:56 w l lt 2 # almost mixed # <v^2> = b(1+b)/<R>^2 + (b/<R>) [ (2+r)/rho_2 - 2(1+b)/<R> ] (b/b_{nm})^{1/2} where b_{nm} = r^2/(1+r) [X(1-X)], low At plot "stat.txt.low" u 2:((-$25*(1.0-$25)/$8/$8 - $25/$8 * ((2.0+0.0101)-2.0*(1.0-$25)/$8)) * (-$25/(0.0101*0.0101/(1.0+0.0101)*($3*(1.0-$3))))**(1.0/2.0)), "stat.txt.low" u 2:48 w l lt 2 # no-mix plot "stat.txt.low" u 2:((-$30*(1.0-$30)/$9/$9 - $30/$9 * ((2.0+0.0101)-2.0*(1.0-$30)/$9)) * (-$30/(0.0101*0.0101/(1.0+0.0101)*($4*(1.0-$4))))**(1.0/2.0)), "stat.txt.low" u 2:50 w l lt 2 plot "stat.txt.low" u 2:((-$35*(1.0-$35)/$10/$10 - $35/$10 * ((2.0+0.0101)-2.0*(1.0-$35)/$10)) * (-$35/(0.0101*0.0101/(1.0+0.0101)*($5*(1.0-$5))))**(1.0/2.0)), "stat.txt.low" u 2:52 w l lt 2 # uniform plot "stat.txt.low" u 2:((-$40*(1.0-$40)/$11/$11 - $40/$11 * ((2.0+0.0101)-2.0*(1.0-$40)/$11)) * (-$40/(0.0101*0.0101/(1.0+0.0101)*($6*(1.0-$6))))**(1.0/2.0)), "stat.txt.low" u 2:54 w l lt 2 plot "stat.txt.low" u 2:((-$45*(1.0-$45)/$12/$12 - $45/$12 * ((2.0+0.0101)-2.0*(1.0-$45)/$12)) * (-$45/(0.0101*0.0101/(1.0+0.0101)*($7*(1.0-$7))))**(1.0/2.0)), "stat.txt.low" u 2:56 w l lt 2 # almost mixed # b = (rho2 r)^2<x^2> / <R>^2 * [1 + <x^2>/<x^2>_{nm} [<R>^2/(rho_1rho_2) - 1]] with <x^2>_{nm} = [X(1-X)], low At plot "stat.txt" u 2:(0.01*0.01*$18/$8/$8 * (1.0 + $18/($3*(1.0-$3)) * ($8*$8/0.99 - 1.0))), "stat.txt" u 2:(-$25) w l lt 2 # no-mix plot "stat.txt" u 2:(0.01*0.01*$19/$9/$9 * (1.0 + $19/($4*(1.0-$4)) * ($9*$9/0.99 - 1.0))), "stat.txt" u 2:(-$30) w l lt 2 plot "stat.txt" u 2:(0.01*0.01*$20/$10/$10 * (1.0 + $20/($5*(1.0-$5)) * ($10*$10/0.99 - 1.0))), "stat.txt" u 2:(-$35) w l lt 2 # uniform plot "stat.txt" u 2:(0.01*0.01*$21/$11/$11 * (1.0 + $21/($6*(1.0-$6)) * ($11*$11/0.99 - 1.0))), "stat.txt" u 2:(-$40) w l lt 2 # plot "stat.txt" u 2:(0.01*0.01*$22/$12/$12 * (1.0 + $22/($7*(1.0-$7)) * ($12*$12/0.99 - 1.0))), "stat.txt" u 2:(-$45) w l lt 2 # almost mixed # b = (rho2 r)^2<x^2> / <R>^2 * [1 + <x^2>/<x^2>_{nm} [<R>^2/(rho_1rho_2) - 1]] with <x^2>_{nm} = [X(1-X)], high At plot "stat.txt" u 2:(0.9*0.9*$18/$8/$8 * (1.0 + $18/($3*(1.0-$3)) * ($8*$8/0.1 - 1.0))), "stat.txt" u 2:(-$25) w l lt 2 # no-mix plot "stat.txt" u 2:(0.9*0.9*$19/$9/$9 * (1.0 + $19/($4*(1.0-$4)) * ($9*$9/0.1 - 1.0))), "stat.txt" u 2:(-$30) w l lt 2 plot "stat.txt" u 2:(0.9*0.9*$20/$10/$10 * (1.0 + $20/($5*(1.0-$5)) * ($10*$10/0.1 - 1.0))), "stat.txt" u 2:(-$35) w l lt 2 # uniform plot "stat.txt" u 2:(0.9*0.9*$21/$11/$11 * (1.0 + $21/($6*(1.0-$6)) * ($11*$11/0.1 - 1.0))), "stat.txt" u 2:(-$40) w l lt 2 # plot "stat.txt" u 2:(0.9*0.9*$22/$12/$12 * (1.0 + $22/($7*(1.0-$7)) * ($12*$12/0.1 - 1.0))), "stat.txt" u 2:(-$45) w l lt 2 # almost mixed