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CMSSW

CMSSW. ROOT. Monte-Carlo methods are used to: - evaluate difficult integrals examples : cross section, decay rate - sample complicated distribution function The simplest pdf (probablity distribution function): uniform

demetrius-kelly
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CMSSW

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  1. P.Miné Monte Carlo

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  3. P.Miné Monte Carlo

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  5. CMSSW ROOT P.Miné Monte Carlo

  6. Monte-Carlo methods are used to: - evaluate difficult integrals examples : cross section, decay rate - sample complicated distribution function The simplest pdf (probablity distribution function): uniform 0 < x < 1 all values of x have the same probabilty f(x) =1 Programs exist to give a value of x at each call: Example: TRandom in ROOT P.Miné Monte Carlo

  7. Random generators simple cases Exponential distribution (decay time of a particle) f(t) = (1 / τ ) exp(- t / τ ) in the interval [a, b] let a’ = exp(- a / τ ) and b’ = exp(- b / τ ) t = - τ ln ( b’ + u ( a’–b’) ) where u is uniform in [0, 1] then dN / dt = (dN / du) (du / dt) = 1 x (1 / τ ) exp(- t / τ ) / (a’ –b’) For [0 , ∞] take simply t = - τ ln u This method works every time the primitive F(x) of f(x) is invertible x = F-1(u) where u is a uniform probability variable P.Miné Monte Carlo

  8. Isotropy in 3D Density is proportional to solid angle dΩ = d(cosθ) dϕ cosθ is uniform in [-1, 1] : take 2u1 -1 ϕ uniform in [0, 2π] Gaussian variable are gaussian independent P.Miné Monte Carlo

  9. The acceptance rejection method (Von Neumann) • Assume that for any x, the probability distribution function f(x) can be calculated and its maximum C is known Take the envelope Ch(x) ,h(x) is uniform f(x) and h(x) are normalized so C > 1 For each value of x, generate the random variable u, with uniform distribution If u C h (x) < f(x) , accept x ; if not , reject Try again P.Miné Monte Carlo

  10. Importance sampling Increase the efficiency if C >> 1 by changing h(x) The best solution: change the variable (Jacobian method) P.Miné Monte Carlo

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  16. ν l W t b P.Miné Monte Carlo

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  19. Proton proton cross section P.Miné Monte Carlo

  20. Parton density functions are obtained experimentally by deep inelastic scattering of leptons P.Miné Monte Carlo

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  22. Practiceon ROOT Dowload from root.cern.ch/root/root_v5.22.00.source.tar.gz gzip -dc root-v5.22.00.source.tar.gz | tar –xf export ROOTSYS=<path>/root ./configure –help ./configure [<arch>] [set arch appropriately if not default] (g)make export PATH=$ROOTSYS/bin:$PATH export LD_LIBRARY_PATH=$ROOTSYS/lib:$LD_LIBRARY_PATH Exercise in tutorials P.Miné Monte Carlo

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