Probability Distributions

Assume we have discrete random variables. Example multivariable, p(N1,,Nk)=p1N1pkNkN!N1!Nk!.

For continuous random variables, we define a cumulative probability distribution function, CPF. P(x)=p(E{,x}). Properties:

Probability density function, PDF. p(x)=dP(x)dx. Note:

Expectation value of a function F(x) is F(x)=dxp(x)F(x).

F(x)=xn are called the moments of a PDF. xn=dxp(x)xn.

Mean: x is the first moment, the first cumulent

Variance: |xx|2=σ2=x2x2 can be expressed in the first and second moments, second cumulent.

The normal distribution is fully specified by the first and second cumulent.

Notes

Ensembles: Snapshots of the system at different times, which are consistent to our boundary conditions. Assume we are looking at a property A and seeing how it evolves over a time t. Aτ=1τt0t0+τA(t)dt is related to the ensemble average.

αAα - Monte Carlo Integration. Aα=αpαAα with microstates.

Fundamental assumption of SM (Ergodicity) Aτ=Aα.

Af=C0f(AA2).

Phases (butterfly collecting) systems contain different symmetries at different boundary conditions.

Phase transitions.

  • Symmetry breaking
  • Order parameter (Not necessarily a number and not necessarily real, consider a vector or a vector with a length but two heads)
  • Only 2 categories
    • Abrupt (1st order) - order parameter changes discontinuously Example: liquid-gas phase transition below the critical point - density change (λ|ηηg|), high T phase transition has λ=0
    • Continuous (2nd order) - non-analytic contionuous at the phase transition. Order parameter change is continuous at the phase transition but non-analytic. Example: Ferromagnetic and paramagnetic phase transitions has M(T)
  • Critical Phenomena
    • Self similarity. Looks the same at all length scales
    • Universality.
    • Power laws. xn, xλx(λx)n This is implied by self-similarity.

F(x)=dxF(x)p(x)

pF(f)df=prob(F(x)[f,f+df])

pF(f)df=ip(xi)dxi pF(f)=sumip(xi)(dxidF)x=xi.

Generator of moments is called characteristic function which is the Fourier transform of the PDF. P~(k)=dxp(x)exp(ikx)=exp(ikx)=n=0(ik)nn!xn=n=0(ik)nn!xn.

(1) P~(k)=n=0(ik)nn!xn.

Generator for cumulents. (2) lnP~(k)=n=1(ik)nn!xnc.

(3) ln(1+ε)=n=1(1)n+1εnn. This gives, xc=x, x2c=x2x2, x3c=x33x2x+2x3.

(1) RHS = exp((2) RHS) and match powers (-ik)m gives xm{qn}nqn=mm!n1qn!(n!)qnxncqn.

. := x

. . := x2 = (. .) + . . = x2+x2

x3=(...)+(..).3+...=x3c+3x2cxc+xc3.

Normal distribution (HW 1.2)

p(x)=12πσ2exp((xμ)22σ2).

p~N(k)=12πσ2exp((xμ)22σ2ikx)dx Completing the square, xx+ba, then we can solve the integral by doubling it and solving it with a polar integral. p~N(k)=exp(ikμk2σ22). lnp~N(k)=ikxck22x2c. So xc=μ, x2c=σ2, xkc=0. x=μ, x2=σ2+λ2, x3=3σ2+λ3, x4=3σ2+6σ2λ2+λ4.

Basis of perturbation theory in QFT and SM, pqexp(dtiS), Z=exp(βEα)=Dϕexp(H(ϕ)).

Author: Christian Cunningham

Created: 2024-05-30 Thu 21:18

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