Probability Distributions

Assume we have discrete random variables. Example multivariable, \(p(N_1,\cdots,N_k) = p_1^{N_1}\cdots p_k^{N_k}\frac{N!}{N_1!\cdots N_k!}\).

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

Probability density function, PDF. \(p(x) = \frac{dP(x)}{dx}\). Note:

Expectation value of a function \(F(x)\) is \(\langle F(x)\rangle = \int_{-\infty}^\infty dx p(x)F(x)\).

\(F(x)=x^n\) are called the moments of a PDF. \(\langle x^n\rangle = \int_{-\infty}^\infty dx p(x)x^n\).

Mean: \(\langle x\rangle\) is the first moment, the first cumulent

Variance: \(\langle |x-\langle x\rangle|^2\rangle=\sigma^2 = \langle x^2\rangle - \langle x\rangle^2\) can be expressed in the first and second moments, second cumulent.

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


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 \(t\). \(\langle A\rangle_\tau=\frac{1}{\tau}\in_{t_0}^{t_0+\tau}A(t)dt\) is related to the ensemble average.

\(\sum_\alpha A_\alpha\) - Monte Carlo Integration. \(\langle A\rangle_\alpha = \sum_\alpha p_\alpha A_\alpha\) with microstates.

Fundamental assumption of SM (Ergodicity) \(\langle A\rangle_\tau = \langle A\rangle_\alpha\).

\(\frac{\partial A}{\partial f} = \mathbb{C}_0f(\langle A-\langle A\rangle^2\rangle)\).

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 \((\lambda - |\eta-\eta_g|)\), high \(T\) phase transition has \(\lambda=0\)
    • 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. \(x^n\), \(x\to \lambda x\to(\lambda x)^n\) This is implied by self-similarity.

\(\langle F(x)\rangle = \int_{-\infty}^\infty dxF(x)p(x)\)

\(p_F(f)df = \text{prob} (F(x)\in[f,f+df])\)

\(p_F(f)df = \sum_i p(x_i)dx_i\) \(p_F(f) = sum_i p(x_i)\left(\frac{dx_i}{dF}\right)_{x=x_i}\).

Generator of moments is called characteristic function which is the Fourier transform of the PDF. \(\tilde{P}(k)=\int dxp(x)\exp(-ikx) = \langle\exp(-ikx)\rangle = \langle\sum_{n=0}^\infty\frac{(-ik)^n}{n!}x^n\rangle=\sum_{n=0}^\infty\frac{(-ik)^n}{n!}\langle x^n\rangle\).

(1) \(\tilde{P}(k)=\sum_{n=0}^\infty\frac{(-ik)^n}{n!}\langle x^n\rangle\).

Generator for cumulents. (2) \(\ln \tilde{P}(k) = \sum_{n=1}^\infty\frac{(-ik)^n}{n!}\langle x^n\rangle_c\).

(3) \(\ln(1+\varepsilon) = \sum_{n=1}^\infty(-1)^{n+1}\frac{\varepsilon^n}{n}\). This gives, \(\langle x\rangle_c = \langle x\rangle\), \(\langle x^2\rangle_c = \langle x^2\rangle-\langle x\rangle^2\), \(\langle x^3\rangle_c = \langle x^3\rangle - 3\langle x^2\rangle\langle x\rangle + 2\langle x\rangle^3\).

(1) RHS = exp((2) RHS) and match powers (-ik)\(^m\) gives \(\langle x^m\rangle \sum_{\{q_n\}}^{\sum nq_n=m}m!\prod_n\frac{1}{q_n!(n!)q_n}\langle x^n\rangle_c^{q_n}\).

. := \(\langle x\rangle\)

. . := \(\langle x^2\rangle\) = (. .) + . . = \(\langle x^2\rangle + \langle x\rangle^2\)

\(\langle x^3\rangle = (. . .) + (. .) . *3 + . . . = \langle x^3\rangle_c+3\langle x^2\rangle_c\langle x\rangle_c + \langle x\rangle_c^3\).

Normal distribution (HW 1.2)

\(p(x) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(\frac{-(x-\mu)^2}{2\sigma^2}\right)\).

\(\tilde{p}_N(k) = \int_{-\infty}^\infty\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(\frac{-(x-\mu)^2}{2\sigma^2}-ikx\right)dx\) Completing the square, \(x\to x+\frac{b}{a}\), then we can solve the integral by doubling it and solving it with a polar integral. \(\tilde{p}_N(k) = \exp\left(-ik\mu -\frac{k^2\sigma^2}{2}\right)\). \(\ln\tilde{p}_N(k) = -ik\langle x\rangle_c - \frac{k^2}{2}\langle x^2\rangle_c\). So \(\langle x\rangle_c = \mu\), \(\langle x^2\rangle_c = \sigma^2\), \(\langle x^k\rangle_c = 0\). \(\langle x\rangle = \mu\), \(\langle x^2\rangle = \sigma^2+\lambda^2\), \(\langle x^3\rangle = 3\sigma^2+\lambda^3\), \(\langle x^4\rangle = 3\sigma^2+6\sigma^2\lambda^2+\lambda^4\).

Basis of perturbation theory in QFT and SM, \(\int p_q\exp(\int dt\frac{i}{\hbar}S)\), \(Z=\sum\exp(-\beta E_\alpha) = \int \mathcal{D}\phi\exp(-\int H(\phi))\).

Author: Christian Cunningham

Created: 2024-05-30 Thu 21:18