# 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:

• Monotomically increasing
• $$P(-\infty) = 0$$
• $$P(\infty) = 1$$

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

• $$p(-\infty) = 0$$
• $$p(\infty) = 0$$
• $$P(\infty) = \int_{-\infty}^\infty p(x) dx = 1$$
• $$p(x)> 0$$
• $$p(x)$$ has dimensions $$[x]^{-1}$$
• No upper bound

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, $$\int p_q\exp(\int dt\frac{i}{\hbar}S)$$, $$Z=\sum\exp(-\beta E_\alpha) = \int \mathcal{D}\phi\exp(-\int H(\phi))$$.

Created: 2023-06-25 Sun 02:31

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