Probability Distributions
Assume we have discrete random variables.
Example multivariable,
For continuous random variables, we define a cumulative probability distribution function, CPF.
- Monotomically increasing
Probability density function, PDF.
has dimensions- No upper bound
Expectation value of a function
Mean:
Variance:
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
Fundamental assumption of SM (Ergodicity)
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
, high phase transition has - 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
- Abrupt (1st order) - order parameter changes discontinuously
Example: liquid-gas phase transition below the critical point - density change
- Critical Phenomena
- Self similarity. Looks the same at all length scales
- Universality.
- Power laws.
, This is implied by self-similarity.
Generator of moments is called characteristic function which is the Fourier transform of the PDF.
(1)
Generator for cumulents. (2)
(3)
(1) RHS = exp((2) RHS) and match powers (-ik)
. :=
. . :=
Normal distribution (HW 1.2)
Basis of perturbation theory in QFT and SM,