OUTLINE
Part I: Deterministic processes
Sec. 1: Single variable
Lec. 1: Construction of Equations of Motion
Math: Taylor series, differentials, and construction of ODEs
Physics: continuity, locality (and possibly symmetry) leading to ODEs
Goal: Justify linear ODEs
m\ddot x + g \dot x = a+bx+cx^2/2+…
1.1.1 Continuity and Taylor series expansions
1.1.2 Small displacements and linear response
1.1.3 Ordinary differential equation (ODE)
1.1.4 The exponential solution
1.1.5 Time reversal symmetry
1.1.6 Energy function; gradient descent and conservation
Lec. 2: Solving 1st order linear ODEs
Math: Series expansion for real exponentials, integration of inverse polynominals
Physics: decay or growth to a steady state
Goal: Solve equations fo the form
\dot x= -bx, \dot x =bx-cx^2, and general \dot x= F(x)
1.2.1 General solution to first order ODEs
1.2.2 Population Growth; logistic equation
1.2.3 Symmetry breaking
1.2.4 Universality; critical slowing down
Lec. 3: Solving 2nd order linear ODEs
Math: Complex numbers, analyticity, Series expansion for complex exponentials, sine and cosine via manipulations of complex exponentials
Physics: Time reversal symmetry and oscillations, conservation of energy
Goal: Solve the equations
\ddot x =-w^2 x, 1st integral of \ddot x=F(x)
1.3.1 General solution for second order ODEs
1.3.2 Complex Exponentials and Simple Harmonic Oscillator (SHO)
1.3.3 Geometric representation of complex exponentials
1.3.4 Addition of complex numbers
1.3.5 Hyperbolic trigonometric functions
1.4.6 Resonance
1.4.7 Transients
Lec. 4: General inhomogeneous linear ODEs
Math: General solutions of homogeneous linear ODEs, special solution of inhomogeneous ODE, and superposition principle
Physics: Forcing, response and transients in damped Harmonic oscillator
Goal: Solve the equations
\ddot x +\gamma\dot x+w^2 x=f\cos( v t)
1.4.1 General solution to linear ODEs
1.4.2 General Damped Harmonic Motion
1.4.3 Forms of Damped Motion
1.4.4 Inhomogeneous linear ODEs
1.4.5 Steady-state solutions to forced harmonic motion
Num 1: Numerical techniques for solving 1D ODEs
Sec. 2: Many variables
Lec. 5: Many variables, vectors
Math: Vectors and index notation, differential/integral calculus of several variables, coordinate systems (cartesian/polar/spherical)
Physics: Coupled equations of motion, energy landscape
Goal: Understand equations of the form
\dot x_i=F_i({\vec x})
2.1.1 Two first order coupled ODEs
2.1.2 Gradient descent in two dimensions
2.1.3 Beyond Gradient descent
2.1.4 Hamiltonian Evolution
Lec. 6: Coupled linear ODEs, Matrices
Math: Matrix representation of equations of motion, rotation matrices
Physics: Equations of motion for a particle in a 2 or 3 dimensional quadratic potential
Goal: Understanding and solving some simple equations of the form
\dot x_i=M_{ij}x_j
2.2.1 Many coupled ODEs
2.2.2 Index notation
2.2.3 Eigenvectors and eigenvalues
2.2.4 Functions of a matrix
Lec. 7: Higher order coupled ODEs
Math: Eigenvalues and eigenvectors, symmetric/non-symmetric
Physics: How both rotation and decay can emerge from linearly coupled degrees of freedom
Goal: Solving some simple examples of the form
\dot x_i=M_{ij}x_j & \ddot x_i=M_{ij}x_j including motions of a particle in a 2 or 3 dimensional quadratic potential
2.3.1 General form
2.3.2 Normal modes
2.3.3 Chain of masses connected by springs
Lec. 8: Exploring symmetries in Matrices
Math: Symmetries and degeneracies, discrete Fourier modes
Physics: Using symmetries to find normal modes
Goal: Normal modes of n-particles connected by springs on a line/ring
2.4.1 Excahnge symmetry
2.4.2 Periodic Chain of blocks
2.4.3 Pinned chain of blocks
Num. 2: Numerical techniques for linear algebra and matrices
Sec. 3: Continuous fields
Lec. 9: Passing from many particles to continuum description
Math: Manipulating scalar fields in 1 dimension
Physics: Field theory formulation, gradient expansion
Goal: Understanding origins of partial differential equations
3.1.1 Continuum limit
3.1.2 Functional derivatives
3.1.3 The gradient expansion
3.1.4 Partial differential equations (PDEs)
3.1.5 Initial conditions, Boundary conditions
Lec. 10: Solving PDEs
Math: Partial differential equations, Gradient operator, Bessel functions
Physics: Modes of a drum
Goal: Finding normal modes of a drum for rectangular and circular shapes
3.2.1 Separable solutions
3.2.2 Quantized modes
3.2.3 Superposition of normal modes
3.2.4 Plucked string
Lec. 11: Fourier series
Math: Fourier series and Fourier integrals, orthogonality, Dirac delta function
Physics: Superposition of waves, diffusion equation
Goal: Some familiarity with Fourier decomposition
3.3.1 Fourier series
3.3.2 Complex exponentials
3.3.3 Fourier integrals
3.3.4 Diffusion
Lec. 12: Scalar field theories in 2 and higher dimensions
Math: Partial differential equations, Gradient operator, Bessel functions
Physics: Modes of a drum
Goal: Finding normal modes of a drum for rectangular and circular shapes
3.4.1 Locality, uniformity, and isotropy
3.4.2 Can you hear the shape of a drum?
3.4.3 Normal modes with a rectangular frame
3.4.4 Radial Laplacian on a circle
3.4.5 Circularly symmetric normal modes
3.4.6 Planar and circular travelling waves
Lec. 13: Circular and spherical coordinates
Math: Laplacian in these coordinates
Physics: Diffusion within a sphere
Goal: Generalized Laplacian
3.5.1 Laplacian on a circle, including angular variations
3.5.2 Spherical coordinates
3.5.3 Diffusion on a sphere
Lec. 14:More vector field theories
Math: Tensor notation, isotropy
Physics: Elastic energy of an isotropic material
Goal: Transverse and longitudinal modes of elastic material
3.6.1 Continuity equation
3.6.2 General change of coordinates
3.6.3 Cylindrical coordinates
3.6.4 Dynamics of vector fields
Num. 3: Numerical approaches for partial differential equations
Part II: Stochastic processes
Sec. 1: Probability
Lec. 15: Introduction to probability
Math: Binomial, multinomial, and Poisson
Physics: Motivate by Brownian motion, use coin flips to perform a random walk, Radioactive decay
Goal: Understand probability, mean and variance
4.1.1 Describing random change
4.1.2 Moments and cumulants
4.1.3 Binomial distribution
4.1.4 Poisson distribution
Lec. 16: One continuous random variable
Math: PDF, Generating functions via Fourier transforms
Physics: Change of variables
Goal: Moments of a Gaussian PDF
4.2.1 Probability density function (PDF)
4.2.2 Change of variables
4.2.3 The characteristic function
4.2.4 The Gaussian distribution
Lec. 17: Many random variables
Math: joint PDF and moments, Bayes’ rule, Matrix representation of joint Gaussian
Physics: Wick’s theorem
Goal: Understanding correlations
4.3.1 Joint probability density function
4.3.2 Joint moments and cumulants
4.3.3 Multi-variable Gaussian
Lec. 18: From probability to certainty
Math: Large N, saddle point,
Physics: Central limit theorem, information, entropy and estimation
Goal: Appreciating large N bimplifications
4.4.1 Sums of random variables
4.4.2 Simplifications for large N
4.4.3 Stirling's approximation
4.4.4 Information and entropy
4.4.5 Maximum entropy estimation
Sec. 2: Evolving probabilities
Lec. 19: Time dependent probabilities
Math: Markovian processes, Master equation, properties of rate matrices, matrix products
Physics: Time evolution of a DNA sequences
Goal: Time course of approach to a steady state
5.1.1 Evolving sequence
5.1.2 Steady state
5.1.3 Evolving binary sequence
5.1.4 The Master equation
5.1.5 Mutating population
Lec. 20: Drift-diffusion equation
Math: Drift-diffusion equation, eneral solution in steady state
Physics: Another perspective on diffusion
Goal: Finding steady states
5.2.1 Drift and diffusion
5.2.2 Steady states
5.2.3 Evolving composition of a population of fixed size
Lec. 21: Brownian motion
Math: Maximum entropy state, Lagrange multiplier
Physics: Einstein relation, fluctuation-dissipation relation
Goal: Understanding fluctuations in equilibrium
5.3.1 Langevin description
5.3.2 Fokker-Planck description
5.3.3 From dissipation to fluctuation
5.3.4 Maximum likelihood estimation
5.3.5 Einstein's relation
Lec. 22: Fluctuating (Gaussian) fields
Math: Fourier decomposition and dynamics
Physics: Fluctuating polymer, fluctuating membrane
Goal: Importance of fluctuations in thermal equilibrium