S.No 
Lecture Name 
1

Lecture 1: Recapitulation of equilibrium statistical mechanics
 Isolated system in thermal equilibrium
 Fundamental postulate of equilibrium statistical mechanics
 Microcanonical ensemble
 Boltzmann's formula for the entropy
 Connection with thermodynamics
 Closed systems and the canonical ensemble
 Canonical partition function

2 
Lecture 2: The Langevin model (Part 1)
 Brownian particle in a
uid
 Langevin model
 Equation of motion including thermal noise
 Conditional and thermal averages
 The need to include a dissipative random force

3 
Lecture 3: The Langevin model (Part 2)
 Mean squared velocity
 Relation between noise strength and friction:
fluctuationdissipation (FD)
theorem
 Velocity autocorrelation function
 Stationarity of the velocity process
 Mean squared displacement and diffusion constant

4 
Lecture 4: The Langevin model (Part 3)
 Variance of the displacement for a free Brownian particle
 Conditional PDF of the velocity: OrnsteinUhlenbeck (OU) distribution
 Langevin equation (LE) for a Brownian particle in a magnetic field
 Velocity autocorrelation tensor in a magnetic field
 Explicit solution for the correlation tensor

5 
Lecture 5: The Langevin model (Part 4)
 Velocity correlation tensor for t ≥ 0 and t ≤ 0
 Symmetric and antisymmetric parts of the correlation tensor
 Diffusion tensor in a magnetic field: longitudinal and transverse parts
 Conditional PDF of the velocity in a magnetic field: modified OU distribution
 Linear response theory: introductory remarks

6 
Lecture 6: Linear response theory (Part 1)
 Classical and quantum equations of motion in Hamiltonian dynamics
 Liouville operator and its hermiticity
 Unitarity of the time evolution operator
 Density matrix; pure and mixed states
 Liouville and von Neumann equations for the density operator
 Expectation value of a physical observable

7 
Lecture 7: Linear response theory (Part 2)
 Equilibrium density matrix in the canonical ensemble
 Time dependent perturbation of a Hamiltonian system
 Firstorder correction to the density operator
 Firstorder correction to the mean value of an observable
 Linear, causal, retarded response
 Definition of the response function

8 
Lecture 8: Linear response (Part 3)
 Equivalent expressions for the response function
 Response to a sinusoidal force and generalized susceptibility
 Symmetry properties of the frequencydependent susceptibility
 Doubletime retarded Green function
 Spectral function and its relation to the generalized susceptibility

9 
Lecture 9: Linear response(Part 4)
 Susceptibility for an oscillator in a
fluid
 Poles of the oscillator susceptibility in the complex frequency plane
 Simplification of the general expression for the response function
 Simplified expression in the classical case
 Kubo canonical correlation in the quantum mechanical case

10 
Lecture 10: Linear response (Part 5)
 Canonical correlation functions
 Response function as a canonical correlation
 Properties of canonical correlations: stationarity, symmetry and reality
 Physical implication of reality property
 Analyticity of the susceptibility in the upper half frequency plane

11 
Lecture 11: Linear response (Part 6)
 Dispersion relations for the real and imaginary parts of the susceptibility
 Asymptotic behavior of the susceptibility and subtracted dispersion relations
 Case of a singular DC susceptibility
 Response function in terms of matrix elements of observables
 Susceptibility in terms of transition frequencies

12 
Lecture 12: Linear response theory (Part 7)
 Spectral function in terms of the transition frequencies of a system
 Master analytic function from the spectral function
 Boundary values of the master function: Retarded and advanced susceptibilities
 Fourier representation of twotime correlation functions
 Fourier representation of twotime anticommutator

13 
Lecture 13: Quiz 1  Questions and answers 
14 
Lecture 14: Linear response theory (Part 8)
 Symmetry or antisymmetry of the response function under timereversal
 Spectral function as the real or imaginary part of the susceptibility
 Equilibrium averages of equaltime commutators and moments of the spectral function
 Highfrequency expansion of the susceptibility

15 
Lecture 15: Linear response theory (Part 9)
 Derivation of the response in the Heisenberg picture
 Differential and integral equations for the timedevelopment operator
 Solution to first order in the perturbation
 Expression for the response function
 General relation between power spectra of the response and
fluctuations

16 
Lecture 16: The dynamic mobility
 Definition of the mobility of a Brownian particle
 Zerofrequency mobility and diffusion constant
 Dynamic mobility as a generalized susceptibility
 Consistency of the Langevin model with linear response theory
 Nondiffusive behaviour of a Brownian oscillator

17 
Lecture 17: FokkerPlanck equations (Part 1)
 Langevin equation (LE) for a general diffusion process
 Corresponding FokkerPlanck equation (FPE) for the conditional PDF
 Case of linear drift and constant diffusion coefficients
 Examples: FPE for the velocity PDF, diffusion equation for the positional
PDF
 FPE for the phase space PDF of a Brownian particle
 Generalization to three dimensions

18 
Lecture 18: FokkerPlanck equations (Part 2)
 FPE for general (nonlinear) drift and diffusion coefficients in the multi
dimensional case
 Kramers' equation for phase space PDF in an applied potential
 Asymptotic form of the phase space PDF
 Diffusion regime (or highfriction limit): Smoluchowski equation for the
positional PDF
 Overdamped oscillator: OU distribution for the positional PDF

19 
Lecture 19: FokkerPlanck equations (Part 3)
 Stationary solution of the Smoluchowski equation
 Thermallyassisted escape over a potential barrier
 Kramers' escape rate formula
 Diffusion in a constant force field: sedimentation

20 
Lecture 20: The generalized Langevin equation (Part 1)
 Inconsistency in the Langevin model: nonstationarity of the velocity
 Divergence of mean squared acceleration
 Generalized Langevin equation and memory kernel
 Frequencydependent friction
 Dynamic mobility in the generalized model

21 
Lecture 21: The generalized Langevin equation (Part 2)
 KuboGreen formula for the mobility: first FD theorem
 Consistency of the model with stationarity and causality
 Crosscorrelation between the noise and the velocity
 Relation between noise autocorrelation and memory kernel: second FD
theorem

22 
Lecture 22: Diffusion in a magnetic field
 Langevin equations for position and velocity with a velocitydependent force
 Smoluchowski equation for positional PDF
 Identification and calculation of the diffusion tensor
 FPE for the radial distance PDF in Brownian motion
 Corresponding LE with a drift term for the radial distance

23 
Lecture 23: The Boltzmann equation for a dilute gas (Part 1)
 Singleparticle phase space
 Equation for number density in the absence of collisions
 Binary collisions and twoparticle scattering
 The collision integral
 The Boltzmann equation

24 
Lecture 24: The Boltzmann equation for a dilute gas (Part 2)
 The equilibrium distribution: sufficiency condition
 Boltzmann's HTheorem
 The equilibrium distribution: necessary condition
 The MaxwellBoltzmann distribution
 Equilibrium distribution in a potential

25 
Lecture 25: The Boltzmann equation for a dilute gas (Part 3)
 Remarks on the HTheorem
 Detailed balance and equilibrium distribution
 Collision invariants and equations of continuity
 Linearization of the Boltzmann equation close to equilibrium

26 
Lecture 26: The Boltzmann equation for a dilute gas (Part 4)
 Single relaxation time approximation to the collision integral
 Relaxation of the velocity
 Equivalence to a KuboAnderson Markov process
 Relaxation of a nonuniform distribution in the position variable

27 
Lecture 27: The Boltzmann equation for a dilute gas (Part 5)
 Relaxation of a nonuniform gas
 Frequencydependent diffusion coefficient
 The diffusion constant
 Shift of the equilibrium velocity distribution under a uniform force

28 
Lecture 28: Quiz 2  Questions and answers 
29 
Lecture 29: Critical phenomena (Part 1)
 Recapitulation of thermodynamics
 Intensive and extensive variables
 Phase diagram for a single component substance
 Liquidgas coexistence line and the critical point

30 
Lecture 30: Critical phenomena (Part 2)
 Extensivity of thermodynamic potentials
 Some convexity properties of thermodynamic potentials
 Divergence of specific heat at the critical point
 Simplest magnetic equation of state
 Fluidmagnet analogy

31 
Lecture 31: Critical phenomena (Part 3)
 Fluidmagnet analogy (contd.): phase diagrams
 Ising model with nearestneighbour interaction
 Mean field theory (MFT) for the Ising model
 Critical temperature in MFT
 Critical exponents in MFT

32 
Lecture 32: Critical phenomena (Part 4)
 Definition of specific heat, order parameter, susceptibility and critical isotherm
exponents
 Difference between actual and MFT values of critical exponents
 Static susceptibility formula
 Correlation length
 Critical exponent for the divergence of the correlation length

33 
Lecture 33: Critical phenomena (Part 5)
 Equation of state in the Ising model
 Magnetization versus magnetic field for different temperatures
 Landau expansion for the free energy
 Criterion for the validity of MFT
 Upper critical dimensionality in the Ising universality class

34 
Lecture 34: Critical phenomena (Part 6)
 Scaling functions
 Relations between critical exponents
 Landau free energy functional
 Equilibrium configuration of the order parameter
 Relaxation to equilibrium configuration

35 
Lecture 35: Critical phenomena (Part 7)
 Timedependent LandauGinzburg equation
 Langevin equation for the order parameter
 FokkerPlanck equation for configuration probability
 Linearized LE and relaxation to equilibrium
 Critical slowing down
 Dynamic scaling hypothesis

36 
Lecture 36: The Wiener process (standard Brownian motion)
 The Wiener process (standard Brownian motion)
 Sample path properties
 Iterated logarithm law and arcsine law
 Functionals of the Wiener process
 Itô calculus: basic rules
 The FeynmanKac formula and generalizations
