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Thermal fluctuations, Langevin dynamics, Brownian motion and diffusion, Fokker-Planck equations, linear response theory, fluctuation-dissipation relations, the Boltzmann equation, critical phenomena, scaling and critical exponents.

Lecture Name
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
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
Lecture 3: The Langevin model (Part 2)
  • Mean squared velocity
  • Relation between noise strength and friction: fluctuation-dissipation (FD) theorem
  • Velocity autocorrelation function
  • Stationarity of the velocity process
  • Mean squared displacement and diffusion constant
Lecture 4: The Langevin model (Part 3)
  • Variance of the displacement for a free Brownian particle
  • Conditional PDF of the velocity: Ornstein-Uhlenbeck (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
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
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
Lecture 7: Linear response theory (Part 2)
  • Equilibrium density matrix in the canonical ensemble
  • Time dependent perturbation of a Hamiltonian system
  • First-order correction to the density operator
  • First-order correction to the mean value of an observable
  • Linear, causal, retarded response
  • Definition of the response function
Lecture 8: Linear response (Part 3)
  • Equivalent expressions for the response function
  • Response to a sinusoidal force and generalized susceptibility
  • Symmetry properties of the frequency-dependent susceptibility
  • Double-time retarded Green function
  • Spectral function and its relation to the generalized susceptibility
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
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
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
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 two-time correlation functions
  • Fourier representation of two-time anticommutator
Lecture 13: Quiz 1 - Questions and answers
Lecture 14: Linear response theory (Part 8)
  • Symmetry or antisymmetry of the response function under time-reversal
  • Spectral function as the real or imaginary part of the susceptibility
  • Equilibrium averages of equal-time commutators and moments of the spectral function
  • High-frequency expansion of the susceptibility
Lecture 15: Linear response theory (Part 9)
  • Derivation of the response in the Heisenberg picture
  • Differential and integral equations for the time-development operator
  • Solution to first order in the perturbation
  • Expression for the response function
  • General relation between power spectra of the response and fluctuations
Lecture 16: The dynamic mobility
  • Definition of the mobility of a Brownian particle
  • Zero-frequency mobility and diffusion constant
  • Dynamic mobility as a generalized susceptibility
  • Consistency of the Langevin model with linear response theory
  • Non-diffusive behaviour of a Brownian oscillator
Lecture 17: Fokker-Planck equations (Part 1)
  • Langevin equation (LE) for a general diffusion process
  • Corresponding Fokker-Planck 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
Lecture 18: Fokker-Planck 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 high-friction limit): Smoluchowski equation for the positional PDF
  • Overdamped oscillator: OU distribution for the positional PDF
Lecture 19: Fokker-Planck equations (Part 3)
  • Stationary solution of the Smoluchowski equation
  • Thermally-assisted escape over a potential barrier
  • Kramers' escape rate formula
  • Diffusion in a constant force field: sedimentation
Lecture 20: The generalized Langevin equation (Part 1)
  • Inconsistency in the Langevin model: non-stationarity of the velocity
  • Divergence of mean squared acceleration
  • Generalized Langevin equation and memory kernel
  • Frequency-dependent friction
  • Dynamic mobility in the generalized model
Lecture 21: The generalized Langevin equation (Part 2)
  • Kubo-Green formula for the mobility: first FD theorem
  • Consistency of the model with stationarity and causality
  • Cross-correlation between the noise and the velocity
  • Relation between noise autocorrelation and memory kernel: second FD
Lecture 22: Diffusion in a magnetic field
  • Langevin equations for position and velocity with a velocity-dependent 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
Lecture 23: The Boltzmann equation for a dilute gas (Part 1)
  • Single-particle phase space
  • Equation for number density in the absence of collisions
  • Binary collisions and two-particle scattering
  • The collision integral
  • The Boltzmann equation
Lecture 24: The Boltzmann equation for a dilute gas (Part 2)
  • The equilibrium distribution: sufficiency condition
  • Boltzmann's H-Theorem
  • The equilibrium distribution: necessary condition
  • The Maxwell-Boltzmann distribution
  • Equilibrium distribution in a potential
Lecture 25: The Boltzmann equation for a dilute gas (Part 3)
  • Remarks on the H-Theorem
  • Detailed balance and equilibrium distribution
  • Collision invariants and equations of continuity
  • Linearization of the Boltzmann equation close to equilibrium
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 Kubo-Anderson Markov process
  • Relaxation of a non-uniform distribution in the position variable
Lecture 27: The Boltzmann equation for a dilute gas (Part 5)
  • Relaxation of a non-uniform gas
  • Frequency-dependent diffusion coefficient
  • The diffusion constant
  • Shift of the equilibrium velocity distribution under a uniform force
Lecture 28: Quiz 2 - Questions and answers
Lecture 29: Critical phenomena (Part 1)
  • Recapitulation of thermodynamics
  • Intensive and extensive variables
  • Phase diagram for a single component substance
  • Liquid-gas coexistence line and the critical point
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
  • Fluid-magnet analogy
Lecture 31: Critical phenomena (Part 3)
  • Fluid-magnet analogy (contd.): phase diagrams
  • Ising model with nearest-neighbour interaction
  • Mean field theory (MFT) for the Ising model
  • Critical temperature in MFT
  • Critical exponents in MFT
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
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
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
Lecture 35: Critical phenomena (Part 7)
  • Time-dependent Landau-Ginzburg equation
  • Langevin equation for the order parameter
  • Fokker-Planck equation for configuration probability
  • Linearized LE and relaxation to equilibrium
  • Critical slowing down
  • Dynamic scaling hypothesis

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 Feynman-Kac formula and generalizations

Basics of equilibrium statistical mechanics

  1. V. Balakrishnan, Elements of Nonequilibrium Statistical Mechanics, Ane Books, Delhi & CRC Press, 2008.(Chapters 1-4, 6, 9, 11-13, 15-17.)
  2. N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Levant Books, Kolkata, India, 2005. (Chapters 1, 5, 8.)
  3. K. Huang, Statistical Mechanics, 2nd edition, Wiley, New York, 1987. (Chapters 3, 4, 16, 17.)
  4. M. Kardar, Statistical Physics of Fields, Cambridge University Press, Cambridge, 2007. (Chapters 3, 4.)
  5. R. Kubo, M. Toda and N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics, Springer-Verlag, Berlin, 1985. (Chapters 1, 2, 4.)
  6. L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd edition, Pergamon, New York, 1980. (Chapter 12.)
  7. G. F. Mazenko, Nonequilibrium Statistical Mechanics, Wiley-VCH, Weinheim, 2006. (Chapters 1, 2, 7, 8.)
  8. H. Risken, The Fokker-Planck Equation, Springer-Verlag, New York, 1996. (Chapters 2-4, 6.)
  9. H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, Oxford, 1989. (Chapters 1, 3, 5, 6, 10-12.)

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