PH6607: Nonequillibrium Statistical Physics

Course Information

L-T-P-C3-0-0-3
Prerequisites None

Syllabus

Module 1: Introduction

Lecture hours: 5 · Tutorial hours: 0 · Lab hours: 0

Recap of equilibrium statistical mechanics needed for non-equilibrium statistical mechanics, Fluctuations and irreversibility in macroscopic systems, approach to the equilibrium state, Boltzmann transport equation, linear irreversible thermodynamics, Review of critical phenomena and scaling, dynamic scaling (Edwards-Wilkinson and KPZ models, time-dependent Ginzburg-Landau models) .

Module 2: Microscopic Approach - Langevin Equation and Fluctuation-Dissipation Relation

Lecture hours: 6 · Tutorial hours: 0 · Lab hours: 0

The Langevin equation, the Maxwellian distribution of velocities, equation of motion of the Brownian particle, The fluctuation-dissipation relation, conditional and complete ensemble averages, Conditional mean squared velocity, velocity correlation time, stationarity of a random process, Velocity autocorrelation in three dimensions, time reversal property of the correlation matrix .

Module 3: Dynamics of a Brownian Particle

Lecture hours: 6 · Tutorial hours: 0 · Lab hours: 0

Free particle, mean squared displacement in equilibrium, time scales in the Langevin model, Equilibrium autocorrelation function, conditional autocorrelation function, fluctuations in the displacement, Velocity-position cross-correlation, Particles in an external potential, Langevin equation in an external potential, the Brownian oscillator .

Module 4: Fokker-Planck Equation

Lecture hours: 8 · Tutorial hours: 0 · Lab hours: 0

Diffusion process and diffusion equation, the fundamental Gaussian solution, diffusion in three dimensions, Smoluchowski equation for the oscillator, Fokker-Planck equation and its relation with the stochastic differential equation, Kramers' escape rate formula, the Ornstein-Uhlenbeck distribution .

Module 5: Green-Kubo Formulas

Lecture hours: 8 · Tutorial hours: 0 · Lab hours: 0

Diffusion constant in terms of the velocity autocorrelation, generalization to three dimensions, The mobility, relation between diffusivity and the static mobility, the dynamic mobility, Kubo-Green formula for the dynamic mobility, application to the Brownian oscillator .

Module 6: Generalised Langevin Equation

Lecture hours: 3 · Tutorial hours: 0 · Lab hours: 0

Memory kernel, frequency dependent friction, Maxwell model for visco-elastic fluids, Generalized fluctuation-dissipation theorem .

Module 7: Nonequilibrium Fluctuations

Lecture hours: 6 · Tutorial hours: 0 · Lab hours: 0

Brownian ratchets, Brownian motors, Thermodynamics at the molecular level, Fluctuation theorems, Crooks' theorem, Jarzynski equality .

Study Materials

Textbooks

  1. Elements of Nonequilibrium Statistical Mechanics, V. Balakrishnan, Springer, ISBN: 978-3-030-62232-9 / 978-3-030-62233-6
  2. Statistical Mechanics of Nonequilibrium Liquids, Denis J. Evans and Gary P. Morriss, Cambridge University Press, ISBN: 978-0521857918

References

  1. NPTEL Video Lectures on Non-equilibrium Statistical Mechanics, V. Balakrishnan
  2. An Introduction to Dynamics of Colloids, J. Dhont, Elsevier Science, ISBN: 0444820094
  3. Stochastic Processes in Physics and Chemistry, N. G. Van Kampen, ISBN: 0444529659
  4. A Modern Course in Transport Phenomena, D. C. Venerus, H. C. Oettinger, Cambridge University Press, ISBN: 1107129206