Nonequilibrium Statistical Physics
Introduction
Equilibrium thermodynamics overview [nln6]
Thermal equilibrium and nonequilibrium [nln1]
Levels of description in statistical physics [nln2]
Random variables
Probability: Intuition - Ambiguity - Absurdity - Puzzles
- Two bus companies: regular versus random schedules
- Pick the winning die [nex2]
- Educated guess [nex4]
- Coincident birthdays [nex82]
- Win the new car or take the goat! [nex11]
- Three-cornered duel [nex13]
- Bad luck: waiting for the big one [nex74]
- Bertrand's paradox
- Random quadratic equations [nex12]
- Out of Africa [nex84]
Elements of probability theory
- Elements of set theory [nln4]
- Sample space, events
- Probability axioms
- Complement, intersection, union
- Joint probability
- Symmetry and elementary events
- Conditional probability
- Bayes' theorem
- Statistical independence
- Statistical uncertainty and information [nln5]
Applications and illustrations
- Set identities [nex88]
- Conditional probability [nex90]
- Subtlety of statistical independence [nex1]
- Field of sets [nex75]
- Successive random picks [nex91]
- Random train connections [nex92]
- Heads or tails [nex93]
- Event or complement? That is the question [nex9]
- Quantity and quality [nex76]
- Diagnosis of a rare disease [nex77]
- Equal events? [nex89]
- Statistical concept of uncertainty [tex47]
- Information and the reduction of ignorance [tex48]
- Information of sequenced messages [tex61]
Probability distributions
- Stochastic variables
- Moments, variance, standard deviation
- Moment expansion and characteristic function
- Cumulant expansion
- Factorial moments and cumulants, generating
function
- Transformation of random variables
- Multivariate distributions
- Stability of probability distributions
- Renormalization group transformation
- Linearized RG transformation
- Scaling properties
Applications
- Reconstructing probability distributions [nex14]
- Chebyshev inequality [nex6]
- Law of large numbers [nex7]
- Propagation of statistical uncertainty [nex24]
- Variances and covariances [nex20]
- Binomial and Poisson distributions [nex15]
- Moments and cumulants of the Poisson distribution
[nex16]
- Binomial and Gaussian distributions: De
Moivre
- Laplace theorem [nex21]
- Pascal distribution [nex22]
- Central limit theorem
- Robust probability distributions [nex19]
- Stable probability distributions [nex81]
- Random number sequences with stable probability
distributions
[nln3]
- Exponential distribution
- Erlang distribution
- Eigenfunctions of linearized RG transformation
for
index alpha=2 [nex83]
Illustrations
- Random inkjet printer [nex10]
- Random variable transformation Y=X12+X22
[nex8]
- Product of independent random variables [nex79]
- Generating exponential and Lorentzian random
numbers
[nex80]
- Transformation between uniform and Gaussian
distributions
[nex78]
- Statistically independent or merely uncorrelated?
[nex23]
- Gaussian shootist versus Lorentzian shootist
[nex3]
- Maxwell velocity distribution (Maxwell's
derivation)
[nex17]
- Random bus schedules [nex18]
Stochastic Processes
Fundamentals
- Time-dependent probability distributions
- Time-dependent correlation functions
- Classification of processes (factorizing p.,
Markov
p., non-Markov p.)
Markov processes
- General specifications
- Chapman-Kolmogorov equation
- Continuous versus discontinuous processes
(Lindeberg
condition)
- Computer generated sample paths [nsl1] [nsl2]
- Fokker-Planck equation (derived from C.-K.
equation)
- Differential Chapman-Kolmogorov equation
- Jump processes: master equation
- Detailed balance condition
- Diffusion processes: Fokker-Planck equation
- Deterministic processes: drift equation
- Derivation of Fokker-Planck equation from master
equation
- Overview of equations governing the time
evolution
of P(x,t|x0)
- Markov chains
- Master equation with detailed balance
- Birth death processes
- Stationary distribution of birth-death master
equation
- Nonlinear birth-death process
Applications
- Diffusion process and Cauchy process are Markov
processes
[nex26]
- Continuous versus continuous Markov process
[nex27]
- Master equation for a continuous random variable
[nex28]
- Drift equation [nex29]
- Equations of motion for mean value and variance
[nex30]
- Jump moments of the master equation [nex32]
- Fokker-Planck equation with constant coefficients
- Ornstein-Uhlenbeck process
- Stationary solution
- Solution of P(x,t|x0) [nex31]
- Ornstein Uhlenbeck process: general solution
[nex41]
- Regression theorem for autocorrelation functions
[nex39]
- Random walk in one dimension
- Master equation for random walk in one dimension
[nex33]
- Chapman-Kolmogorov equation for biased random
walk
[nex34]
- Random walk in Las Vegas [nex40]
- Poisson process [nex25]
- Shot noise
- Campbell processes [nex37]
- Remodel the non-Markovian house into a Markovian
house
[nex43]
- Mixing marbles red and white [nex42]
- Prey-predator system [nsl3]
- Linear birth-death process
- Catalyst-driven chemical reaction [nex46]
- Air escaping from a leaking container [nex48]
- Air remaining in the leaking container [nex49]
- Master equation yielding Pascal distribution at
stationarity
[nex50]
- Bistable chemical system [nex52]
- Stationary state for nonlinear birth-death
process
[nex51]
- Ultracold neutrons in an ideal Steyerl bottle
[nex47]
- Random frequency oscillator [nex35]
- Free particle with uncertain position and
velocity
[nex36]
- Life expectancy of the young and the old [nex38]
- Random light switch [nex45]
Brownian motion
- Relevant time scales
- Einstein's theory
- Smoluchovski equation
- Einstein relation
- Langevin's theory (on the same level of
contraction)
- Brownian motion as a time series [nsl4]
- Wiener process [nsl5]
- Non-differentiability of Wiener process [nex53]
- Autocorrelation function of Wiener process [nex54]
- Ballistic and diffusive regime of Langevin
solution
- Formal solution of Langevin's equation
- Velocity autocorrelation function of Brownian
particle
[nex55]
- Mean-square displacement of Brownian particle
[nex56],
[nex57]
- Ergodicity
- Intensity spectrum and spectral density
(Wiener-Khintchine
theorem)
- Fourier analysis of Langevin equation
- Generalized Langevin equation: attenuation with
memory
- Fluctuation-dissipation theorem
- Brownian harmonic oscillator [nsl6], [nsl7],
[nex58],
[nex59], [nex60]
- Generalized Langevin equation inferred from
microscopic
dynamics
- Brownian motion: levels of contraction and modes
of
description
- Brownian motion described by Fokker-Planck
equation
- Fokker-Planck equation for Brownian particle:
stationary
solution [nex61]
- Fokker-Planck equation for Brownian particle:
eigenvalue
problem [nex62]
Linear response and equilibrium dynamics
- Response function and generalized susceptibility
- Kubo formula for response function
- Kramers-Kronig dispersion relations
- Causality property of response function [nex63]
- Energy transfer between system and radiation field
- Reactive and absorptive part of response function
[nex64]
- Fluctuation-dissipation theorem
- Spectral representations [nex65]
- Linear response of classical realxator [nex66]
- Linear response of classical oscillator [nex67]
Zwanzig-Mori formalism
- Time-dependence of expectation values (quantum
and
classical)
- Zwanzig's kinetic equation: generalized master
equation
[nex68]
- Projection operator method (Mori formalism)
- Quantum and classical formulation
- Continued-fraction representation of relaxation
function
- Recursion method (algorithmic implementation of
Mori
formalism)
- Relaxation function with uniform
continued-fraction
coefficients [nex69]
- Continued-fraction expansion and moment expansion
- Genralized Langevin equation
- n-pole approximation
- Green's function formalism
- Structure function of harmonic oscillator
[nex71],
[nex72], [nex73]
- Scattering process and dynamic structure factor
- Electron scattering, neutron scattering, light
scattering
- Scattering from a free atom
- Scattering from an atom bound in a harmonic
potential
- Scattering from a harmonic crystal
Some Relevant Textbooks and
Monographs:
- L. E. Reichl: A modern course in statisitical
physics.
Wiley-Interscience, New York 1998.
- R. E. Wilde and S. Singh: Statistical
mechanics.
Fundamentals and modern applications. Wiley, New York 1998.
- C. W. Gardiner: Handbook of stochastic
methods for
physics, chemistry, and the natural sciences. Springer-Verlag, New
York
1985.
- W. Brenig: Statistical theory of heat.
Nonequilibrium
phenomena. Springer-Verlag, New York 1989.
- R. Kubo, M. Toda, and N. Hashitsume: Statistical
physics II. Nonequilibrium statistical mechanics. Springer-Verlag,
New
York 1985.
- E. Fick and G. Sauermann: The quantum
statistics
of dynamic processes. Springer-Verlag, New York 1990.
- J. McLennan: Introduction to nonequilibrium
statistical
mechanics. Prentice Hall 1989.
- S. W. Lovesey: Condensed matter physics.
Dynamic
correlations.Benjamin/ Cummings, Reading 1980.
- David Chandler: Introduction to modern
statistical
mechanics.Oxford University Press 1987
- C. Garrod: Statistical mechanics and
thermodynamics.
Oxford University Press 1995.
- W. Greiner, L. Neise, and H. Stöcker: Thermodyamics
and statistical mechanics. Springer-Verlag, New York 1995.
- M. Plischke and B. Bergersen: Equilibrium
statistical
physics. 2nd edition. World Scientific 1994.
- K. Huang: Statistical mechanics. Wiley,
New
York 1987.
- E. M. Lifshitz and L. P. Pitaevskii: Statistical
physics, part 1. Pergamon, New York 1980.
- R. K. Pathria: Statistical mechanics.
Pergamon,
New York 1972.
- J. Honerkamp: Statistical physics. An
advanced approach
with applications. Springer-Verlag, New York 1998.
- A. Papoulis: Probability, random variables,
and
stochastic processes. McGraw-Hill, New York 1991.
- R. F. Streater: Statistical dynamics. A stochastic
approach
to nonequilibrium thermodynamics. Imperial College Press, London
1995.
- R. Balescu: Statistical dynamics. Matter out of
equilibrium.
Imperial College Press, London 1997.
Do you have a question about any of the problems
[nex]?
Do you need a hint?
Do you wish to suggest additional problems?
Do you have a question about the lecture notes [nln,nsl]?
Did you find any mistakes?
[nsl] pages with no link contain figures from books. Ask
how to access these pages.
Drop a note to gmuller@uri.edu.
Last updated 02/14/07