Nonequilibrium Statistical Physics
Free and easy access to complete set of presentable lecture
notes and exercises is available on URI
Digital Commons (downloadable pdf files covering entire
sections).
1. Introduction: Contents and Maps
- Table of contents [ntc]
- Equilibrium thermodynamics overview [nln6]
- Thermal equilibrium and nonequilibrium [nln1]
- Levels of description in statistical physics [nln2]
- Contraction - memory - time scales [nln15]
- Markov process: map of specifications [nln16]
- Brownian motion: panoramic view [nln23]
- Linear response and equilibrium dynamics [nln24]
- Stage for recursion method [nln79]
- Modules of recursion method [nln80]
2. Probability: Intuition - Ambiguity - Absurdity - Puzzles
3. Elements of Probability Theory with
Applications
- Event or complement? That is the question [nex9]
- Successive random picks [nex91]
- Heads or tails [nex93]
- Quantity and quality [nex76]
- Diagnosis of a rare disease [nex77]
- Subtlety of statistical independence [nex1]
- Random train connections [nex92]
- Random inkjet printer [nex10]
- Information and the reduction of ignorance [tex48]
- Information of sequenced messages [tex61]
4. Random Variables: Concepts
- Probability distributions [nln46]
- Characteristic function, moments, and
cumulants [nln47]
- Cumulants expressed in terms of moments [nex126]
- Generating function and factorial moments [nln48]
- Multivariate distributions [nln7]
- Transformation of random variables [nln49]
- Sums of independent exponentials [nex127]
- Propagation of statistical uncertainty [nex24]
- Chebyshev's inequality [nex6]
- Law of large numbers [nex7]
- Binomial, Poisson, and Gaussian distribution [nln8]
- Binomial to Poisson distribution
[nex15]
- De Moivre - Laplace limit theorem [nex21]
- Central limit theorem [nln9]
- Multivariate Gaussian distribution
- Robust probability distributions [nex19]
- Stable probability distributions [nex81]
- Exponential distribution [nln10]
- Waiting time problem [nln11]
- Pascal distribution [nex22]
5. Random Variables: Applications
- Reconstructing probability distributions [nex14]
- Probability distribution with no mean value [nex95]
- Variances and covariances [nex20]
- Statistically independent or merely
uncorrelated? [nex23]
- Sum and product of uniform distribution [nex96]
- Exponential integral distribution [nex79]
- Generating exponential and Lorentzian random
numbers [nex80]
- Random chords (Bertrand's paradox) [nex5]
- From Gaussian to exponential distribution [nex8]
- Transforming a pair of random variables [nex78]
- Gaussian shootist versus Lorentzian shootist [nex3]
- Moments and cumulants of the Poisson
distribution [nex16]
- Maxwell velocity distribution [nex17]
- Random bus schedules [nex18]
- Life expectancy of the young and the old [nex106]
- Life expectancy of the ever young [nex38]
- Random frequency oscillator [nex35]
6. Stochastic Processes: Concepts
- Time-dependent probability distributions [nln50]
- Correlation functions and characteristic functions
- Degrees of memory [nln51]
- Markovian or non-Markovian I [nln52]
- Markovian or non-Markovian II [nln53]
- Contraction - memory - time scales [nln15]
- Markov process: general attributes [nln54]
- Diffusion process and Cauchy process [nln55]
- Stationarity, normalization, consistency, Markovian
nature [nex26]
- Computer generated sample paths [nsl1]
- Continuous versus discontinuous processes
(Lindeberg condition) [nex97]
- Differential Chapman-Kolmogorov equation [nln56]
- Fokker-Planck equation (drift and diffusion
processes) [nln57]
- Drift equation (deterministic processes) [nex29]
- Master equation (jump processes) [nex28]
- Non-differentiability of sample paths [nex99]
- Predominantly small jumps [nln58]
- Time evolution of mean and variance [nln59]
- Master equation with finite jump moments [nex32]
- Equations of motion for mean and variance [nex30]
- Markov process: map of specifications [nln16]
- Approach to a stationary state (detailed
balance) [nex85]
- Markov chains [nln61]
- Master equation with detailed balance
(discrete variables, continuous time) [nln12]
- Regression theorem for autocorrelation
functions [nex39]
- Birth death processes (specifications, models,
levels of description) [nln18]
- Birth and death of single species [nln19]
- Birth-death master equation: stationary state
[nln17]
- Nonlinear birth-death process
7. Stochastic Processes: Applications
- Diffusion process [nex27]
- Cauchy process [nex98]
- Random walk in one dimension [nln60]
- Random walk in one dimension: unit steps at
unit times [nex34]
- Random walk in one dimension: unit steps at
random times [nex33]
- Random walk in one dimension: tiny steps at
frequent times [nex100]
- Random walk in Las Vegas: chance and necessity
[nex40]
- Poisson process [nex25]
- Free particle with uncertain position and
velocity [nex36]
- Fokker-Planck equation with constant
coefficients [nex101]
- House of the mouse: two-way doors only [nex102]
- House of the mouse: some one-way doors [nex103]
- House of the mouse: one-way doors only [nex104]
- House of the mouse: mouse with inertia [nex105]
- House of the mouse: mouse with memory [nex43]
- Mixing marbles red and white [nex42]
- Random traffic around city block [nex86]
- Modeling a Markov chain [nex87]
- Ornstein-Uhlenbeck process [nln62] [nex31]
[nex41]
- Predator-prey system: deterministic,
stochastic, observational [nsl3]
- Populations with linear birth and death rates
I [nex44]
- Populations with linear birth and death rates
II [nex112]
- Populations with linear birth and death rates
III [nex130]
- Catalyst-driven chemical reaction: stationary
state [nex46]
- Catalyst driven chemical reaction: dynamics [nex107]
- Catalyst driven chemical reaction: total rate
of reactions [nex108]
- Air in leaky tank I: generating function [nex48]
- Air in leaky tank II: probability distribution
[nex109]
- Air in leaky tank III: detailed balance [nex49]
- Air in leaky tank IV: evolution of mean and
variance [nex110]
- Pascal distribution and Planck radiation law [nex50]
- Effects of nonlinear death rate I:
Malthus-Verhulst equation [nex111]
- Effects of nonlinear death rate II:
stationarity and fluctuations [nex51]
- Modified linear birth rate I: stationarity [nex113]
- Modified linear birth rate II: evolution of
mean and variance [nex114]
- Modified linear birth rate III: generating
function [nex115]
- Modified linear birth rate IV: probability
distribution [nex116]
- Bistable chemical system [nex52]
- Ultracold neutrons in an ideal Steyerl bottle
[nex47]
- Random light switch [nex45]
8. Brownian Motion
- Early Landmarks [nln63]
- Relevant time scales (collisions, relaxation,
observations) [nln64]
- Einstein's theory [nln65]
- Diffusion equation analyzed [nln73]
- Release of Brownian particle from box
confinement [nex128]
- Smoluchowski equation [nln66]
- Einstein's fluctuation-dissipation relation [nln67]
- Smoluchowski vs Fokker-Planck [nln68]
- Fourier's law for heat conduction [nln69]
- Thermal diffusivity [nex117]
- Shot noise [nln70]
- Campbell's theorem [nex37]
- Critically damped ballistic galvanometer [nex70]
- Langevin's theory [nln71]
- White noise
- Brownian motion and Gaussian white noise [nln20]
- Wiener process [nsl4]
- Autocorrelation function of Wiener process [nex54]
- Attenuation without memory [nln21]
- Formal solution of Langevin equation [nex53]
- Velocity correlation function of Brownian
particle I [nex55]
- Mean-square displacement of Brownian particle
[nex56], [nex57],
[nex118]
- Ergodicity [nln13]
- Intensity spectrum and spectral density
(Wiener-Khintchine theorem) [nln14]
- Fourier analysis of Langevin equation
- Velocity correlation function of Bownian
particle II [nex119]
- Generalized Langevin equation [nln72]
- Attenuation with memory [nln22]
- Velocity correlation function of Brownian
particle III [nex120]
- Brownian harmonic oscillator [nln75]
- Brownian harmonic oscillator VII: equivalent
specifications [nex129]
- Brownian harmonic oscillator I: Fourier
analysis [nex121]
- Brownian harmonic oscillator II: position
correlation function [nex122]
- Brownian harmonic oscillator III: contour
integrals [nex123]
- Brownian harmonic oscillator IV: velocity
correlations [nex58]
- Brownian harmonic oscillator V: formal
solution for velocity [nex59]
- Brownian harmonic oscillator VI:
nonequilibrium correlations [nex60]
- Langevin dynamics from microscopic model [nln74]
- Brownian motion: levels of contraction and
modes of description [nln23]
9. Linear Response and Equilibrium Dynamics
- Overview [nln24]
- Many-body system perturbed by radiation field [nln25]
- Response function and generalized
susceptibility [nln26]
- Kubo formula for response function [nln27]
- Symmetry properties [nln30]
- Kramers-Kronig dispersion relations [nln37]
- Causality property of response function [nex63] *
- Energy transfer between system and radiation
field [nln38]
- Reactive and absorptive part of response
function [nex64]
- Fluctuation-dissipation theorem (quantum and
classical) [nln39] (2)
- Moment expansion [nln78]
- Spectral representations [nex65]
*
- Linear response of classical relaxator [nex66]
- Dielectric relaxation in liquid water [nln76]
- Linear response of classical oscillator [nex67]
- Scattering process and dynamic structure
factor [nln89]
- Scattering from free atoms [nln93] *
- Scattering from atoms bound to lattice [nln94]
- Scattering from a harmonic crystal [nln95]
- Magnetic resonance or scattering [nln97]
10. Zwanzig-Mori Formalism
- Introduction [nln28]
- Time-dependence of expectation values (quantum
and classical) [nln77] *
- Zwanzig's kinetic equation: generalized master
equation [nln29] * [nex68]
- Projection operator method (Mori formalism) [nln31]
- Kubo inner product [nln32]
*
- Projection operators [nln33]
- First and second projections [nln34] * [nln35]
- Continued-fraction representation of
relaxation function [nln36]
- n-pole approximation [nln87] *
- Relaxation function with uniform
continued-fraction coefficients [nex69]
- Link to Green's function formalism [nln88]
- Structure function of harmonic oscillator [nex71], * [nex72],
[nex73]
11. Recursion Method: Concepts
- Stage for recursion method [nln79]
- Modules of recursion method [nln80]
- Representations of recursion method [nln81]
- Orthogonal expansion of dynamical variables [nln82]
- Gram-Schmidt orthogonalization I [nln83]
*
- Relaxation function and spectral density [nln84]
- Moment expansion vs continued fraction I [nln85]
- Link to generalized Langevin equation [nln86]
- Orthogonal expansion of wave functions [nln90]
- Gram-Schmidt orthogonalization II [nln91]
- Structure function [nln92]
- Moment expansion vs continued fraction II [nln96]
- Genetic code of spectral densities [nln98]
- Spectral Lines from finite sequences of
continued-fraction coefficients [nln99]
- Spectral densities with bounded support [nln100]
- Bandwidth and gap in spectral density [nln101]
- Spectral densities with unbounded support [nln102]
- Unbounded support and gap [nln103]
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.
- 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.
- Gerd Röpke: Nonequilibrium statistical physics.
Wiley-VCH, 2013.
- R. Mahnke, J. Kaupuzs, and I. Lubashevsky: Physics of
stochastic processes. Wiley-VCH, 2009.
- V. S. Viswanath and G. Müller: The recursion method.
Application to many-body dynamics.
Springer-Verlag, New York 1994.
Do you have a question about any of the problems [nex]?
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Last updated 04/26/18