Stochastic Methods
Instructor:
Massimiliano Di Ventra (Physics)
Recommended texts: No single book will be followed. Different sources will be used for
different subjects. Examples include:
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“Electrical Transport in Nanoscale Systems”, M. Di Ventra (Cambridge University
Press, 2008)
“Stochastic Processes in Physics and Chemistry”, N. G. Van Kampen (Elsevier, 2001)
“Handbook of Stochastic Methods”, C. W. Gardiner (Springer, 2004)
“An Introduction to Stochastic Processes”, D.S. Lemons (Johns Hopkins, 2002)
“Elements of Stochastic Process Simulation”, B. S. Gottfried (Prentice Hall, 1983)
“Numerical Methods for Stochastic Processes”, N. Bouleau, D. Lépingle (Wiley, 1993)
“Stochastic processes” S.M. Ross (Wiley, 1995)
Prerequisites: Probability theory and basics of Quantum Mechanics
Course description:
In almost every aspect of science and engineering as well as in disparate disciplines
such as medicine and finance, we encounter quantities which, at any given time, can assume
values that cannot be determined in advance with absolute certainty. Such quantities are called
random variables, and their time evolution is called a random or stochastic process. In many
instances it is indeed just an approximation/assumption to model any process in science and
engineering as a sure or deterministic process. In other words, stochastic processes are the
norm, not the exception, in everyday life.
The course Stochastic Methods will introduce students to different random processes,
their theoretical description and the numerical methods employed to study them. Examples of
stochastic processes will be taken from both classical and quantum processes. The topics that
will be covered include:
1. Introduction to random variables
• Expected values; covariance; correlation and statistically independent variables;
moment-generating functions; Poisson distribution; Normal variable theorems.
2. Stochastic processes
• Definition; examples of stochastic processes in science and engineering
3. Markov processes
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The Markov property; Markov chains
4. The Master equation
• Derivation; the long-time limit; detailed balance
5. One-step processes
• The Poisson process; random walk; non-linear one-step processes
6. The Fokker-Planck equation
• Derivation; Brownian motion; examples of Brownian motion in everyday life
7. The Langevin approach
• Langevin treatment of the Brownian motion; relation to Fokker-Planck
equation; non-Gaussian white noise; Colored noise
8. Monte-Carlo methods
• Introduction; Monte-Carlo simulation of Brownian motion and of Langevin
equations
9. Fluctuations
• Diffusion noise; the method of compounding moments; the Boltzmann equation
10. Stochastic differential equations
• Definitions; solutions in special cases; nonlinear stochastic differential equations
11. Quantum stochastic processes
• Density matrix; information entropy; open quantum systems; quantum master
equations; stochastic Schrödinger equation.
Grading:
The students will be assigned a research project which will end in a short
(4-page) write-up followed by an oral presentation during a workshoptype of exam (10 minutes of presentation +10 of questions from all course
participants).