Lecture

Random Walk and Diffusion: Basics and Applications

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In course

PHYS-403: Computer simulation of physical systems I

The two main topics covered by this course are classical molecular dynamics and the Monte Carlo method. ...

Description

This lecture covers the fundamentals of random walk and diffusion, starting with the binomial distribution and its application to one-dimensional random walk. It then explores the link between random walk and diffusion, transitioning from discrete to continuous spatial and temporal variables. The lecture concludes with the diffusion equation derived through the master equation.

Instructor

Alfredo Pasquarello

Alfredo Pasquarello studied physics at the Scuola Normale Superiore of Pisa and at the University of Pisa, obtaining their respective degrees in 1986. He obtained a doctoral degree at the EPFL in 1991 with a thesis on Multiphoton Transitions in Solids . Then, he moved to Bell Laboratories at Murray Hill (New Jersey), where he carried out postdoctoral research on the magnetic properties of carbon fullerenes. In 1993, he joined the Institute for Numerical Research in the Physics of Materials (IRRMA), where his activity involved first-principles simulation methods. In 1998, he was awarded the EPFL Latsis Prize for his research work on disordered silica materials. Succeeding in grant programs of the Swiss National Science Foundation, he then set up his own research group at IRRMA. In July 2003, he is appointed Professor in Theoretical Condensed Matter Physics at EPFL. Currently, he leads the Chair of Atomic Scale Simulation.

Official source

Related concepts (20)

Negative binomial distribution

In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of successes (denoted ) occurs. For example, we can define rolling a 6 on a dice as a success, and rolling any other number as a failure, and ask how many failure rolls will occur before we see the third success ().

Random variable

A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' can be misleading as it is not actually random nor a variable, but rather it is a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads and tails ) in a sample space (e.g., the set ) to a measurable space (e.g., in which 1 corresponding to and −1 corresponding to ), often to the real numbers.

Poisson distribution

In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. It is named after French mathematician Siméon Denis Poisson ('pwɑːsɒn; pwasɔ̃). The Poisson distribution can also be used for the number of events in other specified interval types such as distance, area, or volume.

Random walk

In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line which starts at 0, and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas (see Brownian motion), the search path of a foraging animal, or the price of a fluctuating stock and the financial status of a gambler.

Binomial theorem

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial (x + y)n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. For example, for n = 4, The coefficient a in the term of axbyc is known as the binomial coefficient or (the two have the same value).

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