Partition functions | Entropy and information

Partition function (mathematics)

The partition function or configuration integral, as used in probability theory, information theory and dynamical systems, is a generalization of the definition of a partition function in statistical mechanics. It is a special case of a normalizing constant in probability theory, for the Boltzmann distribution. The partition function occurs in many problems of probability theory because, in situations where there is a natural symmetry, its associated probability measure, the Gibbs measure, has the Markov property. This means that the partition function occurs not only in physical systems with translation symmetry, but also in such varied settings as neural networks (the Hopfield network), and applications such as genomics, corpus linguistics and artificial intelligence, which employ Markov networks, and Markov logic networks. The Gibbs measure is also the unique measure that has the property of maximizing the entropy for a fixed expectation value of the energy; this underlies the appearance of the partition function in maximum entropy methods and the algorithms derived therefrom. The partition function ties together many different concepts, and thus offers a general framework in which many different kinds of quantities may be calculated. In particular, it shows how to calculate expectation values and Green's functions, forming a bridge to Fredholm theory. It also provides a natural setting for the information geometry approach to information theory, where the Fisher information metric can be understood to be a correlation function derived from the partition function; it happens to define a Riemannian manifold. When the setting for random variables is on complex projective space or projective Hilbert space, geometrized with the Fubini–Study metric, the theory of quantum mechanics and more generally quantum field theory results. In these theories, the partition function is heavily exploited in the path integral formulation, with great success, leading to many formulas nearly identical to those reviewed here. However, because the underlying measure space is complex-valued, as opposed to the real-valued simplex of probability theory, an extra factor of i appears in many formulas. Tracking this factor is troublesome, and is not done here. This article focuses primarily on classical probability theory, where the sum of probabilities total to one. (Wikipedia).

Video thumbnail

Split function example. Calculus for Beginners - Dr Chris Tisdell Live Stream

Split functions follow one rule and the swtich to another rule, depending on where you are on a x-axis. I define what a split function is (also called a picewise defined function) and discuss and example. In mathematics, functions are an important tool for understanding how things depend

From playlist Calculus for Beginners

Video thumbnail

Partitions of a Set | Set Theory

What is a partition of a set? Partitions are very useful in many different areas of mathematics, so it's an important concept to understand. We'll define partitions of sets and give examples in today's lesson! A partition of a set is basically a way of splitting a set completely into disj

From playlist Set Theory

Video thumbnail

What is a split function?: Dr Chris Tisdell Live Stream

Split functions follow one rule and the swtich to another rule, depending on where you are on a x-axis. I define what a split function is (also called a picewise defined function) and discuss and example. In mathematics, functions are an important tool for understanding how things depend

From playlist Calculus for Beginners

Video thumbnail

Abstract Algebra | Partitions and Equivalence Relations

We prove that there is a one-to-one correspondence between partitions of a set and equivalence relations on a set. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/

From playlist Abstract Algebra

Video thumbnail

Discrete Math - 2.3.4 Useful Functions to Know

Ceiling function, floor function and factorial function. Textbook: Rosen, Discrete Mathematics and Its Applications, 7e Playlist: https://www.youtube.com/playlist?list=PLl-gb0E4MII28GykmtuBXNUNoej-vY5Rz

From playlist Discrete Math I (Entire Course)

Video thumbnail

13 Equivalence sets

We have seen an example of partitioning in the previous video. These partitioned sets are called equivalence sets or equivalence classes. In this video we look at some notation.

From playlist Abstract algebra

Video thumbnail

Steffen Borgwardt: The role of partition polytopes in data analysis

The field of optimization, and polyhedral theory in particular, provides a powerful point of view on common tasks in data analysis. In this talk, we highlight the role of the so-called partition polytopes and their studies in clustering and classification. The geometric properties of parti

From playlist Workshop: Tropical geometry and the geometry of linear programming

Video thumbnail

Rational Functions

In this video we cover some rational function fundamentals, including asymptotes and interecepts.

From playlist Polynomial Functions

Video thumbnail

Partitions, Dyson, and Ramanujan - George Andrews

George Andrews The Pennsylvania State University September 27, 2013 More videos on http://video.ias.edu

From playlist Dreams of Earth and Sky

Video thumbnail

Rogers-Ramanujan Identities | Part 3: More on partition generating functions.

We look more at partition generating functions, focusing on what each part of the generating function means in terms of the allowed partitions given some restriction. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/

From playlist Rogers-Ramanujan Identities

Video thumbnail

Alternating Sign Triangles by Arvind Ayyer

Program : Integrable? ?systems? ?in? ?Mathematics,? ?Condensed? ?Matter? ?and? ?Statistical? ?Physics ORGANIZERS : Alexander Abanov, Rukmini Dey, Fabian Essler, Manas Kulkarni, Joel Moore, Vishal Vasan and Paul Wiegmann DATE & TIME : 16 July 2018 to 10 August 2018 VENUE : Ramanujan L

From playlist Integrable​ ​systems​ ​in​ ​Mathematics,​ ​Condensed​ ​Matter​ ​and​ ​Statistical​ ​Physics

Video thumbnail

Analysis III - Integration: Oxford Mathematics 1st Year Student Lecture:

The third in our popular series of filmed student lectures takes us to Integration. This is the opening lecture in the 1st Year course. Ben Green both links the course to the mathematics our students have already learnt at school and develops that knowledge, taking the students to the next

From playlist Oxford Mathematics 1st Year Student Lectures

Video thumbnail

[Discrete Mathematics] Integer Partitions

We talk about the number of ways to partition an integer. Visit our website: http://bit.ly/1zBPlvm Subscribe on YouTube: http://bit.ly/1vWiRxW *--Playlists--* Discrete Mathematics 1: https://www.youtube.com/playlist?list=PLDDGPdw7e6Ag1EIznZ-m-qXu4XX3A0cIz Discrete Mathematics 2: https://

From playlist Discrete Math 2

Video thumbnail

The hardest "What comes next?" (Euler's pentagonal formula)

Looks like I just cannot do short videos anymore. Another long one :) In fact, a new record in terms of the slideshow: 547 slides! This video is about one or my all-time favourite theorems in math(s): Euler's amazing pentagonal number theorem, it's unexpected connection to a prime numbe

From playlist Recent videos

Video thumbnail

Yang-Lee Zeros of Integrable Field Theories by Giuseppe Mussardo

PROGRAM: INTEGRABLE SYSTEMS IN MATHEMATICS, CONDENSED MATTER AND STATISTICAL PHYSICS ORGANIZERS: Alexander Abanov, Rukmini Dey, Fabian Essler, Manas Kulkarni, Joel Moore, Vishal Vasan and Paul Wiegmann DATE : 16 July 2018 to 10 August 2018 VENUE: Ramanujan Lecture Hall, ICTS Bangalore

From playlist Integrable​ ​systems​ ​in​ ​Mathematics,​ ​Condensed​ ​Matter​ ​and​ ​Statistical​ ​Physics

Video thumbnail

Ulrike von Luxburg and Solveig Klepper: Clustering with tangles

CONFERENCE Recording during the thematic meeting : " Machine Learning and Signal Processing on Graphs" the November 10, 2022 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mat

From playlist Mathematics in Science & Technology

Video thumbnail

Real Analysis - Part 50 - Properties of the Riemann Integral for Step Functions

Support the channel on Steady: https://steadyhq.com/en/brightsideofmaths Or support me via PayPal: https://paypal.me/brightmaths Or via Ko-fi: https://ko-fi.com/thebrightsideofmathematics Or via Patreon: https://www.patreon.com/bsom Or via other methods: https://thebrightsideofmathematics.

From playlist Real Analysis

Video thumbnail

Additive number theory: Extremal problems and the combinatorics of sum. (Lecture 4) by M. Nathanson

Program Workshop on Additive Combinatorics ORGANIZERS: S. D. Adhikari and D. S. Ramana DATE: 24 February 2020 to 06 March 2020 VENUE: Madhava Lecture Hall, ICTS Bangalore Additive combinatorics is an active branch of mathematics that interfaces with combinatorics, number theory, ergod

From playlist Workshop on Additive Combinatorics 2020

Video thumbnail

Dyson's Rank, Harmonic Weak Maass Form, and Recent Developments - Kathrin Bringmann

Kathrin Bringmann University of Cologne September 27, 2013 More videos on http://video.ias.edu

From playlist Dreams of Earth and Sky

Video thumbnail

Determine if a Relation is a Function

http://mathispower4u.wordpress.com/

From playlist Intro to Functions

Related pages

Differential operator | Path integral formulation | Exponential family | Invariant measure | Karush–Kuhn–Tucker conditions | Power set | Generalized coordinates | Jacobian matrix and determinant | Trace (linear algebra) | Probability density function | Probability space | Markov logic network | Group (mathematics) | Fisher information | Hopfield network | Gaussian integral | Covariance matrix | Operator (mathematics) | Projective Hilbert space | Berezin integral | Lagrange multiplier | Partition problem | Probability-generating function | Correlation function | Markov random field | Green's function | Information theory | Simplex | Fubini–Study metric | Fredholm theory | Clique (graph theory) | Probability amplitude | Equivalence class | Fisher information metric | Information geometry | Artificial intelligence | Markov property | Riemannian manifold | Probability distribution | Energy | Renormalization group | Probability measure | Lagrangian mechanics | Gibbs measure | Trace class | Integral | Manifold | Partition function (statistical mechanics) | Random variable | Hilbert space | Metric tensor | Normalizing constant | Quadratic form | Scalar potential | Measure space | Probability theory | Ising model | Measure (mathematics) | Ursell function | Conservation of energy | Matrix (mathematics) | Universality (dynamical systems) | Complex projective space | Bures metric | Generating function | Grassmann number