physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...

and

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an

educated guess A guess (or an act of guessing) is a swift conclusion drawn from data directly at hand, and held as probable or tentative, while the person making the guess (the guesser) admittedly lacks material for a greater degree of certainty. A guess is als ...

or an additional assumption made to help solve a problem, and which may later be verified to be part of the solution by its results.

Use

An ansatz is the establishment of the starting equation(s), the theorem(s), or the value(s) describing a mathematical or physical problem or solution. It typically provides an initial estimate or framework to the solution of a mathematical problem, and can also take into consideration the

boundary conditions In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...

(in fact, an ansatz is sometimes thought of as a "trial answer" and an important technique in solving differential equations). After an ansatz, which constitutes nothing more than an assumption, has been established, the equations are solved more precisely for the general function of interest, which then constitutes a confirmation of the assumption. In essence, an ansatz makes assumptions about the form of the solution to a problem so as to make the solution easier to find. It has been demonstrated that machine learning techniques can be applied to provide initial estimates similar to those invented by humans and to discover new ones in case no ansatz is available.

Examples

Given a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

of experimental data that looks to be clustered about a line, a linear ansatz could be made to find the

parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

s of the line by a

least squares The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the re ...

curve fit. Variational

approximation An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...

methods use ansätze and then fit the parameters. Another example could be the mass, energy, and entropy balance equations that, considered simultaneous for purposes of the elementary operations of

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...

, are the ''ansatz'' to most basic problems of

thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...

. Another example of an ansatz is to suppose the solution of a homogeneous linear

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...

to take an exponential form, or a power form in the case of a

difference equation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

. More generally, one can guess a particular solution of a system of equations, and test such an ansatz by directly substituting the solution into the system of equations. In many cases, the assumed form of the solution is general enough that it can represent arbitrary functions, in such a way that the set of solutions found this way is a full set of all the solutions.

References

Bibliography

* * * *{{Citation , last1 = Karbach , first1 = M. , last2 = Hu , first2 = K. , last3 = Müller , first3 = G. , title = Introduction to the Bethe ansatz III. , date = August 1, 2000 , url = http://www.phys.uri.edu/~gerhard/MSS/ms91.pdf , access-date = 2008-10-25 , archive-url = https://web.archive.org/web/20060901073407/http://www.phys.uri.edu/~gerhard/MSS/ms91.pdf , archive-date = September 1, 2006 , url-status = dead Philosophy of physics Concepts in physics Mathematical terminology German_words_and_phrases

Use

Examples

See also

References

Bibliography