Saddle point approximation

In calculus , the saddle point approximation is used to calculate integrals of the form

${\displaystyle I=\lim _{N\to \infty }\int \limits _{-\infty }^{\infty }e^{-Nf(x)}\;\mathrm {d} x}$

to calculate approximately. The method comes from Pierre Simon de Laplace (1774) and is sometimes named after him. It is part of the asymptotic analysis .

If the function is analytical and has a global minimum at , we get: ${\displaystyle f(x)}$ ${\displaystyle x_{0}}$

${\displaystyle I=\lim _{N\to \infty }e^{-Nf(x_{0})}{\sqrt {\frac {2\pi }{Nf''(x_{0})}}}}$

With

${\displaystyle \left.f''(x_{0})={\frac {\partial ^{2}f(x)}{\partial x^{2}}}\right|_{x=x_{0}}>0}$.

The second derivative is positive because there is a minimum here. The result is asymptotical, that is, towards infinity . ${\displaystyle N}$

There can also be finite integration limits. ${\displaystyle \left(a,b\right)}$

The generalization of the saddle point approximation into the complex number plane is also called the saddle point method . This explains the naming after a saddle point .

Alternative formulation

Another sign can also be considered in the exponent:

With a different sign applies to

${\displaystyle I=\lim _{N\to \infty }\int \limits _{-\infty }^{\infty }e^{Nf(x)}\;\mathrm {d} x}$

if there is a global maximum asymptotically: ${\displaystyle x_{0}}$ ${\displaystyle I=\lim _{N\to \infty }e^{Nf(x_{0})}{\sqrt {\frac {2\pi }{N|f''(x_{0})|}}}}$

with . ${\displaystyle \left.f''(x_{0})={\frac {\partial ^{2}f(x)}{\partial x^{2}}}\right|_{x=x_{0}}<0}$

Since there is a maximum here, the second derivative is negative.

Reason

The first case (minimum at ) is considered, the argumentation in the second case is analogous. ${\displaystyle x_{0}}$

For large , the exponential function becomes arbitrarily small outside the vicinity of . Therefore um is expanded into a Taylor series : ${\displaystyle N}$ ${\displaystyle x_{0}}$ ${\displaystyle f(x)}$ ${\displaystyle x_{0}}$

${\displaystyle f(x)\approx f(x_{0})+{\frac {1}{2}}f''(x_{0})(x-x_{0})^{2}}$.

(Because of the global minimum at is ) ${\displaystyle x_{0}}$ ${\displaystyle f'(x_{0})=0}$

Inserting into the integral delivers

${\displaystyle I=\lim _{N\to \infty }\int \limits _{-\infty }^{\infty }e^{-Nf(x_{0})-N{\frac {1}{2}}f''(x_{0})(x-x_{0})^{2}}\mathrm {d} x={\lim _{N\to \infty }e^{-Nf(x_{0})}}\;\int \limits _{-\infty }^{\infty }e^{-N{\frac {1}{2}}f''(x_{0})(x-x_{0})^{2}}\;\mathrm {d} x}$.

The size is therefore the limit value of the product and the non- elementary integral. The latter is closely related to the Gaussian error integral or the Gaussian distribution . The integral is of the form , where holds. It is especially because the minimum in the case of a positive second derivative, which will be important in the following: ${\textstyle I}$ ${\textstyle N\to \infty }$ ${\textstyle e^{-Nf(x_{0})}}$ ${\textstyle \Phi (z)}$ ${\textstyle \int _{-\infty }^{+\infty }e^{-{\frac {N}{2}}f''(x_{0})(x-x_{0})^{2}}\;\mathrm {d} x}$ ${\textstyle \int _{-\infty }^{+\infty }e^{-a(x+b)^{2}}\,\mathrm {d} x}$ ${\textstyle a={\frac {N}{2}}f''(x_{0})>0\ \mathrm {und} \ \ b=-x_{0}}$ ${\displaystyle a>0}$ ${\textstyle x_{0}}$

For everyone with the following relation (e.g. via substitution ) can be shown: ${\displaystyle a}$ ${\displaystyle \Re (a)>0}$ ${\textstyle y(x)={\sqrt {a}}x\ ,\ \mathrm {d} x={\frac {1}{\sqrt {a}}}\mathrm {d} y}$

${\displaystyle \int _{-\infty }^{+\infty }e^{-ax^{2}}\,\mathrm {d} x={\sqrt {\frac {\pi }{a}}}}$

Furthermore, a shift of by the constant does not affect the value of , since the integral can easily be converted into the above integral through the linear substitution with , the value of which is already known. ${\displaystyle y}$ ${\textstyle b}$ ${\textstyle \int _{-\infty }^{+\infty }e^{-a(y+b)^{2}}\,\mathrm {d} y}$ ${\textstyle x(y)=y+b\ ,\ \mathrm {d} x=1\mathrm {d} y}$ ${\textstyle \lim _{y\to \pm \infty }x(y)=\pm \infty }$ ${\textstyle \int _{-\infty }^{+\infty }e^{-ax^{2}}\,\mathrm {d} x}$

So you get with : ${\textstyle a={\frac {N}{2}}f''(x_{0})>0\ \mathrm {und} \ \ b=-x_{0}}$

${\displaystyle \int _{-\infty }^{+\infty }e^{-{\frac {N}{2}}f''(x_{0})(x-x_{0})^{2}}\;\mathrm {d} x={\sqrt {\frac {\pi }{{\frac {N}{2}}f''(x_{0})}}}}$

Thus it follows for (asymptotically): ${\displaystyle I}$

${\displaystyle I=\lim _{N\to \infty }\int \limits _{-\infty }^{+\infty }e^{-Nf(x)}\;\mathrm {d} x={\lim _{N\to \infty }e^{-Nf(x_{0})}}\;\int \limits _{-\infty }^{+\infty }e^{-N{\frac {1}{2}}f''(x_{0})(x-x_{0})^{2}}\;\mathrm {d} x=\lim _{N\to \infty }e^{-Nf(x_{0})}{\sqrt {\frac {2\pi }{Nf''(x_{0})}}}}$

Applications

The saddle point approximation and saddle point methods find various applications in theoretical physics, including in statistical physics in the limiting case of large systems, in quantum field theory when evaluating path integrals or in optics.

One application is the Stirling formula

${\displaystyle N!\approx {\sqrt {2\pi N}}N^{N}e^{-N}\,}$

for big . ${\displaystyle N}$

From the definition of the gamma function it follows

${\displaystyle N!=\Gamma (N+1)=\int _{0}^{\infty }e^{-x}x^{N}\,\mathrm {d} x.}$

With the variable transformation (so that : :) we get: ${\displaystyle x=Nz\,}$ ${\displaystyle \mathrm {d} x=N\mathrm {d} z.\,}$

${\displaystyle {\begin{aligned}N!&=\int _{0}^{\infty }e^{-Nz}\left(Nz\right)^{N}N\,\mathrm {d} z\\&=N^{N+1}\int _{0}^{\infty }e^{-Nz}z^{N}\,\mathrm {d} z\\&=N^{N+1}\int _{0}^{\infty }e^{-Nz}e^{N\ln z}\,\mathrm {d} z\\&=N^{N+1}\int _{0}^{\infty }e^{N(\ln z-z)}\,\mathrm {d} z.\end{aligned}}}$

Now you can use the saddle point approximation in the second form (for maxima) with

${\displaystyle f\left(z\right)=\ln {z}-z}$

with the derivatives

${\displaystyle f'(z)={\frac {1}{z}}-1,}$

${\displaystyle f''(z)=-{\frac {1}{z^{2}}}.}$

The maximum of f is with the value of the second derivative −1. With the saddle point approximation one obtains: ${\displaystyle z_{0}=1}$

${\displaystyle N!\approx N^{N+1}{\sqrt {\frac {2\pi }{N}}}e^{-N}={\sqrt {2\pi N}}N^{N}e^{-N}.}$

generalization

The approach to the saddle point is generalized when considering the complex in the method of steepest descent (English: Method of steepest descent) or the method of the stationary phase (English: Method of stationary phase) (general saddle point method). The aim is the asymptotic evaluation of closed path integrals in the complex number plane ( ) ${\displaystyle z=x+iy}$

${\displaystyle \displaystyle \int _{C}f(z)e^{\lambda g(z)}\mathrm {d} z}$

for big real ones . In this case, it deformed in the complex the integration path so that a stationary point (zero of the first derivative of g) of lying on the path of integration and then proceeds similar to above (with the additional application of the Cauchy's theorem ). In the version of the steepest descent method, the integration path is placed in such a way that the real part u of g has a maximum there. Since the real part of u g a harmonic function is able to and did not have the same sign: there exists a saddle point and its commitment to integration path along the path of the "steepest descent". Hence the name of the method. ${\displaystyle \lambda }$ ${\displaystyle z_{0}}$ ${\displaystyle g(z)}$ ${\displaystyle z_{0}}$ ${\displaystyle \partial ^{2}u \over \partial x^{2}}$ ${\displaystyle \partial ^{2}u \over \partial y^{2}}$

The stationary phase method specifically considers integrals for which the exponent of the exponential function is imaginary along the path:

${\displaystyle \displaystyle \int _{C}f(z)e^{i\lambda u(z)}\mathrm {d} z}$

With a real function u and a large . ${\displaystyle \lambda }$

The method was first published by Peter Debye in 1909 to estimate Bessel functions , but was also used by Bernhard Riemann .

literature

• Jon Matthews, RL Walker: Mathematical Methods of Physics , Addison-Wesley, Chapter 3.6 (Saddle Point Methods)
• Philip M. Morse , Herman Feshbach : Methods of Theoretical Physics , Volume 1, McGraw Hill, 1953, Chapter 4.6
• Arthur Erdélyi : Asymptotic Expansions , Dover 1956
• Nicolaas Govert de Bruijn : Asymptotic Methods in Analysis , Wiley 1961

Web links

• Lecture notes Gajjar, Method of Steepest Descent, pdf

Individual evidence

1. ↑ Saddle point method . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .