What is Correlation function: Definition and 51 Discussions

A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables representing the same quantity measured at two different points, then this is often referred to as an autocorrelation function, which is made up of autocorrelations. Correlation functions of different random variables are sometimes called cross-correlation functions to emphasize that different variables are being considered and because they are made up of cross-correlations.
Correlation functions are a useful indicator of dependencies as a function of distance in time or space, and they can be used to assess the distance required between sample points for the values to be effectively uncorrelated. In addition, they can form the basis of rules for interpolating values at points for which there are no observations.
Correlation functions used in astronomy, financial analysis, econometrics, and statistical mechanics differ only in the particular stochastic processes they are applied to. In quantum field theory there are correlation functions over quantum distributions.

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  1. ab_kein

    I Two-body correlation function computation

    I'm studying how to compute excess entropy in molecular dynamics (MD). I've found it is needed to compute the two-body correlation function (neglecting high-order terms), the details can be found, for example, in this article. So the definition of correlation function (CF for short) is ##C(t...
  2. gremory

    A Computing Correlation functions

    Hello, recently I'm learning about correlation functions in the context of QFT. Correct me with I'm wrong but what i understand is that tha n-point correlation functions kinda of describe particles that are transitioning from a point in space-time to another by excitations on the field. So, what...
  3. T

    A Brownian Motion (Langevin equation) correlation function

    So the Langevin equation of Brownian motion is a stochastic differential equation defined as $$m {d \textbf{v} \over{dt} } = - \lambda \textbf{v} + \eta(t)$$ where the noise function eta has correlation function $$\langle \eta_i(t) \eta_j(t') \rangle=2 \lambda k_B T \delta_{ij} \delta(t -...
  4. W

    Coding of Spin Spin correlation function

    I tried to code spinoperators who act like $S_x^iS_x^j$ (y and z too) and to apply them to the states, which works fine. I am not sure about how to code the expectation value in the product Space. Has anyone pseudo Code to demonstrate that?
  5. Danny Boy

    A Quantum Ising model correlation function query

    In this paper, on quantum Ising model dynamics, they consider the Hamiltonian $$\mathcal{H} = \sum_{j < k} J_{jk} \hat{\sigma}_{j}^{z}\hat{\sigma}_{k}^{z}$$ and the correlation function $$\mathcal{G} = \langle \mathcal{T}_C(\hat{\sigma}^{a_n}_{j_n}(t_n^*)\cdot\cdot\cdot...
  6. A

    A Quantum vacuum fluctuation in BEC

    Hi, I have some problems with visualization (I'm trying to understand Jeff Steinhauer's experiment, but my questions are general). Why the quantum vacuum fluctuations are guaranteed by the underlying pointlike atoms composing a BEC? And if vacuum fluctuations generate excitations (i.e...
  7. Danny Boy

    A Correlation functions of quantum Ising model

    I have a single technical question regarding a statement on page 7 of the paper "Dynamical quantum correlations of Ising models on an arbitrary lattice and their resilience to decoherence". The paper up until page 7 defines a general correlation function ##\mathcal{G}## of a basic quantum Ising...
  8. G

    A Correlation function of a Klein-Gordon field

    First, let me introduce the notation; given a Hamiltonian ##H## and a momentum operator ##\vec{P}##, and writing ##P=(H,\vec{P})##. Let ##|\Omega\rangle## be the ground state of ##H##, ##|\lambda_\vec{0}\rangle## an eigenstate of ##H## with momentum 0, i.e. ##\vec{P}|\lambda_\vec{0}\rangle=0##...
  9. P

    Mean, variance and correlation function of Langevin equation

    Homework Statement I have simulated Langevin equation (numerically in Matlab) for some specific conditions, so I have obtained the solution ##X(t)##. But now, with the solution I have obtained, I have to calculate ## <X(t)|x_0>, <X^2(t)|x_0>-(<X(t)|x_o>)^2 ## and the conditional correlation...
  10. A

    I Why is there this extra term for this correlation function?

    Let's say we have a Dirac field ##\Psi## and a scalar field ##\varphi## and we want to compute this correlation function $$<0|T \Psi _\alpha (x) \Psi _\beta (y) \varphi (z_1) \varphi (z_2)|0>$$ $$= \frac {1}{i} \frac{\delta}{\delta \overline{\eta}_\alpha(x)} i \frac{\delta}{\delta \eta_\beta(y)}...
  11. J

    Density-density correlation function for spinless Fermions

    Homework Statement Consider the groundstate of a one-dimensional, non-interacting system of spinless fermions. Let ##a^†(x)## and ##a(x)## be the creation and annihilation operators for a fermion at the point ##x##, so that the density operator is ##n(x) = a^†(x)a(x)##. Show that the...
  12. C

    A Help with Correlation/Green's Function of Rotated Variables

    Hello (I'm reposting this from stack exchange, and thought this site may be more appropriate, so if you see it that's why), I'm working through this paper, and have encountered "a little algebra shows that...", yet I'm not familiar enough with the topic at hand to figure this out. Here is the...
  13. B

    I How does one actually compute a correlation function?

    I have been reading about weak gravitational lensing and I am trying to calculate the dispersion ##\langle M_{ap}^2\rangle## of the aperture mass for a singular isothermal sphere acting as a lens for distant objects. I need some guidance on how to actually carry out the calculation of the power...
  14. rocdoc

    About the first-order electric field correlation function.

    For a fixed point in space, the first-order electric field correlation function may be given as (Possibly incorrectly, see my "second" post to this thread)! $$\langle\vec E^*(t)\vec E(t+\tau)\rangle = {\frac {1} {T}} \int_T\vec E^*(t)\vec E(t+\tau)dt~~~~~(1)$$ Where T is a very large time and *...
  15. C

    A Calculating asymptotic behavior of a correlation function

    I was hoping to get some assistance in reproducing a calculation from https://arxiv.org/abs/0803.1292 (https://journals.aps.org/pra/abstract/10.1103/PhysRevA.78.012304 for the published version). A certain four-spin correlation function for Kitaev's spin-1/2 model on the honeycomb lattice is...
  16. B

    B Correlation function in QFT

    Hi I would be happy if anyone helped me understand what the physical meaning of n-particle correlation function in QFT is ?
  17. K

    A Correlation properties of quasi-thermal light

    Hello! I'm measuring the second order correlation function g(2) of quasi-thermal light, generated by focusing a 633 nm laser onto a rotating sandpaper-like surface at 45 deg. Part of the dispersed light is collected into a fiber and split with a fiber beam splitter. A simple HBT setup. g(2) is...
  18. S

    I Two point correlation function in Peskin book

    Hello! I am reading Peskin's book on QFT and I reached a part (in chapter 4) where he is analyzing the two-point correlation function for ##\phi^4## theory. At a point he wants to find the evolution in time of ##\phi##, under this Hamiltonian (which is basically the Klein-Gordon - ##H_0## - one...
  19. J

    Flux-flux correlation function under Feynman's path integral

    This is a chemically inspired problem, but the path is fully quantum mechanics and a bunch of integrals. How does one calculate fully quantum mechanical rate ($\kappa$) in the golden-rule approximation for two linear potential energy surfaces? Attempt: Miller (83) proposes...
  20. ChrisVer

    A QFT phi3 Feynman diagrams and correlation function

    I have some difficulty understanding how to go about with this problem: I came up with several graphs, you can see them in the attached picture (they are up to ~g^4 order). I am not sure about the self-interaction diagrams, but I think they are considered in the connected graphs (they are not...
  21. S

    A Free theory time-ordered correlation function with two internal fields

    Wick's theorem allows one to write a free theory time-ordered ##n##-point correlation function as a product of free theory time-ordered ##2##-point correlation function. The procedure involves the pairwise Wick contraction of fields such that external fields are not paired up each...
  22. S

    A Why can the ##1##-point correlation function be made to vanish?

    The ##1##-point correlation function in any theory, free or interacting, can be made to vanish by a suitable rescaling of the field ##\phi##. I would like to understand this statement. With the above goal in mind, consider the following theory: $$\mathcal{L} =...
  23. T

    A Qn on photon statistics (second order correlation function)

    I am trying to better understand the concept of second order coherence G2(τ) (in particular G2(0)) and a few questions have arisen. Note that I am trying to get a physical idea of what is happening so I would appreciate it if your responses can keep the math to the minimum possible. :) How do...
  24. P

    Second order correlation function for non ideal laser

    Hi, it is known that second order correlation function (g2) is a constant( =1) for ideal laser or single frequency light sources. So, what is the second order correlation function for non ideal laser? Is it still a constant or something related to the coherence time of the laser?
  25. enrikofermi

    Direct correlation function

    Hi all. I'm learning something about critical phenomena and I have one problem. I'm bad with Fourier transforms so I don't know how from 7.37 we have 7.38. I have tryed everything I knew, but fruitless. I have attached picture of my problem. Does anybody has any idea how I can solve this?
  26. S

    Help to 2-point correlation function in $ \phi^{4} $-theory

    Hi everyone! After a few slow days in the office I thought I would like to derive the 2-point correlation function within statistical(Euclidian) ## \phi^{4} ##-theory but I ran into some problems. For the sake of clarity I will show you from where I startet and ask questions when I need help...
  27. B

    Two-point correlation function in path integral formulation

    Suppose that I have already calculated the two-point correlation function for a Lagrangian with no interations using the path integral formulation. \langle \Omega | T[\phi(x)\phi(y)] | \Omega \rangle = \frac{ \int \mathcal{D}\phi \phi(x)\phi(y) \exp[iS_0] }{ \int \mathcal{D}\phi \exp[iS_0] }...
  28. F

    Correlation function poles in Peskin's derivation of LSZ formula

    Hi folks, Been trying to fill some of the more formal gaps in my knowledge by tackling the more technical stuff in P&S Chapter 7. Their derivation of the LSZ formula is quite different to those of books like, say, Srednicki, as they basically Fourier transform the whole argument as I...
  29. mhsd91

    Correlation Function of the Ornstein-Uhlenbeck Process

    Basically, The Ornstein-Uhlenbeck (OU) process (and its time-integral) decribes the velocity of a brownian particle. The OU process is Stationary (in time), Stochastic AND Markovian. Now, I've done an exact, one dimensional, numerical simulation of the OU process similar to D. T. Gillespie in...
  30. R

    Correlation function of damped harmonic oscillator

    The model of damped harmonic oscillator is given by the composite system with the hamiltonians ##H_S\equiv\hbar \omega_0 a^\dagger a##, ##H_R\equiv\sum_j\hbar\omega_jr_j^\dagger r_j##, and ##H_{SR}\equiv\sum_j\hbar(\kappa_j^*ar_j^\dagger+\kappa_ja^\dagger...
  31. V

    Auto correlation function for water molecules in a box

    I have simulated a box water molecules (1500 water molecules) using molecular dynamics method in NpT ensemble. I got the time auto-correlation function (C(t) vs t, time) where its function line is parallel to x-axis. Which means the C(t) is zero. I understand the water molecules are...
  32. S

    Second-order correlation function

    Hi all, I'm studying for my Quantum Optics exam and still have problems with the second-order correlation function. The question concerned is question 3b, 3c etc which can be found here: http://www.arago.utwente.nl/comms/sotn/tentamendatabase/optics/qo/351500_Quantum_Optics_2010-11-03.pdf...
  33. N

    Photocurrent shot noise correlation function

    Hi I have stumpled upon the following expression for the correlation function of the photocurrent shot noise for a photodiode <\delta i(t)\delta i(t+\tau)> = \frac{e^2\eta}{h\nu}P\delta(\tau) where η is the quantum efficiency and P the power in the signal. δ(τ) denotes the Dirac delta, which...
  34. S

    Calculating velocity correlation function

    Hi, I have been trying to write a code to calculate velocity correlation function so that I can obtain a frequency spectrum for my system on taking a Fourier transform. For testing my code I generated a sine wave data and calculated vcf and took dft. I got correct frequency value but the peak...
  35. S

    What is the interpretation of the vacuum correlation function in QFT

    I'm currently trying to make some intuitive sense out of Quantum field theory, but I'm not really understanding the vacuum. Consider a real (or complex, with + in the right places) scalar particle (a Klein-Gordon field). Now consider the propagator (or correlation function) G(x-y)=...
  36. M

    Two point correlation function

    Hi I have two images and I want to compare the "structures" of them at different scales. I remember from cosmology that the two point correlation function was used to extract similar structural information from the CMB, generating a graph of structure Vs scale. Then at certain length scales you...
  37. N

    Why correlation function decay as power law at critical temperature point?

    Please teach me this: In QTF theory of Schroeder,chapter 13.1 saying: Just at t=0(t=\frac{T-T_{c}}{T_{c}}),the correlation should decay as power law. Define the exponent \eta by the formula: G(x)=\frac{1}{x^{d-2+\eta}} where d is Euclidien space dimension. I do not understand why at...
  38. N

    Why connected correlation function depends on (x-y) only when <Phi(x)> equal constant

    Please teach me this: Why the connected correlation functions depend on the x-y(the difference of two space-time coordinates of two points) only in case the expectation of field(<phi(x)>) is constant(the translation invariance of vacuum state). Thank you very much in advanced.
  39. N

    Why 2-point correlation function diagram have only two external legs?

    Please teach me this: Why 2-point correlation function diagrams have only two external lines.Because of Wick theorem they would have many external legs. Another thing is whether are there odd number point correlation functions(meaning they are different from 0). Thank you very much in advanced.
  40. T

    Fourier transform of the auto correlation function to get energy

    Homework Statement The Fourier transform of the auto correlation function is the energy spectral density (ESD) of a signal. Here is the "apparent" proof: \int e^{-jwT} [ \int g(t)g(t+T)dt] dT => \int g(t)[ \int g(T+t)e^{-jwT}dT] dt What happened here? Why did the second integral change from...
  41. M

    Correlation Function for a 2-D field

    I have a 2-dimensional field of values (they are actually heights of a surface) and I want to compute a correlation function or some sort of correlation parameter. I have seen something similar done with galaxies and you end up with something like the probability of finding a galaxy at a...
  42. 1

    Realspace and redshift space correlation function questions

    why is the redshift space correlation function smaller than the realspace correlation function at small scales and the opposite on large scales?
  43. C

    [Matlab] Correlation function for henon map

    Hi! I need to write MATLAB script, which will be plotting corretation function for two-dimmensial system. Henon map is not my system, but is very popular, so solusion for henon map can be very helpful for me. Have anybody know code for have plot like this...
  44. R

    Why Correlation Function of Order Parameter Density Affects Magnetization

    Does anyone know why: <m(k)m(p)>=(2 \pi)^3 \delta(k+p)|m(k)|^2 where m(k) and m(p) are the Fourier transform of the order parameter density and the angled brackets <> stand for an ensemble average? For example, the magnetization M is given by: M=<\int d^3r m(r)>
  45. S

    How to draw a correlation function in excel

    Homework Statement For my homework i have to compute a correlation function in excel from available data. Data goes somewhat like this: 1 xxx yyy 2 xxx yyy 3 xxx yyy I now I have to calculate...
  46. T

    Joint Probability From Correlation Function

    Dear all I have the following problem. Given a set of correlated binary variables, can I determine the joint probability from the correlation function? {Xi} is a set of binary variables Pr(Xi=1) = p and Pr(Xi=0) = q for all i Corr(Xi Xj) = cij cij is symmetric Now how can I determine the...
  47. maverick280857

    Four Point Correlation function from Generating Functional

    Hi everyone, I'm working through Section 9.2 (Functional Quantization of Scalar Fields) from Peskin and Schroeder. I have trouble understanding the absence of a term in equation 9.41 which I get but the authors do not. Define \phi_i \equiv \phi(x_i), J_{x} \equiv J(x), D_{xi} \equiv...
  48. T

    Order parameter, correlation function and length

    Hello people, I don't understand the relation between those concepts, I tried to read what's written in wikipedia about them but no use. Can someone please elaborate more about the relation between them in some intuitive manner? Thank you :)
  49. S

    Exam question - correlation function

    Homework Statement This is a question from an exam in fractals and chaos. The paper is attached, the question is the first one, part (b). Homework Equations In the paper The Attempt at a Solution Am I expected to compute continuous correlation functions with respect to epsilon...
  50. W

    Correlation function, 1d polymer

    Homework Statement 1D polymer, fixed segment length a If the angle between segment j and j+1 is 0, the energy is 0 If the angle is pi the energy is +2J. Compute the correlation function <s_i s_{i+n}>, where s_j = \pm 1 denotes the direction of segment j Find the persistence length Lp...