- #1
Frank Castle
- 580
- 23
I thought I understood the concept of a correlation function, but I having some doubts.
What exactly does a correlation function quantify and furthermore, what is a correlation length.
As far as I understand, a correlation between two variables ##X## and ##Y## quantifies how much the two variables are related to one another, i.e. a statistical measure of how much they covary (fluctuate together).
With this in mind, does a correlation function, ##\langle\xi(x)\chi(y)\rangle## ##^{(\ast)}## of two spatially dependent variables ##\xi(x)## and ##\chi(x)## measure the extent to which the values of the variables, at spatially separated points ##x## and ##y##, are related? In this sense, if the value of ##\langle\xi(x)\chi(y)\rangle## is close to 1 does this indicate a co-dependency between the two variables at spatially separated points, i.e. that if ##\xi## has a particular value at ##x## then ##\chi## will have a very similar value at the point ##y##?!
As an example in physics, for spins on a lattice, does the correlation function between spins at different lattice points quantify the extent to which spins at different lattice points are co-aligned?! Another example would be anisotropies in the CMB - does the correlation function between temperature fluctuations in the CMB quantify the extent to which temperature fluctuations at different points in the CMB are similar in value?!
As for correlation length, does this quantify the length at which two variables cease to be similar in value? For example, if one has a surface that has a spatially varying height, does the correlation length quantify the length at which the height of the surface starts to significantly deviate from a given value?! Another example, in the case of spins on a lattice, does the correlation length quantify the length at which spins are no longer similarly aligned (i.e. the length at which they cease to be correlated)?!##^{(\ast)}## [Does the notation ##\langle\;\rangle## indicate the one is taking the expectation value, for example, does ##\langle\xi(x)\chi(y)\rangle =\mathbb{E}\left[\xi(x)\chi(y)\right]##?]
What exactly does a correlation function quantify and furthermore, what is a correlation length.
As far as I understand, a correlation between two variables ##X## and ##Y## quantifies how much the two variables are related to one another, i.e. a statistical measure of how much they covary (fluctuate together).
With this in mind, does a correlation function, ##\langle\xi(x)\chi(y)\rangle## ##^{(\ast)}## of two spatially dependent variables ##\xi(x)## and ##\chi(x)## measure the extent to which the values of the variables, at spatially separated points ##x## and ##y##, are related? In this sense, if the value of ##\langle\xi(x)\chi(y)\rangle## is close to 1 does this indicate a co-dependency between the two variables at spatially separated points, i.e. that if ##\xi## has a particular value at ##x## then ##\chi## will have a very similar value at the point ##y##?!
As an example in physics, for spins on a lattice, does the correlation function between spins at different lattice points quantify the extent to which spins at different lattice points are co-aligned?! Another example would be anisotropies in the CMB - does the correlation function between temperature fluctuations in the CMB quantify the extent to which temperature fluctuations at different points in the CMB are similar in value?!
As for correlation length, does this quantify the length at which two variables cease to be similar in value? For example, if one has a surface that has a spatially varying height, does the correlation length quantify the length at which the height of the surface starts to significantly deviate from a given value?! Another example, in the case of spins on a lattice, does the correlation length quantify the length at which spins are no longer similarly aligned (i.e. the length at which they cease to be correlated)?!##^{(\ast)}## [Does the notation ##\langle\;\rangle## indicate the one is taking the expectation value, for example, does ##\langle\xi(x)\chi(y)\rangle =\mathbb{E}\left[\xi(x)\chi(y)\right]##?]
Last edited: