Ursell function
Encyclopedia
In statistical mechanics
, an Ursell function or connected correlation function, is a cumulant
of
a random variable
. It is also called a connected correlation function as it can often be obtained by summing over
connected Feynman diagram
s (the sum over all Feynman diagrams gives the correlation function
s).
If X is a random variable, the moment
s sn and cumulants (same as Ursell functions) un are related by the exponential formula
:
(where E is the expectation
).
The function was named after Harold Ursell, who introduced it in 1927.
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
, an Ursell function or connected correlation function, is a cumulant
Cumulant
In probability theory and statistics, the cumulants κn of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. The moments determine the cumulants in the sense that any two probability distributions whose moments are identical will have...
of
a random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
. It is also called a connected correlation function as it can often be obtained by summing over
connected Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
s (the sum over all Feynman diagrams gives the correlation function
Correlation function
A correlation function is the correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points...
s).
If X is a random variable, the moment
Moment (mathematics)
In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...
s sn and cumulants (same as Ursell functions) un are related by the exponential formula
Exponential formula
In combinatorial mathematics, the exponential formula states that the exponential generating function for structures on finite sets is the exponential of the exponential generating function for connected structures.The exponential formula is a power-series version of a special case of Faà di...
:
(where E is the expectation
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
).
The function was named after Harold Ursell, who introduced it in 1927.