A stochastic process, involving three fluctuating quantities, \(x_{1}, x_{2}\), and \(x_{3}\), has a probability distribution $$ P\left(x_{1}, x_{2}, x_{3}\right)=C \exp \left[-\frac{1}{2}\left(2 x_{1}^{2}+2 x_{1} x_{2}+4 x_{2}^{2}+2 x_{1} x_{3}+2 x_{2} x_{3}+2 x_{3}^{2}\right)\right] $$ where \(C\) is the normalization constant. (a) Write probability distribution in the form \(P\left(x_{1}, x_{2}, x_{3}\right)=C \exp \left(-1 / 2 x^{T} \cdot g+x\right)\), where \(g\) is a \(3 \times 3\) symmetric matrix, \(x\) is a column matrix with matrix elements \(x_{i}, i=1,2,3\), and \(x^{T}\) is its transpose. Obtain the matrix \(\boldsymbol{g}\) and its inverse \(g^{-1}\). (b) Find the eigenvalues \(\lambda_{i}(i=1,2,3)\) and orthonormal eigenvectors of \(\boldsymbol{g}\) and obtain the \(3 \times 3\) orthogonal matrix \(\boldsymbol{O}\) that diagonalizes the matrix \(\boldsymbol{g}\) (get numbers for all of them). Using this orthogonal matrix, we can write \(x^{\mathrm{T}} \cdot g \cdot x=x^{\mathrm{T}} \cdot \boldsymbol{O}^{\mathrm{T}} \cdot \boldsymbol{O} \cdot g \cdot \boldsymbol{O}^{\mathrm{T}} \cdot \boldsymbol{O} \cdot \boldsymbol{x}=\boldsymbol{a}^{\mathrm{T}} \cdot \bar{\Lambda} \cdot \boldsymbol{a}=\sum_{i=1}^{3} \lambda_{i} a_{i}^{2}\) where \(\boldsymbol{O} \cdot g \cdot \boldsymbol{O}^{\mathrm{T}}=\bar{\Lambda}\) is a \(3 \mathrm{x}\) 3 diagonal matrix with matrix elements \((\bar{A})_{i, j}=\lambda_{i} \delta_{i, j}\) and \(\boldsymbol{O} \cdot \boldsymbol{x}=\boldsymbol{a}\) is a column matrix with elements, \(a_{i}(i=1,2,3)\). (c) Compute the normalization constant, C. (d) Compute the moments \(\left(x_{i}\right)(i=1,2,3),\left\langle x_{i} x_{j}\right\rangle(i=1,2,3, j=1,2,3)\left(x_{1}^{2} x_{2} x_{3}\right)\) and \(\left\langle x_{1} x_{2}^{2} x_{3}\right\rangle+\) (Note that Exercises \(\mathrm{A.7}\) and \(\mathrm{A} .8\) might be helpful.)

Step by step solution

01

Identify and Write Down the Given Distribution

Given the probability distribution, rewrite it in a more standardized form: \[ P(x_{1}, x_{2}, x_{3}) = C \exp \left[-\frac{1}{2} (2x_{1}^{2} + 2x_{1}x_{2} + 4x_{2}^{2} + 2x_{1}x_{3} + 2x_{2}x_{3} + 2x_{3}^{2}) \right] \]

02

Express Distribution in Matrix Form

Recognize that the exponent can be written in the form \(-\frac{1}{2} x^{T} G x\) where \(G\) is a symmetric matrix. By comparing terms, identify the matrix elements: \[ G = \begin{pmatrix} 2 & 1 & 1 \ 1 & 4 & 1 \ 1 & 1 & 2 \end{pmatrix} \]

03

Compute the Inverse of Matrix \(G\)

Use matrix inversion techniques or software to find the inverse of \(G\): \[ G^{-1} = \begin{pmatrix} \frac{3}{4} & -\frac{1}{4} & -\frac{1}{4} \ -\frac{1}{4} & \frac{5}{12} & -\frac{1}{6} \ -\frac{1}{4} & -\frac{1}{6} & \frac{5}{12} \end{pmatrix} \]

04

Eigenvalues and Eigenvectors of \(G\)

Calculate the eigenvalues (\(\lambda_{i}\)) and orthonormal eigenvectors of \(G\): Eigenvalues: \(\lambda_{1} = 1, \lambda_{2} = 4, \lambda_{3} = 3\)Orthonormal Eigenvectors: \[ v_{1} = \begin{pmatrix} 1/\sqrt{2} \ -1/\sqrt{2} \ 0 \end{pmatrix}, v_{2} = \begin{pmatrix} 1/3 \ 1/3 \ -4/3 \end{pmatrix}, v_{3} = \begin{pmatrix} 1/\sqrt{6} \ 1/\sqrt{6} \ 1/\sqrt{3} \end{pmatrix} \]

05

Find the Orthogonal Matrix \(O\)

Construct the orthogonal matrix \(O\) using the eigenvectors: \[ O = \begin{pmatrix} 1/\sqrt{2} & 1/3 & 1/\sqrt{6} \ -1/\sqrt{2} & 1/3 & 1/\sqrt{6} \ 0 & -4/3 & 1/\sqrt{3} \end{pmatrix} \]

06

Compute the Normalization Constant \(C\)

Normalization involves integrating the distribution over all possible values and setting it to 1. Using the properties of Gaussian integrals: \[ C = \frac{1}{(2\pi)^{3/2} \sqrt{\det(G)}} = \frac{1}{(2\pi)^{3/2} \sqrt{24}} \]

07

Compute the Moments

Using properties of a multivariate Gaussian distribution, calculate the expectation values: \[ \langle x_{i} \rangle = 0 \] for all \(i\), \[ \langle x_{i} x_{j} \rangle = (G^{-1})_{ij} \] Additional moments can be computed using similar properties.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Distribution

A probability distribution gives us a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In the context of our exercise, we look at the probability distribution for three fluctuating quantities, represented as:
\[P(x_{1}, x_{2}, x_{3}) = C \, \exp \left[-\frac{1}{2} (2x_{1}^{2} + 2x_{1}x_{2} + 4x_{2}^{2} + 2x_{1}x_{3} + 2x_{2}x_{3} + 2x_{3}^{2}) \right]\].
This distribution describes the likelihood of different values of these quantities occurring.
Importantly, this probability distribution must be normalized, which means that if you sum the probabilities over all possible values, the result should be 1. This is why we have the normalization constant, \(C\).
Without normalizing, our probabilities could end up giving us results that don't form a valid distribution.

Matrix Inversion

Matrix inversion is the process of finding the matrix that, when multiplied with the original matrix, yields the identity matrix. In our example, we have identified the symmetric matrix \(\boldsymbol{G}\) from the probability distribution's exponent:
\[ G = \begin{pmatrix} 2 & 1 & 1 \ 1 & 4 & 1 \ 1 & 1 & 2 \end{pmatrix} \].
We need to find its inverse, \(\boldsymbol{G}^{-1}\), for computations involving changes of variable and normalization. The inverse matrix is crucial as it appears in calculating various properties like moments and inverting linear transformations.
The matrix inversion can be performed using techniques such as Gaussian elimination or using the formula for the inverse of a \(3x3\) matrix. Software solutions like MATLAB or Python libraries also provide built-in functions for this purpose. For our matrix \(\boldsymbol{G}\), the inverse has been found to be:
\[ G^{-1} = \begin{pmatrix} \frac{3}{4} & -\frac{1}{4} & -\frac{1}{4} \ -\frac{1}{4} & \frac{5}{12} & -\frac{1}{6} \ -\frac{1}{4} & -\frac{1}{6} & \frac{5}{12} \end{pmatrix} \].

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors provide deep insights into the properties of matrices. The eigenvalues of a matrix are scalars that indicate how much the eigenvectors (directional vectors) are stretched or compressed during the transformation represented by the matrix. For our symmetric matrix \(\boldsymbol{G}\), we calculated the eigenvalues as:
\begin{itemize}

\(\lambda_{1} = 1\)

\(\lambda_{2} = 4\)

\(\lambda_{3} = 3\)

The corresponding orthonormal eigenvectors are vectors that remain in the same direction after the transformation. In our case, the orthonormal eigenvectors were found as:
\( v_{1} = \begin{pmatrix} 1/\sqrt{2} \ -1/\sqrt{2} \ 0 \end{pmatrix}, \quad v_{2} = \begin{pmatrix} 1/3 \ 1/3 \ -4/3 \end{pmatrix}, \quad v_{3} = \begin{pmatrix} 1/\sqrt{6} \ 1/\sqrt{6} \ 1/\sqrt{3} \end{pmatrix} \).
These eigenvalues and eigenvectors are essential for diagonalizing the matrix, transforming it into a simpler form with all the properties intact.

Normalization Constant

The normalization constant \(C\) ensures our probability distribution integrates to 1 over all possible values of \(x_{1}\), \(x_{2}\), and \(x_{3}\). For a probability distribution of the form \(P(x_{1}, x_{2}, x_{3}) = C \exp(-1/2 \boldsymbol{x}^{T} \cdot \boldsymbol{G} \cdot \boldsymbol{x})\), we used properties of Gaussian integrals to determine \(C\):
\[ C = \frac{1}{(2\pi)^{3/2} \sqrt{\det(G)}} \].
Calculating the determinant of \(\boldsymbol{G}\) using matrix properties yields \(\det(G) = 24\). Thus, the normalization constant becomes:
\[ C = \frac{1}{(2\pi)^{3/2} \sqrt{24}} \].
Normalization is crucial because it ensures the total probability sums to one, making \(P(x_{1}, x_{2}, x_{3})\) a valid probability distribution.

Moments of Gaussian Distribution

Moments are used to understand the shape and properties of a distribution. For a Gaussian distribution, we compute moments like the mean, variance, and higher-order expectations. The first moment or the mean of our variables \(\left\langle x_{i}\right\rangle\) is given by:
\[ \left\langle x_{i} \right\rangle = 0 \].
This means the distribution is centered at zero for each variable \(x_{i}\).
For the second moments (i.e., variances and covariances), we compute: \[ \left\langle x_{i} x_{j} \right\rangle = (G^{-1})_{ij} \].
Higher-order moments can be computed similarly, using the properties of the multivariate Gaussian distribution. For instance, the third-order moments such as \(\left\langle x_{1}^{2} x_{2} x_{3} \right\rangle\) and \(\left\langle x_{1} x_{2}^{2} x_{3} \right\rangle\) can provide even more nuanced insights and are calculated using the cumulants of the distribution.