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 Ch. 3 Sec. 4

3.3 Random Sequences

Subsections

3.3.1 Basics

So far: random numbers, preferably no serial correlations $\langle x_{n}x_{n+k} \rangle$.

Now: sequences of random numbers with given serial correlations.

Distribution in function space

Let $\{x(t)\}$ be an ensemble of functions of time $t$. Then

$\begin{eqnarray} P_{1}(x;t) & \equiv & {\cal P} \{x(t) \leq x\} \;\;\;\; {\rm and} \;\;\;\; p_{1}(x;t) \equiv \frac{\textstyle dP_{1}(x;t)}{\textstyle dx} \end{eqnarray}$

are the probability distribution and the respective density.

EXAMPLE: Let $x_{0}(t)$ be a deterministic function of time, and assume that the quantity $x(t)$ at any time $t$ be Gauss distributed about the value $x_{0}(t)$:

$p_{1}(x;t)=\frac{\textstyle 1}{\textstyle \sqrt{\textstyle 2\pi \sigma ^{2}}} e^{\textstyle -\frac{1}{2} \left[x-x_{0}(t)\right]^{2}/ \sigma ^{2}}$

A random process is called a random sequence if the variable $t$ may assume only discrete values $\{t_{k}; \; k=0,1,\dots \}$. In this case one often writes $x(k)$ for $x(t_{k})$.

Higher Distributions

The foregoing definitions may be generalized in the following manner:

$\begin{eqnarray} \!\! P_{2}(x_{1},x_{2};t_{1},t_{2}) & \equiv & {\cal P} \{ x(t_{1})\leq x_{1}, x(t_{2})\leq x_{2} \} \\ \vdots \\ \!\! P_{n}(x_{1},\dots,x_{n};t_{1},\dots,t_{n}) & \equiv & {\cal P} \{ x(t_{1})\leq x_{1},\dots,x(t_{n})\leq x_{n} \} \end{eqnarray}$

Thus $P_{2}(..)$ is the compound probability for the events $x(t_{1})\leq x_{1}$ and $x(t_{2})\leq x_{2}$. These higher order distribution functions and the corresponding densities

$\begin{eqnarray} p_{n}(x_{1},\dots,x_{n};t_{1},\dots,t_{n}) & = & \frac{\textstyle d^{n}P_{n}(x_{1}, \dots , x_{n};t_{1}, \dots,t_{n})}{\textstyle dx_{1} \dots dx_{n}} \end{eqnarray}$

describe the random process in ever more - statistical - detail.

Stationarity:

A random process is stationary if

$\begin{eqnarray} P_{n}(x_{1},\dots,x_{n};t_{1},\dots,t_{n}) & = & P_{n}(x_{1},\dots,x_{n};t_{1}+t,\dots,t_{n}+t), \end{eqnarray}$

This means that the origin of time is of no importance:

$\begin{eqnarray} p_{1}(x;t)=p_{1}(x) \;\;\;\; & {\rm and} & \;\;\;\; p_{2}(x_{1},x_{2};t_{1},t_{2})=p_{2}(x_{1},x_{2};t_{2}-t_{1}) \end{eqnarray}$

Autocorrelation:

$\begin{eqnarray} \langle x(0)x(\tau) \rangle &\equiv& \int \limits_{\textstyle a}^{\textstyle b}\!\int \limits_{\textstyle a}^{\textstyle b} x_{1} x_{2} p_{2}(x_{1},x_{2};\tau)dx_{1} dx_{2}, \end{eqnarray}$

For $\tau \rightarrow 0$ the autocorrelation function (acf) approaches the variance $\langle x^{2} \rangle$. For finite $\tau$ it tells us how rapidly a particular value of $x(t)$ will be "forgotten".

Gaussian process:

The random variables $x(t_{1}), \dots , x(t_{n})$ obey a multivariate Gaussian distribution. The covariance matrix elements are $\langle x(0)x(t_{j}-t_{i}) \rangle$, i.e. the values of the autocorrelation function at the specific time displacement:

$\begin{eqnarray} p_{2}(x_{1}, x_{2};\tau) & = & \frac{\textstyle 1}{\textstyle \sqrt{\textstyle (2\pi)^{2} S_{2}(\tau)}} e^{\textstyle -\frac{1}{2}Q} \end{eqnarray}$

with

$\begin{eqnarray} Q & \equiv & \frac{\textstyle \langle x^{2} \rangle x_{1}^{2} -2 \langle x(0)x(\tau ) \rangle x_{1}x_{2} + \langle x^{2} \rangle x_{2}^{2}}{\textstyle S_{2}(\tau)} \end{eqnarray}$

and
$\begin{eqnarray} S_{2}(\tau) &\equiv& | S_{2}(\tau)| = \langle x^{2} \rangle^{2} - \langle x(0)x(\tau) \rangle^{2} \end{eqnarray}$

Markov Process:

A stationary random sequence $\{x_{n}; n=0, 1 \dots\}$ has the Markov property if its "memory" goes back only one time step:

$\begin{eqnarray} p(x_{n} | x_{n-1} \dots x_{1})&=&p(x_{n}|x_{n-1}) \end{eqnarray}$

where the conditional density
$\begin{eqnarray} p(x_{n}|x_{n-1};\tau) & = & \frac{\textstyle p_{2}(x_{n-1},x_{n})}{\textstyle p_{1}(x_{n-1})} \end{eqnarray}$

is the density of $x_{n}$ under the condition that $x(n-1)=x_{n-1}$.
Thus all statistical properties of the process are contained in $p_{2}(x_{n-1},x_{n})$.

An even shorter memory would mean that successive elements of the sequence were not correlated at all.

Gaussian Markov processes:

To describe them uniquely not even $p_{2}(\dots)$ is needed. If the autocorrelation function $\langle x(n) x(n+l) \rangle$ is known, $p_{2}(..)$ and consequently all statistical properties of the process follow.

Note: The acf of a stationary Gaussian Markov process is always an exponential:

$\begin{eqnarray} \langle x(0)x(\tau) \rangle &=& \langle x^{2} \rangle e^{- \beta \tau} \end{eqnarray}$

or
$\begin{eqnarray} \langle x(n)x(n+k) \rangle &=& \langle x^{2} \rangle e^{- \beta \Delta t k} \end{eqnarray}$

How to produce a Markov sequence? See below...

3.3.2 Generating a stationary Gaussian Markov sequence

Solve the stochastic differential equation

$\begin{eqnarray} \dot{x}(t) &=& - \beta x(t) + s(t) \end{eqnarray}$

with a stochastic "driving" process $s(t)$, assumed to be uncorrelated Gaussian noise, i.e. Gauss distributed about $\langle s \rangle =0$, with $\langle s(0) s(t) \rangle = A \delta(t)$.

The general solution to this equation reads

$\begin{eqnarray} x(t)&=&x(0)e^{-\beta t} + \int \limits_{\textstyle 0}^{\textstyle t} e^{-\beta (t-t')} s(t') dt' \end{eqnarray}$

Inserting $t=t_{n}$ and $t=t_{n+1} \equiv t_{n}+\Delta t$ one finds that

$\begin{eqnarray} x(t_{n+1})&=&x(t_{n}) e^{-\beta \Delta t} + \int \limits_{\textstyle 0}^{\textstyle \Delta t} e^{-\beta (\Delta t-t')} s(t_{n}+t') dt' \end{eqnarray}$

At any time $t$, the values of $x(t)$ belong to a stationary Gauss distribution with $\langle x^{2} \rangle$ $=A/2\beta$, and the process $\{ x(t_{n}) \}$ has the Markov property.

The integrals

$\begin{eqnarray} z(t_{n}) &\equiv& \int \limits_{\textstyle 0}^{\textstyle \Delta t} e^{-\beta (\Delta t-t')} s(t_{n}+t') dt' \end{eqnarray}$

are elements of a random sequence, with $z$ Gauss distributed with zero mean and $\langle z(t_{n})z(t_{n+k}) \rangle = 0$ for $k \neq 0$. Their variance is

$\begin{eqnarray} \langle z^{2} \rangle &=& \frac{A}{2\beta}(1-e^{-2\beta\Delta t}) \end{eqnarray}$

Here is the resulting recipe for generating a stationary, Gaussian Markov sequence:

"Langevin Shuffle":

Let the desired stationary Gaussian Markov sequence $\{x(n);\;n=0,\dots\}$ be defined by the autocorrelation function $\langle x(n)x(n+k) \rangle = (A / 2 \beta) e^{-\beta k\Delta t}$ with given parameters $A$, $\beta$ and $\Delta t$. Choose a starting value $x(0)$, either as $x(0)=0$ or from a Gauss distribution with $\langle x \rangle =0$ and $\langle x^{2} \rangle =A/2\beta$.
• Draw $z(n)$ from a Gaussian distribution with $\langle z \rangle =0$ and $\begin{eqnarray} \langle z^{2} \rangle & = & \frac{A}{2\beta}(1-e^{-2\beta \Delta t}) \end{eqnarray}$
• Construct $\begin{eqnarray} x(n+1)& = & x(n)e^{-\beta\Delta t} + z(n) \end{eqnarray}$
The random sequence thus produced has the desired properties.

If $\beta \Delta t \ll 1$, replace the exponential by its linear Taylor approximation. The iteration prescription then reads $x(n+1)=x(n)(1-\beta\Delta t) +z'(n)$, where $z'(n)$ is Gaussian with $\langle {z'}^{2} \rangle =A \Delta t (1-\beta \Delta t)$.

Employ the above procedure to generate a Markov sequence $\{ x_{n}\}$ with a given $\beta$. Check if the sequence shows the expected autocorrelation.

3.3.3 Wiener-Lévy Process (Unbiased Random Walk)

With $\beta=0$ in the above differential equation, we find

$\begin{eqnarray} x(n+1) &=& x(n)+z(n) \end{eqnarray}$

where $z(n)$ is Gaussian with
$\begin{eqnarray} z(n) &\equiv& \int \limits_{\textstyle 0}^{\textstyle \Delta t} s(t_{n}+t')dt' \;,\;\;\;\; \langle z \rangle =0 \; , \;\;\;\; \langle z^{2} \rangle =A\Delta t \end{eqnarray}$

Since $z$ and $x$ are uncorrelated, we have
$\begin{eqnarray} \langle [x(n)]^{2} \rangle &=& nA\Delta t \end{eqnarray}$

EXAMPLE: Let $x$ be one cartesian coordinate of a diffusing particle. Then $\langle [x(n)]^{2} \rangle$ is the mean squared displacement after $n$ time steps. In this case we may relate the coefficient $A$ to the diffusion constant according to $A=2D$.

 Wiener-Lévy process: Let $A \Delta t$ be given. Choose $x(0)=0$. Pick $z(n)$ from a Gauss distribution with variance $A \Delta t$. Compute $\begin{eqnarray} x(n+1)&=&x(n)+z(n) \end{eqnarray}$ The random sequence thus produced is a nonstationary Gaussian process with variance $<[x(n)]^{2}>=n A \Delta t$.

500 random walkers set out from positions $x(0)$ homogeneously distributed in the interval $[-1,1]$. The initial particle density is thus rectangular. Each of the random walkers is now set on its course to perform its own one-dimensional trajectory, with $A\Delta t = 0.01$. Sketch the particle density after 100, 200, ... steps.

It is not really necessary to draw $z(n)$ from a Gaussian distribution. If $z(n)$ comes from an equidistribution in $[-\Delta x/2,\; \Delta x/2]$, the "compound" $x$-increment after every $10-15$ steps will again be Gauss distributed (central limit theorem).

We may even discretize the $x$-axis: $z=0, \; \pm \Delta x$ with equal probability $1/3$: after many steps, and on a scale which makes $\Delta x$ appear small, the results will again be the same.

To simulate 2- or 3-dimensional diffusion, apply the above procedure independently to 2 or 3 coordinates.

3.3.4 Markov Chains (Biased Random Walk)

A Markov sequence of discrete $x_{\alpha}$ is called a Markov chain.

We generalize the discussion to "state vectors" $\{ x_{\alpha}, \alpha=1, \dots M \}$. The conditional probability

$\begin{eqnarray} p_{\alpha \beta} & \equiv & {\cal P} \{ x(n)=x_{\beta} | x(n-1)= x_{\alpha} \} \end{eqnarray}$

is called transition probability between the states $\alpha$ and $\beta$

Let $M$ be the total number of possible states. The $M \times M$ -matrix $P \equiv \{ p_{\alpha \beta}\}$ and the $M$-vector $p$ consisting of the individual probabilities $p_{\alpha} \equiv {\cal P} \{ x= x_{\alpha} \}$ determine the statistical properties of the Markov chain uniquely.

A Markov chain is reversible if

$\begin{eqnarray} p_{\alpha} p_{\alpha \beta} & = & p_{\beta}p_{\beta \alpha} \end{eqnarray}$

- Meaning?

The $M^{2}$ elements of the matrix $P$ are not uniquely defined by the $M(M-1)/2$ reversibility conditions. $\Longrightarrow$ For a given distribution density $p$ there are many reversible transition matrices. $\Longrightarrow$ "Asymmetrical rule" (N. Metropolis):

N. Metropolis' asymmetric rule:

Let $Z$ be the number of states $x_{\beta}$ accessible from $x_{\alpha}$, and let the a priori access probability be $\pi_{\alpha \beta } = 1/Z$. Then
$p_{\alpha \beta} = \pi_{\alpha \beta } \;\;\; {\rm if} \;\; p_{\beta} \geq p_{\alpha}$
$p_{\alpha \beta} = \pi_{\alpha \beta } \frac{\textstyle p_{\beta}}{\textstyle p_{\alpha}} \;\;\; {\rm if} \;\; p_{\beta} < p_{\alpha}$

$\Longrightarrow$ $p_{\alpha \beta}$ is reversible!

3.3.5 Monte Carlo Method

Central theorem:
If the stationary Markov chain characterized by $p \equiv \{ p_{\alpha} \}$ and $P \equiv \{ p_{\alpha \beta} \}$ is reversible, then each state $x_{\alpha}$ will be visited, in the course of a sufficiently long chain, with the relative frequency $p_{\alpha}$.
$\Longrightarrow$ Here is yet another recipe for generating random numbers with a given probability density $p$:

Random numbers a la Metropolis:

Let $p \equiv \{ p_{\alpha}; \; \alpha=1,2,\dots \}$ be the vector of probabilities of the events $x=x_{\alpha}$. To generate a random sequence $\{ x(n)\}$ in which the relative frequency of the event $x(n)=x_{\alpha}$ approaches $p_{\alpha}$:
• After the $n$-th step, let $x(n)=x_{\alpha}$. Draw a value $x_{\beta}$ from a region around $x_{\alpha}$, e.g. according to $x_{\beta}=x_{\alpha}+ \left( \xi -0.5 \right) \Delta x$ where $\xi$ $\in (0,1)$.
• If for $p_{\beta} \equiv p(x_{\beta})$ we have $p_{\beta} \geq p_{\alpha}$, then let $x(n+1)=x_{\beta}$.
• If $p_{\beta} < p_{ \alpha }$, then pick a random number $\xi$ $\in (0,1)$; if $\xi < p_{\beta}/p_{\alpha}$, let $x(n+1)=x_{\beta}$; else put $x(n+1)=x_{\alpha}$.

Adjust the parameter $\Delta x$ such that approximately one out of two trial moves leads to a new state, $x(n+1)=x_{\beta}$.

Warning: The random numbers thus produced are serially correlated: $\langle x(n)x(n+k) \rangle \neq 0$.

Let $p(x)=A \, \exp[-x^{2}]$ be the desired probability density. Apply Metropolis' prescription to generate random numbers with this density. Confirm that $\langle x(n)x(n+k) \rangle \neq 0$.
Advantage of Metropolis' method: only $p_{\beta}/p_{\alpha}$ is needed, not $p_{\alpha}$.
$\Longrightarrow$ Statistical-mechanical Monte Carlo simulation: only relative thermodynamic probabilities needed!