(x2 ) 1 p(x, t) = J27rDt exp - 2Dt .

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(8.31 )

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THE MASER, THE LASER, ANO THEIR CAVITY QED COUSINS

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Then (x) = 0 and we find that

(F(t )

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= 0,

(8.32)

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as we would expect from a random force that pushes as much as it pulls. What about the self-correlation of F [i.e., (F(t)F(t ' ))]1 Using (8.29) again, we can write this as

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(F(t)F(t ' ) = atat (x(t)x(t ' ). ' Let us calculate (x(t)x(t' ). For t > t', we can write this average as (x(t)x(t' )

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(8.33)

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(8.34)

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where p(x, tlx' , t') is the conditional probability of finding the particle at position x at time t given that it was at position x' at the earlier time t'. This conditional probability is just the probability of the particle diffusing to x at time t having started at x' at time t'. You can easily show that this probability is given by8

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p(x, tlx', t') = J27r(tl_ t')D exp (

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(8.35)

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Substituting (8.31) and (8.35) in (8.34) and doing the integrals (Problem 8.3), we find that (8.36) (x(t)x(t ' = Dt' for t > t'.

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Swapping t and t', we find that

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(x(t)x(t ' )) = Dt

for t'

> t.

(8.37)

Equations (8.36) and (8.37) are equivalent to the single equation

(x(t)x(t ' ) = Dt' 8(t - t')

+ Dt8(t' -

(8.38)

where 8(u) is the Heaviside step function, which is unity for positive u and vanishes when u is negative. Substituting (8.38) in (8.33), we obtain

(F(t)F(t ' ) = D ~ {8(t - t')

= D8(t - t').

+ !8(t' -

t) -: t ' 8(t - t ' ),}

(8.39)

(t-t/)6(t-t')=O

In other words, F is a very special force: it fluctuates so much that it is only instantaneously correlated to itself. This is all that we need for introducing noise in our simple laser model. If, however. you wish to know a bit more about Langevin equations, read the next subsection.

SIn fact. the probability p(x, t) can also be viewed as a conditional probability of finding the particle at x at time t given that it started at the origin at time O. To see this. just take x, = 0 and t' = 0 in (8.35) and compare with (8.31).

HOW TO ADD NOISE

8.2.4 Ito's and Stratonovich's stochastic calculus

Langevin's equation was the first example of what mathematicians now call a stochastic differential equation. As it stands, however, (8.29) can lead to ambiguities. Let us try, for example, to derive Langevin's assumption that (x(t)F(t ) vanishes using only (8.29), and (8.39). Integrating (8.29), and taking x(O) = 0, we can write

(x(t)F(t ) =

Using (8.39), we obtain

lot dt' (F(t)F(t' ). lot dt' 8(t - t').

(8.40)

(x(t)F(t ) = D

(8.41)

The argument of the delta function only vanishes at the upper limit of this integral. What is the value of an integral like this If we rewrite this integral, changing its upper limit to some instant after t and multiplying the integrand by the step function 8(t - t'), we see that the problem here is equivalent to that discussed in Appendix F: how to determine the value ofthe product 8(u)8( u). To prevent such ambiguities, the idea of a stochastic differential equation had to be put into a more rigorous basis than that provided by Langevin's pioneering paper. Here we discuss briefly this more rigorous formulation. We will see that there are an infinite number of ways in which one can construct a rigorous stochastic calculus where equations such as (8.41) are no longer ambiguous. But the two most popular alternative ways are those proposed by Ito [312] and Stratonovich [580]. In Ito calculus, (8.41) vanishes as Langevin had assumed originally. On the other hand, the ordinary calculus chain rule changes. In Stratonovich calculus, the ordinary rules of calculus are maintained, but (8.41) is taken to be D /2. A completely rigorous mathematical presentation of stochastic calculus in the usual axioms-theorem-proof style would be completely out of touch with the spirit of this book and its intended readership. Instead, we have a more concrete discussion of stochastic calculus based on an approach developed by Gillespie [232]. This discussion differs from Gillespie's only because we deal explicitly with the inconsistency problem that leads to Stratonovich's and Ito's calculus, whereas [232] does not. The reader interested in a more mathematical approach can find some useful references in the recommended reading section at the end of the chapter, where even some books about applications of stochastic calculus in quantitative finance are mentioned. Any differential equation can be rewritten as a system of first-order differential equations. Let us look at the simplest case, where there is just one equation in this system and see how we can generalize it to make it stochastic. The general form of a first-order differential equation is

dt x(t)

A (x(t), t) ,

(8.42)