Linear noise approximation

Thus we have found the distribution of the fluctuations around the macroscopic value. They have been computed to order Q 1/2 relative to the macroscopic value n, which will be called the linear noise approximation. In this order of Q the noise is Gaussian even in time-dependent states far from equilibrium. Higher corrections are computed in X.6 and they modify the Gaussian character. However, they are of order 2 1 relative to n and therefore of the order of a single molecule. 

This is a linear Fokker-Planck equation whose coefficients depend on time through . This approximation was christened in section 1 linear noise approximation . The solution of (4.1) was found in VIII.6 to be a Gaussian510, so that it suffices to determine the first and second moments of On multiplying (4.1) by and 2, respectively, one obtains... 

Summary. The aim of solving the master equation with initial condition (2.1) has now been achieved in the linear noise approximation by the following three steps. 

The linear noise approximation may therefore also be called the Gaussian approximation , but it should be clear that the O-expansion derives the Gaussian character rather than postulat-... 

Thus in the linear noise approximation the average obeys the macroscopic law. 

This is the general formula (in the linear noise approximation) for the autocorrelation function of the fluctuations in a stable stationary state. Hence it is possible to write down the fluctuation spectrum in an arbitrary system without solving any specific equations. This fact is the basis of the customary noise theory. 

Even in the present case, however, it is possible to manufacture a nonlinear Fokker-Planck equation, and a corresponding Lange vin equation, which as far as the linear noise approximation is concerned reproduce the same results as found here. But any features they contain beyond that approximation are spurious. For instance, one cannot conclude from this manufactured Fokker-Planck equation that the stationary distribution is given by (VIII. 1.4). 

Exercise. Write the linear noise approximation of the solution P(X, t) with the delta initial value (2.8) explicitly in terms of and the solution of (4.2). 

Exercise. Apply the linear noise approximation to the M-equation (VI.3.12).S, )... 

Exercise. Verify that the linear noise approximation always leads to an Ornstein-Uhlenbeck process for the fluctuations in a stable stationary state. 

Exercise. Show that the terms [Pg.262]

Exercise. Verify that (VIII.5.6) is nothing but the linear noise approximation of (VIII.5.1). 

In this section we examine the higher orders beyond the linear noise approximation. They add terms to the fluctuations that are of order relative to the macroscopic quantities, i.e., of the order of a single particle. They also modify the macroscopic equation by terms of that same order, as has been anticipated in (V.8.12) and (4.8). These effects are obviously unimportant for most practical noise problems, but cannot be ignored in two cases. First, they tell us how equilibrium fluctuations are affected by the presence of nonlinear terms in the macroscopic equation, in particular how... 

While the linear noise approximation led to the linear Fokker-Planck equation (4.1) we now see that the higher powers oi Q 1,2 give rise to three modifications. 

It has been shown that the lowest order of the Q-expansion yields the macroscopic equation, and the next order the linear noise approximation, provided that the stability condition (X.3.4) holds. This condition is violated, albeit marginally, when a0() = 0. In this case the O-expansion takes an entirely different form its lowest approximation is a nonlinear Fokker-Planck equation. 

Consider the solution of the M-equation that at t = 0 consists of a delta peak located at some point (0O, ij/0) on the macroscopic limit cycle. The shape of the probability distribution is (in linear noise approximation) governed by the equation... 

This equation determines (in linear noise approximation) the fluctuations about the solution )) of the Boltzmann equation. 

Although the example in the previous section had a linear master equation the identities (1.2) and (1.3) are general. They can be evaluated when the equation (1.1) can be solved. For nonlinear systems this can be done by means of the O-expansion. It turns out, however, that one has to go beyond the linear noise approximation in order to find a correlation between the jump events. Unfortunately this makes the calculations rather formidable. We shall here treat an example which has been constructed to be as simple as possible. 

After the kinetic model for the network is defined, a simulation method needs to be chosen, given the systemic phenomenon of interest. The phenomenon might be spatial. Then it has to be decided whether in addition stochasticity plays a role or not. In the former case the kinetic model should be described with a reaction-diffusion master equation [81], whereas in the latter case partial differential equations should suffice. If the phenomenon does not involve a spatial organization, the dynamics can be simulated either using ordinary differential equations [47] or master equations [82-84]. In the latter case but not in the former, stochasticity is considered of importance. A first-order estimate of the magnitude of stochastic fluctuations can be obtained using the linear noise approximation, given only the ordinary differential equation description of the kinetic model [83-85, 87]. 

Elf, J. Ehrenberg, M. Fast evaluation of fluctuations in biochemical networks with the linear noise approximation. Genome Res 2003,13 2475-2484. 

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