Nonlinear master equation

Even if de Broglie never proposed a nonlinear master equation for the quantum mechanics, he always thought that quantum mechanics needs, for a more complete description, a nonlinear approach, and his double solution theory is, as we have seen, a good example of it. Here we shall propose a possible way to tackle this problem by means of an example. Let us select... 

Writing the solution of the nonlinear master equation in the general form... 

Looking at this equation, one sees that in free space, where V = 0, the nonlinear master equation transforms into the usual linear Schrodinger equation with a nonnull potential ... 

Under such conditions, for that particular form of the phase, given by (23), it is possible to write a set of solutions for the nonlinear master equation... 

THE MEAN FIELD APPROACH NONLINEAR MASTER EQUATION... 

Prigogine Nicolis and co-workers proposed a nonlinear master equation based on a mean-field approach and on the hypothesis of translational invariance C93. This equation has the form... 

Potentiometric titration curves are used to determine the molecular weight and fQ or for weak acid or weak base analytes. The analysis is accomplished using a nonlinear least squares fit to the potentiometric curve. The appropriate master equation can be provided, or its derivation can be left as a challenge. 

One basic difficulty with the nonlinear equation arises from the following. Consider a physical situation where a source of particles is composed of many emitters, each emitting a particle at a time. If considered alone, each particle would be described by a localized wave /,- solution of the master equation. Now, what happens if, instead of emitting the particles one by one, the source emits many particles at the same time If the master equation were a linear equation, like the usual Schrodinger equation, the answer would be trivial. The general solution would be simply the sum of all particular solutions. 

The theory described so far is based on the Master Equation, which is a sort of intermediate level between the macroscopic, phenomenological equations and the microscopic equations of motion of all particles in the system. In particular, the transition from reversible equations to an irreversible description has been taken for granted. Attempts have been made to derive the properties of fluctuations in nonlinear systems directly from the microscopic equations, either from the classical Liouville equation 18 or the quantum-mechanical equation for the density matrix.19 We shall discuss the quantum-mechanical treatment, because the formalism used in that case is more familiar. 

The basic remark is that linearity of the macroscopic law is not at all the same as linearity of the microscopic equations of motion. In most substances Ohm s law is valid up to a fairly strong field but if one visualizes the motion of an individual electron and the effect of an external field E on it, it becomes clear that microscopic linearity is restricted to only extremely small field strengths.23 Macroscopic linearity, therefore, is not due to microscopic linearity, but to a cancellation of nonlinear terms when averaging over all particles. It follows that the nonlinear terms proportional to E2, E3,... in the macroscopic equation do not correspond respectively to the terms proportional to E2, E3,... in the microscopic equations, but rather constitute a net effect after averaging all terms in the microscopic motion. This is exactly what the Master Equation approach purports to do. For this reason, I have more faith in the results obtained by means of the Master Equation than in the paradoxical result of the microscopic approach. 

The operator Q in the master equation is linear. It can be seen that the transition probability of a Markov process cannot obey a nonlinear equation of the form (1.4). The argument is similar to the one used by D. Polder, Philos. Mag. 45, 69 (1954). 

This interpretation of the master equation means that is has an entirely different role than the Chapman-Kolmogorov equation. The latter is a nonlinear equation, which results from the Markov character, but contains no specific information about any particular Markov process. In the master equation, however, one considers the transformation probabilities as given by the specific system, and then has a linear equation for the probabilities which determine the (mesoscopic) state of that system. 

The first one is that this particular form of H can also be used to prove the approach to equilibrium in the case of Boltzmann s kinetic equation for dilute gases. The Boltzmann equation is nonlinear and a different technique is needed to prove that all solutions tend to equilibrium. This technique is based on (5.6) other convex functions cannot be used. Incidentally, the Boltzmann equation is not a master equation for a probability density, but an evolution equation for the particle density in the six-dimensional one-particle phase space ( /i-space ). The linearized Boltzmann equation, however, has the same structure as a master equation (compare XIV.5). 

Remark. From the linear integro-differential equation for P(y, t) we have derived a nonlinear equation for y(t). Thus the essentially linear master equation may well correspond to a physical process that in the laboratory would be regarded as a nonlinear phenomenon inasmuch as its macroscopic equation is nonlinear. This is not paradoxical provided one bears in mind that the distinction between linear and nonlinear is defined for equations. It is wrong to apply it to a physical phenonemon, unless one has agreed upon a specific mathematical description of it. Newton s equations for the motion of the planets are nonlinear, but the Liouville equation of the solar system is linear. This connection between linear and nonlinear equations is not a matter of approximation the linear Liouville equation is rigorously equivalent with the nonlinear equations of motion of the particles. Generally any linear partial... 

Remark. The distinction between linear and nonlinear one-step processes has more physical significance than appears from the mathematical distinction between linear and nonlinear functions r(n) and g(n). In many cases n stands for a number of individuals, such as electrons, quanta, or bacteria. The master equation for pn is linear in n when these individuals do not interact, but follow their own individual random history regardless of the others. A nonlinear term in the equation means that the fate of each individual is affected by the total number of others present, as is particularly clear in example (iv) above. Thus linear master equations play a role similar to the ideal gas in gas theory. This state of affairs is described more formally in VII.6. 

As the solution of this M-equation is trivial, it follows that the solutions of (6.9) can immediately be found in the form of (6.6). This explains why we were able to solve linear one-step master equations explicitly (although not all of them can be interpreted in this way). On the other hand, when the coefficients in a one-step master equation are nonlinear functions of n (as in VI.9) it means that the molecules interact, and the solution is essentially more difficult. 

The Fokker-Planck equation is a special type of master equation, which is often used as an approximation to the actual equation or as a model for more general Markov processes. Its elegant mathematical properties should not obscure the fact that its application in physical situations requires a physical justification, which is not always obvious, in particular not in nonlinear systems. 

Suppose one is faced with a one-step problem in which the coefficients rn and g are nonlinear but can be represented by smooth functions r(n), g(n). Smooth means not only that r(n) and g(n) should be continuous and a sufficient number of times differentiable, but also that they vary little between n and n+ 1. Suppose furthermore that one is interested in solutions pn(t) that can similarly be represented by a smooth function P(n, t). It is then reasonable to approximate the problem by means of a description in which n is treated as a continuous variable. Moreover, since the individual steps of n are small compared to the other lengths that occur, one expects that the master equation can be approximated by a Fokker-Planck equation. The general scheme of section 2 provides the two coefficients, but we shall here use an alternative derivation, particularly suited to one-step processes. 

The diffusion approximation (1.5) is the nonlinear Fokker-Planck equation (VIII.2.5). In fact, we have now justified the derivation in VIII.2 by demonstrating that it is actually the first term of a systematic expansion in Q 1 for those master equations that have the property (1.1). Only under that condition is it true that the two coefficients... 

Summary. The special class of master equations characterized by (1.1) will be said to be of diffusion type. For such master equations the -expansion leads to the nonlinear Fokker-Planck equation (1.5), rather than to a macroscopic law with linear noise, as found in the previous chapter for master equations characterized by (X.3.4). The definition of both types presupposes that the transition probabilities have the canonical form (X.2.3), but does not distinguish between discrete and continuous ranges of the stochastic variable. The -expansion leads uniquely to the well-defined equation (1.5) and is therefore immune from the interpretation difficulties of the Ito equation mentioned in IX.4 and IX.5. 

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. 

It has the same form as the nonlinear Langevin or ltd equation (IX.4.5) but is supposed to have a small but non-zero tc. Show that the corresponding master equation (5.7) in the limit xc -> 0 takes the Stratonovich form (IX.4.8) rather than the Ito form (IX.4.12). This confirms by explicit calculation what has been argued in IX.5. ... 

An important property of the stochastic version of compartmental models with linear rate laws is that the mean of the stochastic version follows the same time course as the solution of the corresponding deterministic model. That is not true for stochastic models with nonlinear rate laws, e.g., when the probability of transfer of a particle depends on the state of the system. However, under fairly general conditions the mean of the stochastic version approaches the solution of the deterministic model as the number of particles increases. It is important to emphasize for the nonlinear case that whereas the deterministic formulation leads to a finite set of nonlinear differential equations, the master equation... 

The possible states for substrate are 11 and 6 for the complex. R is a 66-dimensional matrix and the initial condition for the master equation is pio,o (0) = 1. Figures 9.27 and 9.28 show the associated probabilities for each state as functions of time for the substrate and the complex, respectively. As previously, the full markers are the expected values and the solid lines the solution of the deterministic model. Notably, the expectation of the stochastic model does not follow the time profile of the deterministic system. This is the main characteristic of nonlinear systems. 

The discrete nature of master equations often impedes an analytic treatment. That holds in particular for master equations with nonlinearities or... 

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