Boltzmann Equations, Functional Analysis and Stochastic Processes
Abstract
A new approach to the Boltzmann equation yields a fairly general corresponding solution technique which actually applies to the linear case.
I show that linear Boltzmann equations can be solved by inspecting the operatorial properties of its members and that many new cases will be solvable as soon as some questions posed here will be given an answer by functional analysis.
As an example, I also report, up to a quadrature, the solution of the problem of neutron slow-down by light moderator which to my knowledge had never been solved earlier.
On the other side, by studying such equations it is natural to give a general theory of them and have a general understanding of their meaning.
I evidence that integro-differential equations (IDEs) represent the most general description of those phenomena in which the use of probability densities is required: stochastic processes.
By the use of an example (many others can be easily thought) I show that the Fokker-Planck equation, which is now considered the mathematical support of stochastic processes, can not cope with all kind of physics and must be replaced by IDEs.
I finally show how the Boltzmann equation applies to those random phenomena (these will be better explained in future articles) that have nothing in common with quantum physics and is, therefore, very general.
I also illustrate how to build IDEs for uncertain processes and then eventually solve them.
The reader realizes that Transport Theory has only been the first field to need stochastic formulation but is far from being the last.
It is showed that "thinking stochastic" is a very powerful tool and has immediate applications to many areas of research such as Financial Markets, Behavior Science and Game Theory.
Every time there is some uncertain process there is a Boltzmann equation governing it.
1. Introduction to the Equation
As everybody knows, the Boltzmann Equation first appeared in Transport Theory.
The original form of the equation for this problem is very complex as it contains terms that take into account the scattering of particles (both linear and non-linear), the action of some potential, the spatial drift of particles, the sources and the removal effects.
In more complex phenomena, such as Plasma Physics or Galaxy Formation in the Universe one should also consider the particle-to-particle interaction potential which complicates the equation to unmanageable levels.
This is the reason why hardly anybody feels at ease with such equations: too many terms saying too many things in too difficult ways.
The transport equation, in its original form (which ages back to more than a hundred years) written by Boltzmann describes the evolution of a function f depending on the three spatial coordinates, on the three components of particle speed and on time such that
represents the number of particles per element of phase-space at time t.
Written in this way (few people use to solve equations in seven variables) the unknown seems too complicated and is misleading to understanding and solving the problem.
This is the reason why, first of all, I want to focus the reader’s attention on more physical quantities.
From Eq. 1, we have, for the total particle number
N at instant t,
Eq. 2
and, dividing f by N, we obtain a new function
F (still depending on the position in phase-space plus time) whose integral over phase space is normalized to unity:
Eq. 3
Eq. 4
One could ask if it is a good business to change one unknown with two and moreover why this notation is better than the older one.
The answer to the first question is that the Boltzmann equation brings into account different processes which in truth take place separately from each other and can be then investigated apart.
For example, scattering does not modify the number of particles but changes their velocity distribution while the drift of particles changes the number of particles in a certain region but neither their overall number nor their velocity spectrum.
Moreover, the biggest difficulty of the equation is its integro-differential nature being the integrals evaluated on phase space.
The total particle number however does not depend on either position or velocity and obeys an ODE which is deducted by integrating both members of the Boltzmann equation on the entire phase-space.
In reply to the second question I want to help the reader understand that the function
F brings with it all the integral nature of the problem and will reveal to be the probability density (PD) of one particle to be in a phase-space point at time t.This is in truth the most important development in this passage which links once and forever (how, will later be clear) IDEs to stochastic processes.
Consider also that
F is not affected by removal effects but only density and spectrum changes in particle distribution and is so (in truth, this will not hold in the non-linear case) independent of the particle number.In addition, it allows a considerable ease of algebra to work with normalized functions.
For all these reasons I invite the reader to join my new notation.
1.1 Stochastic Nature of Transport Theory
Let us start from a very simple problem: the time evolution of the distribution of the velocity modulus of a set of particles supposed to interact with a host medium but not with themselves. The main quantity here is the function
F where
Eq. 5
is the fraction of particles at a certain instant with speed modulus between v1 and v2.
Suppose that at the initial instant of time t0 there is a certain distribution of speeds. This changes by the effect of scattering (in the linear case) against the host medium.
At a new instant of time t1 the Boltzmann equation (we suppose here to have the solution) forecasts a new distribution of speeds which will be generally different from the starting one.
Now, suppose the lucky circumstance in which none of the particles has encountered a particle of the host medium in that time interval and so has not taken part into any scattering event. In this case the distribution of speeds given by the solution of the equation is different from the true one because with no scattering there is no change in
F.So is the Boltzmann equation to be considered false? The answer is surely no but something has got to be cleared. The process under study here is not deterministic because assigning cross-sections does not imply that scattering events will take place in a deterministic way. Uncertainty remains and this means that the process here can evolve in many different ways that we call "realizations". All realizations are good in this kind of physics and, as it happens in quantum mechanics, we can only study their relative probability.
I must also say that, in truth, even if there were scattering events taking place at exact instants of time a stochastic formulation could not be avoided since quantum mechanics states that in a scattering process only transition probabilities can be determined and even if your particle is in a state of fixed speed before scattering you will not be able to know its speed after this event.
So there is no way to go ahead but to accept that sometimes there are in nature (in truth, not only in nature but also in artificial processes) phenomena that bring a certain amount of randomness with them and for these only a stochastic approach can be attempted.
The reason why this had not been considered earlier in transport theory is that we use to deal with big numbers of particles; of the same order of magnitude of the Avogadro number. This implies that deviations from the average realization of the process are so small that can be neglected.
However I want here to give a valid tool for inspecting even the case of a single particle and in order to do this no other ways can be followed.
To give a better idea of what is going on I will show now how to pull out a Boltzmann equation for a certain stochastic process. Moreover I will take as example the process of random-walk which is well known to conduct to the Fokker-Planck’s equation.
True physics shows that without making some approximations, done in the original formulation (in order, also, to obtain a Fokker-Planck), the random walk problem naturally leads to the Boltzmann equation.
Before going on, however, I want to answer the two big question that the reader is probably now asking: 1) what does the Boltzmann equation then say? 2) how to build it in the general case?
- The Boltzmann equation describes the evolution of the average realization of the stochastic process in study where averaging here is: on all possible events.
- In order to obtain an equation for a certain random process write down as the process evolves in time by averaging all possible events with their relative PD.
The example will probably explain much better.
1.2 Random-walk
Suppose you have a particle which undergoes kicks at fixed discrete instants of time. Each kick moves it along a real axis of an unknown quantity s. This quantity is uncertain but different values are not equally probable; on the contrary, they are distributed with a certain PD
which is defined on the entire real axis and whose integral on a certain interval is the probability that the particle has moved of a quantity between the lower and the upper integration limits.
Even if we know the position of the particle at the initial time, as soon as it is first moved, we can no longer know its position in a deterministic way. This means that, as the first random events takes place, all the process becomes stochastic.
Let us suppose however that at the t=1 the particle is in a fixed position at x=x0 and let us write the equation for its PD. At the initial time the PD for the particle is
Eq. 6
since we exactly know the position.
Now, if the particle undergoes an event that moves it of a quantity s in the positive direction of the real axis then its PD becomes
which is still a deterministic knowledge since the particle continues to be localized.
However we do not know exactly the amount of the displacement and so the PD for the particle at instant t=1 has to be evaluated as an average on all possible events of motion being the weight given by the function K. In formulae this reads
Eq. 7
and in general we have that
By using the property of commutation of the convolution product we now obtain
which seems a sort of Boltzmann equation. Therein, indeed, we have a quantity changing at discrete instants of time by the action of an integral operator.
I have intentionally used discrete-time to evidence once more how the link between stochastic processes and integral operators goes much beyond transport theory.
Moreover, notice that the equation above is "exact" since in deducting it I made no approximations.
I have illustrated a discrete-time processes in which one writes down the PD at a certain time as a function (functional is the correct wording and will soon be explained) of the PD at the previous instant.
The underlying process here is in truth somewhat "artificial" since it is hard to imagine something in nature that acts with such precise clocking.
Suppose, on the contrary, as it mostly happens in nature, that your PD changes due to events whose taking place is not assured but also has some probability per unit time
n of happening. In this case the correct equation would have been
If, for example, the particle had to be moved by wind blows, then Eq. 10 would have been a more accurate description than Eq. 9 since wind blows in a non-deterministic way also and so it is impossible to exactly know when a blow will arrive.
Let us now write Eq. 10 by using again the commutation of the convolution product obtaining
Now, by taking the power series in s for
F in Eq. 11 and truncating it to the second order you will obtain a Fokker-Planck’s which is so understood to be a physical approximation. I will not do this since my intention here is to continue on the way of integral physics.Another reason for using Eq. 9 and Eq. 10 instead of Fokker-Planck’s is that they are very easy to solve. I will not annoy the reader with long algebraic manipulations but only outline the way to follow for the general initial condition
Eq. 12
In order to solve either Eq. 9 or Eq. 10 follow these steps:
1) take the Fourier transforms of both sides on the variable x using the property of the convolution product;
2) solve the obtained equation for the transform which is respectively a difference equation or a PDE;
3) insert the initial condition;
4) invert the obtained time-dependent transform.
By now indicating the Fourier transforms with
Eq. 13
Eq. 14
Eq. 15
Eq. 16
the solutions of Eq. 9 and Eq. 10 are expressed respectively by
Eq. 17
Eq. 18
It may be useful sometimes, depending on the specific problem, to reuse here the property of the convolution product when inverting the transform. This expresses the solution in the form of a convolution which is more general if it has to be used with different initial conditions.
Finally, from now, on I will (and encourage you to do the same) adopt the following definition of Boltzmann equations: equations in which a distribution of probability changes during time by the effect of a stochastic process; furnished with an initial condition.
Under this classification I will comprise all kinds of probability distribution: with continuous, discrete and even finite states. At the same time no conceptual distinction will be done (but for the way of solving) between discrete (integro-difference) and continuous time (integro-differential) problems.
However while concepts are the same, equations will look very different. You will see that integral operators will figure in processes with continuous random variables while matrices and tensors will govern the formulation of discrete-state problems.
Although this is confusing now it has the great benefit of fixing one recipe in order to write equations in all cases.
On the contrary, solving these equations will require different techniques that will range from functional analysis (which applies to integral processes that we want to treat in detail here) to tensor calculus so prepare for a big usage of mathematical tools.
2. Functional Calculus and Solutions
Eq. 9 and Eq. 10 represent a particular kind of Boltzmann equations since they deal with a very simplified model of probability transition: motion. This is the reason for the convolution product in the case of random-walk. However, it has to be emphasized that in the general case a one-variable discrete-time stochastic equation takes the form
and its corresponding continuous time equation takes the form
where, in both, K is a general integral operator only obeying the following rules:
Eq. 21
since it moves probability density which is a positive quantity, and
Eq. 22
since the integral of probability density must continue to be unity.
The reader has probably understood now that in truth stochastic processes may also take place on the real semi-axis or even on some bound intervals. This is absolutely true and in those cases expect to see integrals that extend to the corresponding intervals of definition for the process. We will soon have an example of this by inspecting a case in neutron transport theory where the variable is the modulus of speed which is defined only on the positive real axis.
So, here comes functional calculus. In my view, the biggest difficulty to solving the Boltzmann equation has been the way of writing and thinking to it.
As they are written, Eq. 19 and Eq. 20 are very far from being understood since a well-intentioned mathematician would ask (as I did my first time) first: which term is the leading one and needs so to focus attention on?
With my right hand on the Bible the answer absolutely is: the integral term or, better, the functional term. In fact, if you can manage this term the solution is yours.
Let us see how with a bit of functional calculus.
2.1 Some Concepts about Operators
Without making any long mathematical formulation I will briefly define here functionals as a sort of generalization of functions.
In mathematics we are told at first that functions are some black boxes into which we put a value and are returned a new value which is generally different from the former.
Functionals work quite the same but for the fact that in the black box we put a function and are returned something. This "something" may be either a value, or a function or even a set of functions but in the cases interesting to us it will merely be a function.
The derivation with respect to variable x is a functional since if you take a function f(x) and let the functional work on it you obtain a new function which is here known to be f’(x).
In the equations reported we have already seen functionals but in an unaware manner. The integral term, in fact, is something that takes a function and returns a new function being the new function depending on the entire shape of the "argument" function.
In truth here we have more than a simple functional since our operator takes a PD and returns a new PD without ever taking or giving negative values. Moreover this operator is easily seen to be a linear functional since, as stated by the property of integration, by letting it operate on a linear combination of functions it would return the linear combination of the single results.
These simple features already fix the way to the solution.
Let me first give an easier notation by denoting functionals in equations with hatted capital letters and their linear action on functions with the (empty circle) operatorial product. The last equation then becomes
and is perfectly equivalent to the previous but for the fact that focuses attention on operators rather than variables. Derivation as said is also a linear functional but since it is written in explicit form I will not use for it the operator notation.
Now, forget for a while that you are working with PDs and imagine that the function
F(x,t) (x,t) you are looking for has a special feature: under the action of the operator K it reproduces itself multiplied by a constant l. As many know, in mathematics such a function is called an eigenfunction for the operator K with eigenvalue l. In this case the equation is almost solved since it becomes
which is a partial differential equation of the first order whose solution technique I expect to be well-known to the reader. The general integral for this equation is
Eq. 25
By using the initial condition
Eq. 26
we finally obtain a solution for the Boltzmann equation
Eq. 27
It is easily seen however that the solution does not continue to be normalized (unless it has unit eigenvalue) to unity since there is the exponential factor. We will see, in fact, that a solution to a stochastic process is always composed of several eigenfunctions in order to maintain normalization being the eigenfunctions either a continuous, discrete or even finite-number set.
It is not much of a job to understand now that once you have decomposed your initial condition into eigenfunctions these evolve in time each apart from the others (this does not hold in the non-linear case however) and the solution is the mere recombination of the evolving members. Recombination here means both "summation" or "integration" depending on the set of eigenfunctions you have; in great analogy with quantum mechanics.
This was, in fact, the way that drove me during my "laurea thesis" and which then brought me to extend a well-known concept of the quantum world: the Feynman Propagator.
2.2 Integral Representation and Feynman Propagator
Let us consider again the equation written in operator form and let us suppose that we already know the eigenfunctions for the operator K. A common case is that in which these form a continuous set parametrized by their eigenvalue.
In mathematical form
of which a valid example is
Eq. 29
Now, if you can decompose your initial condition into a superposition of eigenfunctions on the domain of eigenvalues
you have the solution since only the coefficients of the superposition will be changed by the operator but not the eigenfunctions.
As you can see, here functions work the same as vectors in linear spaces and in truth here begins functional analysis: the vectors in spaces with infinite dimensions are functions.
In these spaces a vector (function) is no more a linear combination of those vectors (functions) that form the base because these are infinite in number and usually form a continuous set.
It happens here that linear combination is replaced by integral superposition which is also called in mathematics: integral representation.
By finding the eigenfunctions of a certain operator and decomposing the initial condition into a superposition of these you perform a diagonalization of the operator; being this concept exactly the same of its algebraic predecessor.
It is now obvious that the solution will then have the form
Put this equation into the Boltzmann equation (Eq. 23) and use the operator properties mentioned above; then group the integrals and obtain
Eq. 32
that can only be satisfied if the term in squared parenthesis is identically zero. This condition is easily satisfied by solving the resulting PDE (which is of the type of Eq. 24) and then using the initial condition for the coefficients
Eq. 33
The complete solution for the PDE is therefore
Substitute Eq. 34 into Eq. 31 and the work is done.
Not so difficult, after all, but however the mosaic is still not complete.
In fact, finding eigenfunctions and decomposing the initial condition remain open problems. These two steps consist respectively in solving an integral equation and defining an integral transform along with its inversion formula. I will report here a special case in which the eigenfunctions are easily understood and the integral transform is already known and treated on books. There is no space to go in detail on these aspects here but however something more will be cleared in the conclusions. At the same time the problem of decomposition is already solvable by those people who have experience in complex variables and integral equations.
It can be showed that q(
l) is an integral on the domain of the variable x
Eq. 35 is nothing less than the inversion formula of Eq. 30. Here I have used the prime for the variable of integration since I have to put Eq. 35 into the equations containing x and want to avoid any confusion between variables.
Now, by substituting Eq. 35 into Eq. 34 and then this into Eq. 31 we obtain the general expression for the solution
Eq. 36
that with some rearrangements is written
Eq. 37
The expression in squared parenthesis here
Eq. 38
is called the Feynman Propagator (FP) for the stochastic process in question. By adopting this notation the solution takes the form
Eq. 39
or, with operator notation,
Eq. 40
You can see that the FP is independent of the initial condition and moreover if you know the FP you have the solution for every initial condition at the price of an integral.
In fact, the FP is only dependent on the form of the equation.
All these results could have been obtained in an easier way by directly applying deeper functional calculus but when grabbing new concepts it is better to begin with the traditional notation and then switch to the new one. This limits the difficulties to the substance only.
Before going to apply results I want to make a last generalization regarding eigenfunctions which should not surprise much. The reader will probably have realized that all the developments done do not change if Eq. 28 is replaced by
Eq. 41
where the eigenvalue is a function of the parameter on which we make the decomposition. In this case the only difference is that we integrate over k instead of
l itself and substitute to l its expression in terms of k but nothing changes conceptually.Actually too many formal aspects have been treated here and too many explanations seem to be postponed; so it is time to put theory at work.
2.3 Elastic Slow-down on Light Moderator
A long-standing problem has been that of solving the Boltzmann equation for neutron slow-down by a light moderator. We want here to focus on the speed spectrum rather than the spatial distribution of neutrons so we will assume that there is homogeneity in the spatial distribution and isotropy in the speed distribution.
Moreover we will expect the collision frequency to be constant at all speeds which implies a 1/v behavior for neutron-moderator cross sections. This is not always true but my aim here is to focus on the integral properties of the equation. However this complications can be taken into account with no conceptual but only analytical complication.
Finally I assume that scattering only takes place between neutrons and moderator particles which are initially at rest.
With all these assumptions the equation takes the form
Eq. 42
where the integral kernel is
Eq. 43
and
a is a parameter between zero and one only depending on the masses while q is the Heaviside step function.It is easily seen that the functions
Eq. 44
are eigenfunctions (w is here any speed parameter and only adjusts dimensions) for the integral operator in question and their eigenvalues are
Eq. 45
which, using operators, is expressed by
Eq. 46
Notice that here the eigenfunctions were known and have not been determined by following any defined mathematical path.
You can also see that these eigenfunctions are not parametrized by the eigenvalues which poses us in the case of the generalization concluding the previous paragraph. In addition, to ease notation, I will not report in every formula the explicit expression for the eigenvalues.
Suppose we now have the initial condition
Eq. 47
and make the replacement
Eq. 48
We look now for an integral giving us the initial condition as a superposition of eigenfunctions x-k
Eq. 49
where it is obvious that the dependence on v is contained in x. Here c is a constant and its value will be given by the antitransformation process.
This has already been done in mathematics under some conditions satisfied in our case and this representation (the integral reported is the inversion formula) is called Mellin transform.
This is how we bypass also the problem, left open, regarding transforms. The domain
W is here an axis parallel (how to choose it will not be detailed but can be found on every good book about integral transforms) to the imaginary axis and the transformation formula to obtain the function m(k) is
Eq. 50
where it is now
Eq. 51
and so
Eq. 52
which gives the initial condition for the PDE. Here
G is the Euler’s function.The solution of the Boltzmann equation is then expressed as a time-varying Mellin antitransform:
where the function M(k,t) obeys the PDE for the coefficients
Eq. 54
and is solved by
To obtain a solution for the specific initial condition one should now substitute Eq. 55 into Eq. 53 and evaluate the integrals in the complex plane.
Otherwise, to obtain the expression of the FP, and have a general solution valid for all initial conditions, the expression is
Eq. 56
where (notice that w has disappeared) it is easy to isolate the FP
Eq. 57
Unfortunately by substituting now the expression for the eigenvalues one would expect to obtain expressions for the solution and the FP. Evaluating these integrals however is not possible in terms of known functions and I must warn you to expect this to be a very common event here.
We have just crossed the border of integral formulations while actual mathematics and its eigenfunctions are built on differential formulations so expect frequently to encounter new and uncommon sets of functions.
Bessel’s and Elliptic functions, Spherical Harmonics and Hermite polynomials all come from the solution of eigenvalue problems governed by differential equations so be ready for them to serve not much in problems where the main operation is integration.
3. Conclusions
It is now clear that the biggest problem in solving a linear Boltzmann equation is that of finding eigenfunctions and then performing integrals.
Suppose now that in a particular equation the integral operator can be decomposed in the sum of two operators
Eq. 58
whose eigenfunctions are known.
It is easily seen that the eigenfunctions for the sum will differ from the eigenfunctions of both its terms.
However something lets me think that in this and other cases there has to be a way to follow to build the eigenfunctions for K as some integral over the eigenfunctions K’ and K’’. This is actually a subject on which I am working and I am also exploring publications in functional analysis to see if there has already been given an answer to this problem. In any case I will go in detail on this mathematical aspect in a future article.
Similar simplifications seem to be achievable also for the case of the product of operators
Eq. 59
Another problem closely linked to this is that of building general integral transforms given a set of eigenfunctions. The reader has probably realized that Fourier, Laplace and Mellin transforms absolutely do not cover all cases so: what if there where a new set of eigenfunctions that does not fall under any of the mentioned transforms?
This problem would arise as soon as you solve the other one of determining eigenfunctions for your operator.
So these two steps should in some way be done because what happens is that although the linear Boltzmann equation is conceptually understood the mathematical machine has sometimes practical limits to solving it.
This is not such an impressive truth since also in differential equations there are cases impossible to solve but with no theoretical secrets to us.
More complex is the nonlinear case in which a PD interacts with itself. Here the difficulty is very conceptual since one should think about the operator in such equations as to be composed of an integral superposition of operators.
Actually there seems to be no way to follow in order to solve this case, at least with the exposed methods.
Finally, after all this theory, I want to focus how it applies to some phenomena that are not at all related to quantum physics or transport theory.
In fact, having a good tool for random processes is a strong need of modern societies where, if you reflect, very few things are certain.
Uncertainty is a constant factor from meaningless to crucial events in life. For example try to answer these questions in order of importance:
- At what exact time will you reach your office tomorrow?
- Will the sign of tomorrow’s Wall Street’s closing be plus or minus?
- Who will be the next President of the United States?
It is impossible to give an answer to any of these questions without assuming a certain amount of "risk".
Although actual science has developed wonderful tools to deal with deterministic processes there is a strong need for tools to deal with randomness.
The fact that an event is affected by a certain amount of randomness does not mean a stochastic approach will not give results.
Moreover, when dealing with multiple events the average result will tend to that predicted by the stochastic study.
As an example I want to introduce the reader to a first case of nonlinear random process by showing a very simplified model of human behavior in the effort of only opening a first window on the use of stochastic processes in the applied mathematical world.
3.1 Human Behavior in Decision Making
Suppose that a certain group of people is called to take a binary decision on some problem such as electing the president, voting a law or even choosing a phone company among two.
These people interact with themselves at random with a certain frequency
n by exchanging their ideas by exchanging their ideas and so modifying the relative probability
of the two possible choices of being selected.
These relative probabilities define the state vector for the stochastic system in question
Eq. 60
In more complex problems, the state vector may be done of more than two components of course. For example this happens if one wants to study the behavior of consumers in the choice of a product among different n’s which are in competition.
In that case the state vector becomes
Eq. 61
but nothing of the forthcoming theory changes except for the number of items on which to sum.
Here
F1 is defined by being the probability that a person selected at random in the group is inclined to decision 1. The same for decision 2.It is easy to understand that in every moment we will have
Eq. 62
which expresses the conservation of probability.
Now, in the act of exchanging ideas everyone has half of the chances to convince a person of the opposite idea while nothing happens in the encounter between two persons who agree on the choice, of course.
Therefore in the act of meeting between an unknown and a person oriented to decision 1 the relative probabilities for the unknown after the meeting are given by
Eq. 63
while in the meeting event of an unknown with a decision-2-oriented person the situation is
Eq. 64
This defines the two operators of decision-influence
Eq. 65
which act together at the same time in modifying the probabilities.
Since it is also unknown who an unknown person will meet the two operators will have to be weighted with the corresponding probabilities when writing the equations for the two states:
Eq. 66
Eq. 67
that already begins to sound familiar to those expert in tensor calculus. To ease notation, let us now write
Eq. 68
which defines T as being a symmetric tensor
Eq. 69
called decision-influence tensor and done up of two operators. Now, using compact tensor notation, the equations for the stochastic process can be grouped and rewritten as
Eq. 70
which is a tensor (multi-linear) equation describing the evolution of this process.
As it is easily seen, the equation is not non-linear, as it is custom to call Boltzmann equations in mathematics, but multi-linear as tensors correctly express. This is also true for nonlinear Boltzmann equations with integral dependencies where you have a number of integrations (here summations) equal to the degree of appearance of the unknown function.
Moreover, here tensor formalism is not pure ease of notation since it brings all its properties. You can change the base of representation from yes/no to any linear combination of these and the tensor equation will not change as long as you follow the appropriate transformation rules.
The equation reported, as it is easily understood, tends to make one of the two states prevail during time and exactly that one who is prevalent at the beginning. There is therefore no need to study asymptotic solutions in two-state problems. This result does not hold in multi-state problems however where there is also a need for a more precise definition of the decision-influence tensor.
The two-state problem is easy and may seem purely academic however it can be very helpful if you have to make predictions for on decisions that need qualified majorities or if you want to forecast the market share of a certain product at a certain future time.
In an uncertain world the only way of managing is to accept some non-vital errors and make the best possible forecasts.