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[EQUATION] where we have grouped together everything involving the hydrodynamic field into a single effective source term, defined by |
[EQUATION] Eq. ( 24 ) is a linear difference equation. For an infinite lattice, it has the exact solution [EQUATION] where we have defined [MATH] , and we note that [MATH] for [MATH] . The first term above describes the excitation of the kinetic field due to gradients of the hydrodynamic field. The second term is a tra... |
2.4 Self-consistent hydrodynamic difference equation Supposing, for the time being, that the kinetic field is initialized to zero , or that we have waited long enough for transient behavior to be unimportant, we can insert the first term on the right-hand side of Eq. ( 26 ) into the dynamical equation for [MATH] and re... |
[EQUATION] The remarkable thing about this result is that no essential approximations have been made in its derivation. It is an exact difference equation that must be obeyed by a hydrodynamic field [MATH] satisfying the lattice BGK equation. |
2.5 Recovery of a hydrodynamic differential equation Examination of Fig. indicates that the hydrodynamic field changes reasonably slowly in time so that a Taylor expansion in time is justified, but that it can change very rapidly in space after the onset of the shock, rendering a spatial Taylor expansion of questionabl... |
[EQUATION] This is a second-order accurate set of coupled discrete-space, continuous-time equations describing the hydrodynamics of the model. |
To proceed to a spatiotemporal partial differential equation, we may Taylor expand Eq. ( 28 ) about the spatial point [MATH] , retaining spatial derivatives to second order. Because of the symmetry of the differences, it is straightforward to see that the error incurred is of second order. The sum over [MATH] is then a... |
[EQUATION] Ignoring the terms that decay exponentially in time (note [MATH] ), we recover Eq. ( 19 ). Before going on to consider the exact analysis for more general lattice BGK equations, we offer some preliminary observations: |
The usual Chapman-Enskog procedure begins with a Taylor series expansion of the kinetic equation in difference form, and then assembles continuum hydrodynamic equations from that series. This method, by contrast, first derives exact hydrodynamic equations in difference form, and only then (optionally) Taylor expands th... |
Closely related to the above point is the idea that the hydrodynamic difference equation first obtained is not expanded in Knudsen number, and therefore expected to hold in limits that would not ordinarily be considered “hydrodynamic.” Such limits may include situations with steep spatial gradients or long mean-free pa... |
The method yields an exact hydrodynamic equation in difference form that can be Taylor expanded to yield the hydrodynamic equation. The second-order accurate nature of this expansion is manifest. |
The method works only for lattice BGK equations, and relies on the fact that the equilibrium distribution function is a function of the conserved quantities only. (If this were not the case, Eq. ( 24 ) would not be linear in the kinetic field, so we would not be able to solve it exactly.) |
Note that as [MATH] , the effect of the last two terms on the right-hand side of Eq. ( 28 ) is to simply alter the coefficients in front of – or “renormalize” – the other terms in the equation. The penultimate term evaluates to [MATH] so it renormalizes the diffusion term, and the last term is [MATH] so it renormalizes... |
[EQUATION] from which Eq. ( 19 ) follows. General procedure 3.1 Projection operators We now suppose that we have a [MATH] -component discrete-velocity distribution function [MATH] , where [MATH] , where [MATH] is a point on lattice [MATH] , and where [MATH] is the time. The [MATH] hydrodynamic variables are obtained by... |
[EQUATION] The rows of [MATH] and [MATH] are linearly independent, so that knowledge of all the hydrodynamic and kinetic variables is sufficient to reconstruct the distribution function, |
[EQUATION] It is easily verified that the [MATH] matrix [MATH] and the [MATH] matrix [MATH] obey the relations [EQUATION] and [EQUATION] |
In the above presentation, we have adopted the convention of using Greek letters from the beginning of the alphabet ( [MATH] ) to label the hydrodynamic variables, from the middle of the alphabet ( [MATH] ) to label the kinetic variables, and Latin letters ( [MATH] ) to label the distribution function components. We sh... |
Example 1 A triangular grid in two spatial dimensions ( [MATH] ) has the [MATH] lattice vectors [EQUATION] In the widely adopted nomenclature for lattice Boltzmann models, this is called the D2Q6 model . If mass and momentum are conserved, the hydrodynamic and kinetic projection operators may be taken to be |
[EQUATION] and [EQUATION] respectively. Note that the first row of [MATH] corresponds to the mass density [MATH] , while the second and third rows correspond to the momentum density [MATH] . Collectively, we refer to the conserved densities as [MATH] . The corresponding reconstruction matrices are then |
[EQUATION] The identities of Eqs. ( 34 ) through ( 36 ) are then readily verified. 3.2 General form of the lattice BGK equation The general form of the lattice BGK equation, Eq. ( ), may be rewritten |
[EQUATION] where [MATH] is defined as before. Note that the equilibrium distribution function [MATH] is allowed to depend only on the hydrodynamic moments [MATH] . In what follows, it shall prove useful to write the equilibrium distribution in the form |
[EQUATION] Comparing this with Eq. ( 33 ), we see that [MATH] is the kinetic portion of the equilibrium distribution function. Example 2 |
For a fluid with [MATH] , the form generally used for this dependence is the Mach-expanded distribution [EQUATION] where the [MATH] are weights associated with each direction, [MATH] is the rank-two unit tensor and [MATH] is the sound speed defined by the isotropy requirement |
[EQUATION] In particular, for the D2Q6 lattice of Example , it may be verified that [MATH] and [MATH] . The first two terms of Eq. ( 43 ) are then the hydrodynamic portion of the equilibrium distribution function, and |
[EQUATION] is the kinetic portion. By means of the projection operators defined in the previous subsection, it is straightforward to decompose this into coupled evolution equations for the hydrodynamic and kinetic moments, |
[EQUATION] respectively. Eqs. ( 46 ) and ( 47 ) are the generalizations of Eqs. ( 22 ) and ( 23 ), respectively. The equation for the kinetic modes may be written more succinctly as |
[EQUATION] where we have defined [EQUATION] Eqs. ( 48 ) and ( 49 ) are the generalizations of Eqs. ( 24 ) and ( 25 ), respectively. |
3.3 Exact analysis in the general case To proceed as in the example, it is now necessary to find an exact solution to the linear equation Eq. ( 48 ) for the kinetic modes, assuming that the hydrodynamic modes are known. |
Consider a labeled path of [MATH] steps along lattice vectors, whose vertices are labeled by kinetic modes. More specifically, consider a path along the lattice beginning at position [MATH] and mode [MATH] at time [MATH] , and ending at position [MATH] and mode [MATH] at time [MATH] . One such path in the D2Q6 model is... |
A path [MATH] is thus characterized by its sequence of indices [MATH] of the lattice vectors traversed, and also the sequence of kinetic modes [MATH] at the visited vertices. Note that it must be true that |
[EQUATION] and [EQUATION] We use [MATH] to denote the sum over all paths, [MATH] Example 3 A path of length [MATH] in the D2Q6 model of Example is shown in Fig. . The sequence of lattice vector indices pictured is [MATH] |
To a path [MATH] we assign the weight [EQUATION] where there is no understood summation on repeated indices. The exact solution to Eq. ( 48 ) is then |
[EQUATION] Note that if [MATH] can not be connected to [MATH] by a sequence of [MATH] lattice vectors, perhaps because it is too far away, then [MATH] is understood to be the null set. |
Example 4 Eq. ( 54 ) is the generalization of Eq. ( 26 ). To see this, note that our [MATH] example for Burgers’ equation had [MATH] and [MATH] . The projection operators were |
[EQUATION] and [EQUATION] from which it follows that [MATH] . The number of paths connecting [MATH] and [MATH] is then the binomial coefficient [MATH] , resulting in Eq. ( 26 ). |
3.4 Self-consistent hydrodynamic difference equations As with our one-dimensional example, we suppose that the kinetic field is initialized to zero, and we insert Eq. ( 54 ) into Eq. ( 46 ) and rearrange to obtain the exact hydrodynamic difference equation |
[EQUATION] Eq. ( 65 ) is an exact nonlinear hydrodynamic difference equation for the conserved densities, albeit in terms of a diagrammatic summation. We are now free to Taylor expand in either time or space, as appropriate to the phenomenon under consideration. Taylor expansion of Eq. ( 65 ) to second order in the tim... |
[EQUATION] 3.5 Recovery of a hydrodynamic differential equation Finally, to proceed to a hydrodynamic differential equation, we may Taylor expand in space, retaining terms to second order. While this step is not technically difficult, it is tedious enough to warrant splitting the calculation into three parts, correspon... |
3.5.1 Some useful tensors In preparation for the forthcoming analysis, it is useful to define the tensors [EQUATION] where [MATH] denotes an [MATH] -fold outer product. If all of the conserved quantities are scalars, these tensors have [MATH] spatial indices ranging from [MATH] to [MATH] in which they are completely sy... |
When it becomes necessary to refer to these tensors by their spatial indices, we adopt the convention of replacing the number [MATH] in parentheses by the actual list of [MATH] spatial indices. Thus, for example, we have |
[EQUATION] where [MATH] is the [MATH] th spatial component of lattice vector [MATH] . Note that the value of [MATH] , namely [MATH] in this case, can be inferred by the fact that there are three indices in the list. When listing components for [MATH] , we denote the empty list by [MATH] |
If one or more of the conserved quantities are spatial vectors, such as momentum, it is convenient to abuse notation by allowing the hydrodynamic indices to be either scalars or vectors. Each one that is a vector will increase the spatial rank of the tensors by one. For example, if [MATH] is the index of a conserved sc... |
To specify components of these, it may be necessary to use nested subscripts. Thus, we have that [EQUATION] is the [MATH] component of a tensor of spatial rank three that is symmetric under interchange of its rightmost two indices. Note that the notation automatically ensures that the [MATH] symmetric indices will be t... |
Finally, we also define alternative versions of the [MATH] [MATH] and [MATH] tensors, using the same names for economy of notation, but with upper hydrodynamic and lower kinetic indices, |
[EQUATION] All of the above-mentioned considerations about counting independent components likewise apply to these alternative versions. |
Example 5 For the D2Q6 model in Example , supposing that mass and momentum are both conserved, the matrices [MATH] [MATH] [MATH] and [MATH] are given in Eqs. ( 38 ) through ( 40 ). If we denote the hydrodynamic indices by [MATH] , it is straightforward to compute the above tensors. Tabulations of the results for [MATH]... |
To assemble the hydrodynamic equations from Eq. ( 66 ), we label the terms on its right-hand side as ①, ② and ③, and consider each separately. |
3.5.2 Evaluation of first term in Eq. ( 66 The first term on the right of Eq. ( 66 ) can be written [EQUATION] In Appendix , we evaluate this expression for the D2Q6 model. When doing so, we adopt incompressible scaling |
, wherein spatial gradients and Mach number are taken to be order [MATH] , and time derivatives and density fluctuations are taken to be order [MATH] . To leading order, we find that the [MATH] and [MATH] components of the above term are |
[EQUATION] 3.5.3 Evaluation of second term in Eq. ( 66 Similar considerations can be applied to the second term on the right of Eq. ( 66 ) which may be written |
[EQUATION] or [EQUATION] In Appendix , we evaluate this for the D2Q6 model in the limit of incompressible scaling . To leading order, we find that the [MATH] and [MATH] components of the above term are |
[EQUATION] 3.5.4 Evaluation of third term in Eq. ( 66 The third term on the right-hand side of Eq. ( 66 ) involves [MATH] which is given by Eq. ( 49 ), so it is necessary to compute this first. We note that it may be expanded to second-order accuracy to obtain |
[EQUATION] In Appendix , we evaluate this for the D2Q6 model in the limit of incompressible scaling . To second order, we find that its [MATH] and [MATH] components are |
[EQUATION] As noted for the example of Burgers’ equation, the effect of the diagrammatic sum as [MATH] and in the continuum limit is to apply a linear operator to these terms, thereby renormalizing other terms in the equation. The most general form for the third term in Eq. ( 66 ) is then |
[EQUATION] where [MATH] [MATH] and [MATH] are determined by the diagrammatic series. The results are [EQUATION] and [EQUATION] so the third term of Eq. ( 66 ) has no [MATH] component, |
[EQUATION] and [MATH] component equal to [EQUATION] 3.5.5 Assembly of hydrodynamic equations Armed with these terms, we may now assemble the full hydrodynamic equations, |
[EQUATION] Since [MATH] may be ignored in the incompressible limit, the first of these reduces to [EQUATION] while the second gives |
[EQUATION] This simplifies to yield [EQUATION] where the pressure [MATH] is given by the equation of state [EQUATION] and the kinematic viscosity is |
[EQUATION] Eqs. ( 94 ) and ( 96 ) are seen to be the Navier-Stokes equations of viscous, incompressible hydrodynamics. The expressions for the equation of state and the viscosity are well known for the D2Q6 fluid |
Discussion Diagrammatic methods often yield new physical insights, and this one is no exception. Since the [MATH] are the driving terms in the linear equations for the kinetic modes [MATH] , we may think of the [MATH] as the precise combination of hydrodynamic gradients that excite kinetic modes. Thus excited, the kine... |
It is remarkable that the effect of these kinetic excitations is often nothing more than the renormalization of terms already present in the hydrodynamic equation. The diagrammatic sum is necessary to obtain the correct answer for the transport coefficients (advection, diffusion and viscosity in the above examples), bu... |
Conclusions We have described a new method to derive hydrodynamic equations for lattice BGK fluids. It is qualitatively different from the usual Chapman-Enskog analysis, and superior insofar as it results in absolutely exact hydrodynamic equations. Additionally, while more demanding in terms of calculation, the method ... |
We have illustrated this new methodology, first by presenting a simple lattice-BGK model for Burgers’ equation, and second by presenting a more complex lattice-BGK model for a viscous, incompressible fluid. The final step in this method is the extraction of a diagrammatic sum, but we have shown that all that we really ... |
Because the method relegates the Taylor expansion of the propagation operator to an optional step at the end of the analysis, it is possible to derive hydrodynamic equations that are either discrete or continuous in space and/or time. For the examples presented in this work, we showed the result of expanding in time bu... |
Future work will take up the application of this method to complex fluid phenomena, such as multiphase fluids described by the Shan-Chen model |
. It is hoped that this method will yield accurate hydrodynamic equations in the spirit of Halperin and Hohenberg’s Model H for such complex fluids. |
Acknowledgments This work was partially funded by ARO award number W911NF-04-1-0334, AFOSR award number FA9550410176, and facilitated by scientific visualization equipment funded by NSF award number 0619447. The lattice BGK model for Burgers’ equation was worked out in preparation for a presentation given at the Consig... |
Appendix A Appendix: Evaluation of Eq. ( 75 ) for D2Q6 fluid For a mass and momentum-conserving fluid, we wish to compute the [MATH] and [MATH] components of Eq. ( 75 ). The [MATH] component may be written |
[EQUATION] Using the results for the D2Q6 model compiled in Tables and , this becomes [EQUATION] When incompressible scaling is taken into account, the second term is order [MATH] times the first one, so we may write the result, |
[EQUATION] The [MATH] component may be written [EQUATION] Using the results for the D2Q6 model compiled in Tables and , this becomes |
[EQUATION] When incompressible scaling is taken into account, the second term is order [MATH] times the first one, so we may write the result, |
[EQUATION] Appendix B Appendix: Evaluation of Eq. ( 79 ) for D2Q6 fluid For a mass and momentum-conserving fluid, we wish to compute the [MATH] and [MATH] components of Eq. ( 79 ). The [MATH] component may be written |
[EQUATION] Using the results for the D2Q6 model compiled in Tables and , this becomes [EQUATION] All of these terms are negligible when incompressible scaling is applied, so we have |
[EQUATION] Applying incompressible scaling, we are left with the dominant terms [EQUATION] Appendix C Appendix: Evaluation of [MATH] for D2Q6 fluid |
For a mass and momentum-conserving fluid, we wish to compute the [MATH] and [MATH] components of [MATH] , given in Eq. ( 82 ). The [MATH] component may be written |
[EQUATION] Using the results for the D2Q6 model compiled in Tables and , this becomes [EQUATION] Applying incompressible scaling, we are left with the dominant term |
[EQUATION] where we have adopted incompressible scaling in the final step. The [MATH] component may be written [EQUATION] Using the results for the D2Q6 model compiled in Tables and , this becomes [EQUATION] Applying incompressible scaling, we are left with the dominant terms [EQUATION] |
# Source: arxiv 0810.4738 # Title: A Game Theoretical Perspective on the Somatic Evolution of Cancer # Sections: all # Downloaded: 2026-03-03T01:59:27.904136+00:00 |
A game theoretical perspective on the somatic evolution of cancer 0.1 Introduction Environmental and genetic mutations can transform the cells in a co-operating healthy tissue into an ecosystem of individualistic tumour cells that compete for space and resources nowell:1976 Crespi:2005 Merlo:2006 . If we consider a tum... |
Game theory (GT) was introduced by von Neumann and Morgenstern as an instrument to study human behaviour Neumann:1953 Nowak:2006 . A game describes the interactions of two or more players that follow two or more well defined strategies in which the benefit of each player (payoff) results from these interactions Merston... |
To illustrate some of the ideas in EGT let us consider the following example named the Hawk-Dove game Maynard:1982 . In this game we study an imaginary population of individuals and a resource V which affects the reproductive success of the individuals in this population. The population contains two phenotypes that rep... |
Table presents the interactions between the different phenotypes considered in the game. The table should be read following the columns such that the payoff for a Hawk playing another Hawk is [MATH] expressing the fact that they both have to share the resource and that they stand an equal chance of getting injured. The... |
GT has been used to address many problems in biology in which different species or phenotypes within one species compete. Examples of this are the evolution of sex ratios Fisher:1930 , the emergence of animal communication Smith:2003 and fighting behaviour and territoriality Maynard:1982 . A recent focus on the capabil... |
The remaining of this chapter will provide, to the best of our knowledge, all the relevant examples of the application of GT to the study of the somatic cancer evolution and finally hint some of the possible future venues of this method in the context of cancer research. |
0.2 Tumourigenesis Tumour initiation requires the acquisition of a number of phenotypic capabilities such as evasion of apoptosis and independence from environmental signals (see figure ). The evolution of these capabilities, normally acquired when the tumour is still in the avascular stage, are studied in research by ... |
0.2.1 Evasion of apoptosis Problem . Apoptosis or programmed cell death is a mechanism that hinders tumour progression. Cells with a working apoptotic machinery die when genetic abnormalities are detected hanahan:2000 . Thus cells in a malignant cancer have to evolve mechanisms to disable the apoptotic machinery. |
Model . Tomlinson and Bodmer tomlinson:1997b present a model in which three different apoptosis evasion related strategies are considered: |
1. Cells that produce a paracrine growth factor to prevent apoptosis of neighbouring cells. 2. Cells that produce an autocrine growth factor to prevent apoptosis of themselves. |
3. Cells susceptible to paracrine growth factors but incapable of production of factors. The aim of the model is to study the possibility of stable coexistence of the different phenotypes (polymorphism) that could be possible in a tumour when only these three phenotypes are considered. |
In table is the cost of producing the paracrine factor, the benefit of receiving the paracrine factor and the benefit of producing the autocrine factor. |
Results . If is positive then the third strategy displaces the first one from the population. If only the other two strategies are considered then if the benefit of the autocrine factor, provided by , is positive the second strategy will displace the third one. In most relevant situations the model shows a strong selec... |
Remarks . The model is very simple and easy to understand while at the same time it captures the relevant features necessary to study the evolution of the mechanisms to avoid apoptosis. However the authors do not explain the mechanisms by which these phenotypes could appear in a tumour and what would be the biological ... |
0.2.2 Environmental poisoning Problem . Tomlison introduced a further model in which he considers the hypothesis that tumour cells might boost their own replicative potential at the expense of other tumour cells by evolving the capability of producing cytotoxic substances tomlinson:1997a |
Model . Tomlinson speculates with different strategies that cells may adopt to produce or cope with toxic factors. The main model aims to study the polymorphic equilibria when cells can adopt one of the three following strategies: |
1. Cells producing cytotoxic substances against other cells, 2. cells producing resistance to external cytotoxic substances, and |
3. cells producing neither cytotoxins nor resistance. Table shows the payoff table of the game with these phenotypes. Results . Game theoretical analysis and simulations show that production of cytotoxic substances against other tumour cells can evolve in a tumour population and that several cytotoxin related strategie... |
Remarks . Although the author admits that there is little experimental evidence for mutations that cause tumour cells to harm their neighbours, the Warburg effect could fit nicely in the framework presented in this work. The Warburg effect describes the switch of tumour cells from the conventional aerobic metabolism to... |
0.3 Angiogenesis Problem . A very important capability that has to be acquired by tumours on the path to cancer is angiogenesis. Without access to the circulatory system tumours do not grow to sizes bigger than 2mm in diameter Folkman:1992 . Cells capable of angiogenesis produce growth factors that promote the creation... |
Model I . In their interpretation of an angiogenic game Tomlinson and Bodmer tomlinson:1997b consider two strategies: cells denoted as A+ can produce angiogenic factors at a fitness cost and cells denoted as A- that produce no angiogenic factors. In any case cells will get a benefit when there is an interaction involvi... |
Results . The model shows that as long as the benefit j of angiogenesis is greater than the cost i of producing angiogenic factors then both types of strategies will be present in a tumour in proportion to these costs. |
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