Dynamic symmetry-breaking in mutually annihilating fluids with selective interfaces
DDynamic symmetry-breaking in mutually annihilating fluids withselective interfaces
Sauro Succi ∗1,2,3 , Andrea Montessori , and Giacomo Falcucci Center for Life Nano Science@La Sapienza, Istituto Italiano di Tecnologia, 00161 Roma, Italy Istituto per le Applicazioni del Calcolo CNR, via dei Taurini 19, Rome, Italy Institute for Applied Computational Science, Harvard John A. Paulson School ofEngineering And Applied Sciences, Cambridge, MA 02138, United States Department of Enterprise Engineering “Mario Lucertin”,University of Rome Tor Vergata,Via del Politecnico 1, 00133 Rome, ItalyJune 22, 2020
Abstract
The selective entrapment of mutually annihilating species within a phase-changing carrier fluid is exploredby both analytical and numerical means. The model takes full account of the dynamic heterogeneity whicharises as a result of the coupling between hydrodynamic transport, dynamic phase-transitions and chemicalreactions between the participating species, in the presence of a selective droplet interface. Special attentionis paid to the dynamic symmetry breaking between the mass of the two species entrapped within theexpanding droplet as a function of time. It is found that selective sources are much more effective symmetrybreakers than selective diffusion. The present study may be of interest for a broad variety of advection-diffusion-reaction phenomena with selective fluid interfaces.
The spatial dynamics of mutually annihilating species is a subject of wide interdisciplinary concern, with manyapplications in chemistry, condensed matter, material science and even cosmology, with the famous problemof baryogenesis , namely the large asymmetry between matter and antimatter observed in the current Universe[7, 2].A pioneering investigation by Toussaint and Wilczek [20], pointed out that the time asymptotic behaviourof the mutually annihilating species ( A and B for convenience) crucially depends on the initial conditions.The rationale is quite intuitive: if the species mix, they react and both disappear according to the irreversiblereaction A + B → P , where P denotes a set of product species which contains neither A nor B . If, on theother hand, by some transport mechanism, they manage to demix or segregate apart, so that the product oftheir concentrations becomes vanishingly small, then annihilation is quenched, thus spawning a chance for bothspecies to survive much longer than under homogeneous mixing conditions.Besides being of great interest on their own right, the details of such survival may have plenty of applicationsin chemistry, material science or biology, an example in point being the absorption of drugs within liquid dropletsfor microfluidics and drug-delivery applications [21, 4, 10, 13].In this paper we consider a specific mechanism of segregation associated with the growth of droplets within aphase-changing carrier fluid. By postulating a selective transport of the two species across the droplet interface(membrane), we introduce a symmetry-breaking mechanism which is ultimately responsible for the differentialentrapment of the facilitated species (the one with higher transmissivity across the membrane, say A ) withrespect to the inhibited one (say B ). The practical question is: how much mass of both species is entrapped inthe growing and moving droplet as a function of time? Once again, this is interesting per-se as a fundamentaltransport problem in dynamically heterogeneous media, and also for the aforementioned practical purposes. Tothe best of our knowledge, no detailed account of the hydrodynamic complexity associated with a moving andexpanding droplet, in the presence of transport and chemical reaction, has ever been discussed. This is preciselythe aim of the present work, with prospective focus on electroweak baryogenesis. ∗ Electronic address: [email protected],[email protected] ; Corresponding author a r X i v : . [ phy s i c s . c o m p - ph ] J un The transport model
We consider three species A , B and C , where C is a fluid carrier undergoing phase-changes, while A and B arepassively transported by C and mutually annihilate through chemical reactions.The three species k = 1 , , ∂ t ρ k + ∂ a ( ρ k u k,a ) = r k + s k (1)where a = x, y, z runs over spatial dimensions and obeys Einstein’s summation rule.In the above r k and s k denote the density change rate due to chemical reactions and generic sources,respectively.Species A and B share the same mass, which we set to unity by convention, m A = m B = 1, so that thenumber and the mass of the species are the same quantity.The two species annihilate at the following rate: r A = r B = − αρ A ρ B (2)where α is an adjustable reaction parameter.The C field serves as a carrier for the A and B species and obeys a non-ideal Navier-Stokes equation: ∂ t ( ρ C u a ) + ∂ b P C,ab = s C (3)where P C,ab = ρ C u a u b + p C δ ab − σ ab + χ∂ a ρ C ∂ b ρ C (4)is the non-ideal momentum flux tensor, including the contributions of inertia, ideal and non-ideal pressure,dissipation and the capillary forces responsible for the first-order phase transition. The term s C represents anexternal source of mass.Species A and B are passively transported by the C -field and diffuse across it with diffusivity coefficients D A and D B respectively.The A and B species experience a selective permeability of the droplet interface, so that an excess of A over B accumulates around the interface and further penetrates within the expanding droplet.The actual amount of mass engulfed within the droplet resulting from such complex transport process ishighly sensitive to the chemical details, as well as to the hydrodynamic evolution of the system.Our main aim is to investigate the complex transport phenomena which result from the dynamic competitionbetween advection, diffusion and reaction processes taking place in the framework of a phase-changing carrierfluid.In our stylized model, microscopic symmetry breaking between species A and B is accounted for by twomechanisms: i) different values of the diffusivities and ii) different source terms, for species A and B , respectively. The diffusion coefficients are taken in the form: D k ( ρ ) = D ink θ ( ρ − ρ m ) + D outk θ ( ρ m − ρ ) , k = A, B (5)where ρ m = ( ρ l + ρ v ) / θ ( x )is the Heavyside step function. Smoother versions can easily be implemented, but in this work we shall staywith the discontinuous model.The AB symmetry-breaking processes are accounted for by choosing a different ratio between the inner(within the droplet) and outer (outside the droplet) diffusion coefficients of the A and B species, namely, J A = J B , where we have defined the in-out diffusion jump factors as: J k ≡ D ink D outk , k = A, B (6)For convenience, we set the same outer diffusivity for species A and B , namely: D outA = D outB = D (7)In particular, we note that J <
1, i.e. smaller diffusivity inside the droplet than outside, implies a net fluxtowards the interface, due to the diffusive velocity ~u D = −∇ D . This is consistent with the fact that diffusivityis supposed to decrease at increasing carrier density. 2 .2 Selective sources We shall consider the following source terms s A ( x, y ) = s W ( ~r − ~r s ) (8) s B ( x, y ) = ζs W ( ~r − ~r s ) (9)where W ( ~x ) is a piece-wise constant centred around the droplet interface, and 0 ≤ ζ ≤ The transport equations described above are solved by means of a three-species lattice Boltzmann (LB) scheme[19, 1, 8, 5] (see Appendix). The main reason for using LB is its ability of dealing with dynamic phase-transitionsin a much handier way than solving the Navier-Stokes equations of non-ideal fluids.In the sequel, we introduce the simulation set-up and present numerical results for both the selective scenariosdescribed in the previous section.We consider a two-dimensional square with L = 1024 grid-points per side and run the simulations over atimespan of 10 time-steps, thus covering three decades in space and six in time, as it is appropriate for diffusivephenomena. Both species are initialised at the same constant density value throughout the computational domain: ρ A ( x, y ) = ρ B ( x, y ) = ρ = 1 (10)The initial density of the carrier field is defined as follows: ρ C ( x, y ) = ρ in (1 ± η ) , in a ball of radius R (11) ρ C ( x, y ) = ρ out , outside the ball (12)where η is a zero-mean random perturbation with rms δρ/ρ = 0 . ρ = 1, ρ in = 0 . ρ out = 0 .
10 and R = 10.For the phase transition, we choose T /T c = 0 . / . . / .
027 = 10.The values of ρ out and ρ in determine the duration of the growth stage, i.e. the time it takes for the dropletto attain the coexistence values of the liquid ( l ) and vapour ( v ) phases.By mass conservation: ρ in V in + ρ out V out = ρ l V l + ρ v V v = M (13)where V in + V out = V l + V v = V = L is the total volume of the system in two dimensions.Clearly, the volume of the droplet grows at increasing the total mass in the system. More specifically, thefinal value of the volume fraction, i.e. the ratio of the volume of the liquid droplet to the total volume, is givenby: λ ≡ V l V = ρ m − ρ v ρ l − ρ v (14)where ρ m = M/V is the average mass density.The diameter of the liquid droplet is thus given by: D = L r π λ (15)Hence, the maximum droplet diameter, D = L , is attained at the a volume fraction λ max = π/ ∼ . ~u k ( x, y ) = 0 , k = A, B, C (16)Full periodicity is assumed across the four boundaries of the simulation box.3 .2 Chemical rates and diffusivities
Collisional time-scales are fixed at τ A = τ B = τ C = 1, which is the fastest timescale in action. This correspondsto an outer diffusivity and carrier viscosity, D A = D B = ν C = 1 / α = 0 .
1, corresponding to an annihilation timescale τ a = 10 at unit density ρ k = 1.The diffusion timescale across the membrane is τ d = w /D , w being the width of the droplet interface.Given that in LB simulations w ∼ Da ≡ τ d /τ a ∼
15. This means, at unit density, annihilation is about 15 times faster than thediffusive time scale, which also means that annihilation is effective within the droplet interface. At densitiesbelow 1 /
15 the two time scales become comparable, and the interface becomes chemically transparent.
To gain perspective, it is of interest to analyse the homogeneous case, which proves amenable to some analyticalconsiderations. In the homogeneous-symmetric scenario D A = D B = 0, no-phase transitions, no sources andsymmetric initial conditions, both species decay according to the nonlinear homogeneous equation: dρ k dt = − αρ k (17)whose analytical solution reads as follows: ρ k ( t ) = ρ k α ρ k t , k = A, B (18)This yields a τ k /t decay, where τ k ≡ α ρ k , k = A, B (19)is the density-dependent annihilation time-scale.In the presence of a symmetry-breaking membrane, the two species are expected to develop different valuesof τ k , which we refer as to a dynamic symmetry breaking , due to the effect of the selective interface on thespecies density. Such dynamic symmetry breaking is expected to occur as soon as the droplet starts to grow,i.e. it starts to nucleate out of its initial seed of radius R . Both species A and B begin to be entrapped withinthe nucleating droplet and their mass within the droplet grows accordingly, as long as the droplet growth rateexceeds their annihilation rate.As we shall see, such growth is far from monotonic, but characterized instead by large fluctuations, due tothe carrier density waves radiating away from the expanding droplet. Such oscillations do not settle down untilthe droplet condensation has come to an end, i.e at t ∼ τ con ≡ R/ ˙ R , where τ con defines the condensation timeof the droplet, i.e. the time it takes for its mass to reach steady-state.In the long-term, namely at t (cid:29) τ con , the densities of the two species are expected to settle to constantvalues inside and outside the droplet, thus leading to the coexistence of two homogeneous compartments: thedroplet and its surrounding environment.Since the droplet is homogeneous, the mass of the entrapped species is expected to follow again a τ k /t decay,with two different values of τ A and τ B , due to the aforementioned dynamic symmetry breaking.In the sequel, we shall put these qualitative considerations on quantitative grounds based on the result ofextensive numerical simulations. We consider the source-free case s A = s B = 0 and define a quantitative symmetry-breaking indicator in theform of the diffusivity jump factor, J AB = J B /J A .We run three representative cases: J AB = 0 . , . ,
1, the latter denoting the unbroken, symmetric case.For the parameters in point, namely R = 10, ρ in = 0 . ρ out = 0 .
1, the total initial mass of the carrier fluidis M C (0) = 0 . π + 0 . − π ) ∼ , hence the initial carrier density is ρ m ∼ .
1. With ρ l = 0 . ρ v = 0 . λ ∼ .
61, which gives a final droplet volumeof about 6 · lattice units, corresponding to a diameter D ∼
900 lattice units, pretty close to the maximumvalue that can be attained on a lattice of side L = 1024 lattice units.The initial value of the masses is M A (0) = M B (0) = 1024 ∼ , of which only 100 π ∼
314 lies inside theinitial seed droplet.For the present parameters, the droplet is found to reach its final size, D ∼
900 lattice units, after about τ con ∼ . steps, corresponding to an average growth rate ˙ R ∼ ( R f − R ) /τ con ∼ (450 − / . ∼ . c s = 1 / √
3, both in lattice units. This implies that the during thegrowth stage, the expanding droplet emanates trains of density waves radiating away from it. As we shall see,such density waves are well visible in the simulations.In Fig.3 we report the time evolution of the mass A , B and C within the droplet, as well as the order(phase-field) parameter: φ AB ≡ M A − M B M A + M B , (20)which provides a direct measure of symmetry breaking. Indeed, by definition, φ AB = 0 under symmetricconditions, while φ AB = ± A ( B ) components, respectively.As one can see, after a short-term transient, in which both species decrease due to annihilation, the A and B masses start to increase, due to the entrapment within the growing droplet. At about t ∼ , large oscillationsstart to take place, due to the radiation of density(pressure) waves from the growing droplet, which lead to localcondensation and subsequent evaporation of annular rings around the droplet. These rings are well visible inthe three snapshots of the carrier density contours, as reported in the lower insets of panel (a,b), correspondingto three distinct time instants.It is interesting to notice that in the regime of wild oscillations, the species B eventually exceeds species A ,which we tentatively interpret as a dynamic effect of the presence of the rings.Panel b) reports the long-term evolution of the three masses A , B , C , in the time frame 10 < t < . Themain result is that the facilitated species A prevails over B , but only by a comparatively small amount. Indeed,the largest value attained by the order parameter was 0 .
27 for the case J AB = 0 .
01. Once the droplet settlesdown, both species start to decay according to the homogeneous rate 1 /t , although with a slightly differentamplitude, due to the dynamic symmetric breaking which occurred in the condensation stage.The density contours of species C highlight that the equilibrium spherical shape is reached by the dropletafter a very long time-span. Notwithstanding the major shape changes, C mass remains constant, and this issufficient for the homogeneous decay 1 /t to settle down, long before the droplet attains mechanical equilibrium.Finite-size effects are also visible, through the reflection of density waves at the boundary. Indeed, since wework at pretty large values of geometric confinement, D/L ∼ .
9, such boundary effects are inevitable.In all the considered cases, symmetry breaking remains comparatively small at all times, notwithstandingthe large values of the jump coefficients used in the simulations.
In this section, we investigate the effects of selective source terms for the species A and B , by changing theasymmetry source coefficient in the range 0 . ≤ ζ ≤ D A = 1 , D B = 1 and s = 10 − .In Figure 3, we present the time evolution of the mass of species A , B and C within the droplet, for thecase ζ = 0 . A and B masses is observed.As clearly shown in panel 3(b), the main difference is the neat separation in the long run, due to the factthat any symmetry breaking of the source terms leads a secular growth of the facilitated species, M A ( t ) ∼ a t ,versus a homogeneous b/t decay of the unfacilitated one, as we shall discuss shortly.As a result, the mass ratio goes to zero like t − . 5igure 2: Time evolution of the masses of species A , B and C inside the droplet, short term (a) and long term(b) for the cases J AB = 0 .
01 and J AB = 0 . /t .In panel (c) we report the long term evolution of the order parameter φ ab for the cases J AB = 0 .
01 (top) and J AB = 0 . C ,while the density contours of species A and B are reported in panel (d).6imilarly to the case of selective diffusion, the time asymptotic behaviour sets in long before the densityconfiguration in space reaches its mechanical equilibrium, the chief condition being that the mass of the dropletbe stationary in time, regardless of its shape.Since the majority species follows a linear trend M A ( t ) ∼ t , while the minority one obeys a reciprocal trend M B ( t ) ∼ /t , their product remains basically asymptotically constant in time, which is indeed confirmed by thenumerical results.As we shall show in the next section, these results can be interpreted in terms of analytical solutions of thehomogeneous driven case. The equations of the mass evolution within the droplet for the homogeneous (no diffusive fluxes) driven systemread as follows: ˙ M A = − αR AB + S A (21)˙ M B = − αR AB + S B (22)where we have defined the global density overlap R AB = ´ ρ A ( x, y ) ρ B ( x, y ) dxdy , and the global mass inputsper unit time, S k = ´ s k ( x, y ) dxdy , k = A, B .Subtracting the two equations (21), delivers:˙ M A − ˙ M B = S A − S B , (23)which shows that the mass deficit M A − M B grows linearly in time.By multiplying the first by M B , the second by M A and summing them up, we obtain ddt ( M A M B ) = − αR AB ( M A + M B ) + S B M A + S A M B (24)Assuming M B to decay asymptotically to zero, the right hand side is made zero by imposing R AB ∼ S B α (25)Next, we write R AB = ξ AB M A M B V C , which defines ξ AB as the spatial correlation coefficient, V C being thedroplet volume.Further expressing the density as a volume average plus a spatial fluctuation, ρ k = M k /V C + ˜ ρ k , andassuming weak spatial fluctuations, ˜ ρ k V C /M (cid:28)
1, the correlation coefficient is made 1. Thus, the expression(25) finally yields: M A M B = S B V C α = Const. (26)As anticipated earlier on, this is indeed found to be consistent with the numerical observations.Summarizing, the time-asymptotic behaviour of the engulfed masses is given by: M A ( t ) ∼ ( S A − S B ) t (27)and M B ( t ) ∼ S B V C αM A ( t ) . (28)One can solve explicitly also for the non-asymptotic regime, to obtain: M A ( t ) = s V s (1 − ζ ) t p τ /t + 1) (29) M B ( t ) = s V s (1 − ζ ) t p τ /t −
1) (30)where V s = πDw is the shell volume around the interface.In the above, we have set τ = τ s ζ / − ζ (31)with τ s ≡ / ( αs ). For the current simulations, τ s = 10 , fairly close to the droplet equilibration time, τ con ∼ . . 7igure 3: Short (a) and long (b) time evolution of the masses of species A , B and C inside the droplet forthe case of selective sources with ζ = 0 .
99. Panel (c) reports the order parameter φ AB as a function of time forthe cases ζ = 0 .
99 (bottom) and ζ = 0 .
999 (top). Panel (d) reports four snapshots of the density contours ofspecies A (left) and B (right) for the case ζ = 0 .
99. 8he upshot of the above analysis is that the majority species grows asymptotically like (1 − ζ ) t and theminority species decreases like ζ/t . As a result, the mass ratio M A /M B grows quadratically unbounded in time.This is potentially far reaching, since it means that even a minuscule asymmetry in the sources is destinedto give rise to the extinction of the minority species on a times scale proportional to ζ / / (1 − ζ ), which clearlydiverges in the symmetric limit ζ →
1. In this limit, M A = M B = q S B V C α .A similar treatment goes for the non-homogeneous case, provided the source terms are augmented with thecorresponding diffusive fluxes. The results presented so far indicate that selective diffusivity is a weak symmetry breaker, whereas selectivesources are way more effective.It is therefore of interest to speculate whether the present model can be of any use in the context of electro-weak baryogenesis (EWBG) [14].To this purpose, let us remind that, so far, we referred to A and B as generic mutually annihilating speciescarried by a phase-changing fluid C .Baryogenesis implies the identification• A = matter• B = antimatter• C = Higgs fieldAlthough we refrain from making any claim of quantitative relevance to EWBG, it is nonetheless of interestto assess the plausibility of present model towards the basic requirements laid down by Sakharov, back in themid sixties [16].They amount to the following three basic conditions:i) The existence of an explicit baryon-symmetry breaking mechanism,ii) Violation of C and CP invariance,iii) Thermodynamic non-equilibriumAs to i), the baryon symmetry breaking is expressed by the non-unit diffusion jump factor J AB across themembrane, or an explicit symmetry breaking at the level of source terms, i.e ζ = 1.Item ii) states that the system must be invariant upon a reflection in space, say from x to − x across theinterface, and charge conjugation.Our model is electrically neutral, hence item i) is basically a requirement that the density of A at location x be different from the density of B at the mirror location − x , namely ρ A ( x ) = ρ B ( − x ). This is certainly trueonce a non-unit diffusivity jump or source asymmetry factor is in action.Hence the selective models discussed in this work meet both i) and ii) criteria. Finally, thermodynamicnon-equilibrium implies that species A and B must depart from their local thermodynamic equilibrium, whichis certainly true in the presence of density gradients across the interface. Thus, even though we do not claimthat the model discussed in this work has any direct quantitative implications for EWBG, it is nonethelessencouraging to observe that it appears to be conceptually compatible with the basic requirements for baryo-genesis. To proceed towards a quantitative analysis, several aspects need to be explored in more detail. Forinstance, in the EWBG scenario the Higgs droplet expands much faster than the Universe, until it fills it upentirely, whence its alleged pervasiveness at the current day [9].In our model, the droplet stops growing once mass equilibrium is attained, typically for density ratios around10 between the liquid and vapour phases. In addition, our computational Universe is static, as opposed to anexpanding Universe. However, both limitations could be significantly mitigated, if needed.To gain a better understanding of the above issues, it proves useful to inspect the physical time and length-scales of our simulations. The time span goes from the onset of EWBG, t EW BG ∼ − seconds, to the timeof the QCD transition, t QCD ∼ − seconds. With one million timesteps, this fixes the lattice timestep to∆ t = 10 ps. The corresponding lattice spacing is ∆ x = c ∆ t = 3 10 − meters, which means that we deal witha computational Universe of side L = 3 meters, and a Higgs droplet inflating from about 3 mm to 3 meters indiameter.The droplet growth rate in our simulations is ˙ R ∼ .
04 in light speed units, which is about ten times smallerthan the credited wall speed of the true Higgs droplets, estimated at c/ x r M A / M B Figure 4: The ratio M A /M B as a function of ζ at the time when the droplet attains its equilibrium mass( t =12000) (left). The solid line is the fit 2 · e − . ζ (1 − ζ ) . On the right side, we show a typical density profileof the carrier species across the interface, indicated by the horizontal line cutting the left edge of the circulardroplet.Next, let us inspect the values of the matter/antimatter ratio, in our case the ratio M A /M B at the timewhen the droplet reaches its equilibrium mass.In Fig. 4 we report the mass ratio M A /M B at the end of the droplet growth, as a function of the symmetrybreaking parameter ζ .We note that with ζ ∼ .
5, ratios around 10 are obtained, which extrapolate to 10 in the limit ζ → − . Although not visible on the scale of the plot, ζ = 0 .
95 yields amass ratio around 80, nearly two orders of magnitude, in the face of a tiny five percent source asymmetry.Summarizing, it appears reasonable to speculate that, with proper fine-tuning and extensions, the presentmodel could prove useful for computational explorations of the semi-classical aspects of strongly non-equilibriumEWBG scenarios. The inclusion of quantum effects [15] may also be feasible through suitable adaptations ofthe lattice Wigner equation [18].
Summarizing, we have analysed the transport of mutually annihilating species within the flow field of a passivecarrier experiencing a first-order dynamic phase transition. In particular, we analysed the symmetry-breakingeffects on the mass engulfed by the growing droplet as induced by preferential transport over one species overthe other across the droplet interface and also due to an explicit symmetry breaking of the source terms.For the source-free case, the evolution proceeds through three dynamic epochs: a very short initial 1 /t decaydue to annihilation, ii) an intermediate stage associated to the droplet nucleation, in which both masses growwhile undergoing large oscillations, with a minor prevalence of A over B . Finally, a long-term 1 /t decay in which A consistently exceeds B owing to the excess developed in the previous stage. This indicates that, althoughthe annihilation rate reaches down to very small values, it is never exactly zero and perfect separation is neverachieved. As it stands, the model shows that even large diffusivity jump factors across the membrane do notgive rise to any substantial mass asymmetry. Besides suitable customization of the numerical values, it appearslike substantial mass asymmetry requires additional symmetry-breaking mechanism.Indeed, the source-driven scenario appears to be much more effective, since any nonzero asymmetry betweenthe two source terms turns the 1 /t decay of the majority species into a secular linear growth. As a result, inthe long term, the ratio between minority and majority species decays like 1 /t . With suitable adaptations,this present model might be able to provide information on the strongly non-equilibrium spacetime dynamicsof the early stage of electroweak baryogenesis. 10 Figure 5: The 27 discrete velocity lattice in three spatial dimensions (D3Q27).
The research leading to these results was funded by the European Research Council under the European UnionHorizon 2020 Framework Programme (No. FP/2014-2020)/ERC Grant Agreement No. 739964 (COPMAT).One of the authors (SS) acknowledges illuminating discussions with Gian Francesco Giudice, Gino Isidori,Antonio Riotto and David Spergel. He also wishes to thank Fabiola Gianotti for arranging a memorable visitat CERN, during which part of this work was discussed.
10 Appendix: The Lattice Boltzmann formulation
The LB equation takes the following form [19, 8, 12]: f ki ( ~r + ~c i ∆ t, t + ∆ t ) − f ki ( ~r, t ) = − ∆ tτ k ( f ki − f k,eqi ) + F ki ∆ t, k = A, B, H (32)where f i ( ~r ; t ) represents the probability to find a representative particle of species k at the lattice position ~r and time t with the discrete velocity ~c i . The index i runs over the discrete speeds, i = 0 ,
18 for the presentnineteen-velocity three-dimensional lattice.The local equilibria (a truncated version of Maxwell-Boltzmann distribution) encode the mass-momentumconservation laws. At the moment, they are purely classical, but the y can be easily extended to quantumstatistics. In detail, [5]: f A,eqi = w i ρ A (1 + u i ) (33) f B,eqi = w i ρ B (1 + u i ) (34) f C,eqi = w i ρ C (1 + u i + q i /
2) (35)(36)where u i = ~u · ~c i /c s and q i = u i − u , u being the magnitude of the net flow of the carrier fluid, namely ρ~u = X i f Ci ~c i (37)with ρ = X i f Ci (38)the carrier density. Finally, w i is the standard set of weights normalised to unity and c s = P i w i c i /d is thesound speed in d spatial dimensions. In the present lattice c s = 1 / c = 1 inlattice units).The transport properties are controlled by the relaxation rate, according to the standard LB relations,namely: D A = c s ( τ A − ∆ t/ , D B = c s ( τ B − ∆ t/ , ν C = c s ( τ C − ∆ t/
2) (39)Note that A and B equilibria conserve only mass, hence they support mass diffusion, whereas carrier C equilibria conserve momentum as well because the local equilibria contain the self-consistent carrier current,11ee Eq. (37). Consequently the carrier relaxation rate controls momentum diffusivity, also known as kinematicviscosity.For species A and B , the forcing terms are set to zero F Ai = F Bi = 0, so that they obey an ideal equationof state p A,B = ρ A,B c s . Given that c s = 1 / c = ∆ x/ ∆ t = 1 is the light speed.The carrier fluid, however, is subject to self-consistent force resulting from potential energy interactions,according to the standard LB pseudo-potential formulation [17]. Consequently, it obeys a non-ideal equation ofstate of the form (Carnahn-Starling) [3, 11] p C ρ C v T = 1 + r + r + r (1 − r ) − . r (40)where v T = k B T /m and r is the reduced carrier density. This corresponds to a critical temperature T c = 0 . ρ c = 0 .
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