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dimensional KPZ class. This opens up the possibility of analyzing growth models via reaction-diffusion models, which allow much more efficient computer simulations. |
pacs: 05.70.Ln, 05.70.Np, 82.20.Wt Introduction The Kardar-Parisi-Zhang (KPZ) equation Kardar et al. ( 1986 motivated by experimentally observed kinetic roughening has been the subject of large number of theoretical studies Halpin-Healy and Zhang ( 1995 ); Krug ( 1997 Later it was found to model other important physica... |
Kardar ( 1987 and the magnetic flux lines in superconductors Hwa ( 1992 It is a non-linear stochastic differential equation, which describes the dynamics of growth processes in the thermodynamic limit specified by the height function [MATH] |
[EQUATION] Here [MATH] and [MATH] are the amplitudes of the mean and local growth velocity, [MATH] is a smoothing surface tension coefficient and [MATH] roughens the surface by a zero-average Gaussian noise field exhibiting the variance |
[EQUATION] Here [MATH] is used for the dimension of the surface, [MATH] for the noise amplitude and [MATH] denotes average over the noise distribution. In [MATH] dimensions it is exactly solvable Kardar ( 1987 but in higher dimensions only approximations are available (see Barabási and Stanley ( 1995 ). In [MATH] spati... |
Marinari et al. ( 2002 does not support this claim. In one dimension a discrete, restricted solid on solid realization of the KPZ growth is equivalent to the asymmetric simple exclusion process (ASEP) of particles Plischke et al. ( 1987 ); Meakin et al. ( 1986 |
(see Fig. ). In this discrete so-called ’roof-top’ model the heights are quantized and the local derivatives can take the values [MATH] By considering the up derivatives ( [MATH] ) as particles and down ones as holes the roughening dynamics can be mapped onto a driven diffusive system of particles with single site occu... |
The extension of this kind of lattice-gas analogy to higher dimensions has not been considered to our knowledge. Instead hypercube stacking models were constructed Meakin et al. ( 1986 ); Forrest and Tang ( 1990b |
and surface configurations were mapped onto the [MATH] -state Potts spins defined on the substrate lattice itself. Especially [MATH] dimensional surfaces were shown to be related to the six-vertex model with equal vertex energies Baxter ( 1982 |
and to the ground-state configurations of the anisotropic Ising model defined on the triangular lattice Blote and Hilhorst ( 1982 As a consequence the height-height correlation functions can be related to four-spin-correlation functions of the spin system. Very recently the conformal invariance of the isoheight lines h... |
Here we show that a [MATH] dimensional growth model exhibiting KPZ scaling can also be mapped onto a driven lattice gas. This is important from theoretical point of view, because the scaling behavior of driven diffusive system (DDS) has been studied intensively for a long time (for a review see ref. Schmittman and Zia ... |
II Mapping onto lattice gas in two dimensions As a generalization of the [MATH] dimensional roof-top model, where the building blocks are squares let’s put octahedra on the square lattice, such that we get back the [MATH] dimensional model in the [MATH] or [MATH] direction as shown on Figure Surface adsorption or desor... |
The surface built up from the octahedra can be described by the edges meeting in the up/down middle vertexes. The up edges in the |
[MATH] or [MATH] directions are represented by ’ [MATH] ’-s, while the down ones by ’ [MATH] ’ in the model. In this way a single site deposition flips the four edges and means two [MATH] [MATH] [MATH] ’ (Kawasaki) exchanges: one in the [MATH] and one in the [MATH] direction. This can also be understood as a special [M... |
with the generalized Kawasaki updating rules [EQUATION] with probability [MATH] for attachment and probability [MATH] for detachment. We can also call the ’ [MATH] ’-s as particles and the ’ [MATH] -s as holes of the base square lattice. In this way an attachment/detachment update can be mapped onto a single step motio... |
Since the three dimensional space can’t be filled fully by octahedra, holes can occur among them, below the surface. Therefore this approximation of a surface growth may not sound to be faithful and the validity of KPZ growth rules requires confirmation. Note, however that in reality atoms are not cubes either and do n... |
Karmakar et al. ( 2005 ); Silveira and Reis ( 2007 , in which under-surface vacancies may occur KPZ scaling has been reported as well. |
The deterministic part of the KPZ equation ( ), which can be obtained by averaging over the noise can be derived from the surface/dimer model similarly as it was done in |
[MATH] dimension Plischke et al. ( 1987 . If we apply the transformation [EQUATION] we get the Burgers equation for the height profile |
[EQUATION] Our system is represented by two matrices [MATH] and [MATH] of sizes [MATH] , which contain discrete derivatives [MATH] or [MATH] in [MATH] and [MATH] direction, respectively (see Eqs. ( 14 ),( 19 )). In two dimensions we introduce the vector variable [MATH] . This has the value [MATH] |
in case of a dimer and [MATH] for a pair of holes. By setting up the master equation [EQUATION] for the probability distribution [MATH] where the prime index denotes a state as a result of a generalized Kawasaki flip ( ) the transition probability is given by |
[EQUATION] with [MATH] parametrization. This formally looks like the one-dimensional Kawasaki exchange probability (shown in Plischke et al. ( 1987 ), except the cross-product term, which is necessary to avoid surface discontinuity creation. The cross-product as a determinant cancels updates between configurations like... |
of the nonlinear term can be interpreted as follows. For [MATH] positive nonlinearity (positive excess velocity) it is a consequence of growth with voids. |
To obtain Eq. ( ) first one averages over the slope vectors [EQUATION] Then calculating its time derivative using the master equation the cross-product term drops out and one obtains |
[EQUATION] Here one can see the discrete second and first differentials of [MATH] corresponding to the operators of ( ). These differentials are one-dimensional because the dimer dynamics is also one-dimensional. Making a continuum limit in both directions and taking into account the relation of height and slope variab... |
This agreement does not prove the equivalence of KPZ and the dimer model since they are just the first equations in the hierarchy of equations for correlation functions. On the other hand from universal scaling point of view they show the equivalence of the leading order terms. We will show by numerical simulation that... |
III The simulation algorithm In the algorithm we extend the sequence of discrete slopes of the [MATH] ASEP model (Fig. ) to local derivatives at [MATH] sites in [MATH] and [MATH] directions of the surface (see Fig. ). The initially flat surface is presented as a regular sequence of ’ [MATH] ’-s and ’ [MATH] ’-s within ... |
A site [MATH] on the substrate plane is selected randomly. Then, we choose an attachment or detachment attempt according to their probabilities [MATH] and |
[MATH] . Generalized Kawasaki exchanges ( of attachment or detachment are realized if [EQUATION] respectively. Throughout this paper the time is measured by Monte Carlo steps (MCS), i.e. [MATH] jump attempts correspond to one MCS. After certain time intervals data evaluation requires the reconstructions of the surface ... |
IV Results Starting from periodic, vertically striped particle distribution, which corresponds to a flat initial surface we update the particle model by the rules defined in the previous section. At certain time steps we calculate the [MATH] heights from the height differences [MATH] The morphology of a growing surface... |
[EQUATION] In the absence of any characteristic length, growth processes are expected to show power-law behavior of the correlation functions in space and height and the surface is described by the Family-Vicsek scaling Family and Vicsek ( 1985 |
[EQUATION] Here [MATH] is the roughness exponent and characterizes the deviation from a flat surface in the stationary regime [MATH] ), in which the correlation length has exceeded the linear system size [MATH] and [MATH] is the surface growth exponent, which describes the time evolution for earlier (non-microscopic [M... |
[EQUATION] In case of up-down symmetry ( [MATH] [MATH] ) the nonlinear term is dropped, and the KPZ equation ( simplifies to the Edwards-Wilkinson (EW) equation Edwards and Wilkinson ( 1982 Since the upper critical dimension of this equation is: [MATH] mean-field behavior, characterized by [MATH] and logarithmic scalin... |
[EQUATION] as shown in Fig. The prefactor [MATH] obtained by fitting the [MATH] curve in the [MATH] region with the form ( 24 ) is |
[MATH] . This is in agreement with the theoretical estimate for the EW equation [MATH] Nattermann and Tang ( 1992 if take into account the exact value for the stiffness constant (or surface tension): [MATH] This constant was identified by Blote and Hilhorst ( 1982 through the correspondence between the exact calculatio... |
triangular lattice site per surface element and the [MATH] of the octahedron/cube surface fraction, thus the theoretical estimate is: [MATH] |
The saturation values are expected to exhibit logarithmic growth [EQUATION] with the system size Nattermann and Tang ( 1992 . As can be seen in the inset of Fig. this really happens with the prefactor [MATH] which agrees with the theoretical value [MATH] again. |
For pure deposition [MATH] [MATH] , or in case of other general up/down asymmetric cases, we saw power-law increase of the surface width, in agreement with the scaling hypothesis ( 21 (see Fig. ). For the the largest system that we have investigated ( [MATH] we fitted [MATH] in the [MATH] time window with a power-law a... |
The saturation values [MATH] for different system sizes also scale well with ( 22 ) and with the exponent [MATH] of the [MATH] dimensional KPZ class Marinari et al. ( 2000 ); Barabási and Stanley ( 1995 Assuming corrections to scaling of the form |
[MATH] the fitting to our data resulted in very small effect: [MATH] , which marginally overlaps with the value of Marinari et al. ( 2000 but does not support the proposal |
[MATH] of Lässig ( 1998 Using these surface exponents and the scaling law ( 23 we estimated the dynamical exponent to be: [MATH] which is somewhat greater than what one finds for the [MATH] |
dimensional KPZ class in Barabási and Stanley ( 1995 [MATH] ). We think that this is due to the correction to scaling observed in the time dependence discussed above. If we scale the time with the dynamical exponent [MATH] |
we obtain a good scaling collapse of the growth data for different sizes (Fig. in agreement with the ( 21 22 ) law again. Our exponent estimates also satisfy the [MATH] scaling relation within error margin. This implies that the Galilean invariance holds and the lattice model indeed lies in the [MATH] dimensional KPZ u... |
Conclusions and outlook We have pointed out the possibility of mapping of a discrete surface growth processes onto a conserved, driven lattice gas model of oriented dimers, which move perpendicularly in two dimensions. The straight line motion of dimers in the two dimensional space is very similar to the motion of part... |
Ódor ( 2004 2008 ); Hinrichsen and Ódor ( 1999 . Interestingly the [MATH] symmetric surface dynamics maps onto a strongly anisotropic reaction-diffusion model. |
We have found KPZ or EW scaling by numerical simulations, hence we showed that lattice anisotropy and under-surface vacancies are irrelevant. Our simulation results for the |
[MATH] dimensional EW case reproduced the theoretically expected logarithmic scaling, with the correct leading order coefficients. For the KPZ scaling our roughness exponent result is in the middle of the range obtained by various numerical exponent estimates: i.e. between [MATH] |
Ghaisas ( 2006 ); Haselwandter and Vvedensky ( 2006 and the field theoretical value [MATH] Lässig ( 1998 Our [MATH] coincides with that of the numerical study |
Reis ( 2004 and agrees with the renormalization results [MATH] Colaiori and Moore ( 2001 . It overlaps marginally with the simulation results [MATH] |
Marinari et al. ( 2000 as well. Our growth exponent estimate [MATH] matches the results of Ghaisas ( 2006 [MATH] ) and Reis ( 2004 |
[MATH] ), obtained by independent numerical fitting procedures. The dynamical exponent of this study is also in the range provided in Reis ( 2004 |
Our model provides an efficient way of simulations and opens us the possibility to study more complex growth models relevant in recent interest of self-organizing surface nanosystem |
Facsko et al. ( 1999 . An optimized, bit-coded version of our code, which manipulates the two-dimensional bit-field by logical operations runs roughly 10 times faster than the current version and will be published elsewhere. For example the Bradley-Harper Bradley and Harper ( 1988 and the debated Kuramoto-Sivashinsky K... |
Acknowledgments: We thank Zoltan Rácz and Uwe Täuber for the useful comments. Support from the Hungarian research fund OTKA (Grant No. T046129), the bilateral German-Hungarian exchange program DAAD-MÖB (Grant Nos. D/07/00302, 37-3/2008) and from the German Science Foundation (DFG research group 845, project HE2137/4-1)... |
# Source: arxiv 0810.2344 # Title: Exact Hydrodynamics of the Lattice BGK Equation # Sections: all # Downloaded: 2026-03-02T08:58:07.913148+00:00 |
Exact Hydrodynamics of the Lattice BGK Equation Abstract We apply the projection operator formalism to the problem of determining the asymptotic behavior of the lattice BGK equation in the hydrodynamic limit. As an alternative to the more usual Chapman-Enskog expansion, this approach offers many benefits. Most remarkab... |
Introduction Statistical physics is often concerned with the problem of determining a closed set of equations of motion for a relatively small number of macroscopic degrees of freedom, given those for a much larger number of underlying degrees of freedom. This problem occurs in the classical theoretical context of deri... |
, and also in the more modern computational context of so-called multiphysics simulations . These examples span a wide range of difficulty. |
The derivation of the Boltzmann equation assumes excellent separation between three different scales of length and time: 1. The scales associated with a molecular collision are assumed to be the smallest and fastest of all the relevant scales. In particular, the range of intermolecular force is assumed to be much small... |
2. The scales associated with the spatial and temporal intervals between collisions – the mean-free path and the mean-free time, respectively – are assumed to be much larger than the collision duration, but smaller than any hydrodynamic length and time scales. |
3. Hydrodynamic length and time scales are assumed to be longest and slowest. Under these assumptions, an asymptotic approach may be adopted. This is the basis of the Chapman-Enskog expansion |
with which it is possible to derive the Navier-Stokes equations of viscous hydrodynamics. Over the past few decades, it has been recognized that a breakdown of scale separation between scales 1 and 2 is much less catastrophic than a breakdown between scales 2 and 3. In a dense gas or liquid, for example, the mean-free ... |
. The Navier-Stokes equations hold reasonably well for water at standard temperature and pressure, after all, even though scales 1 and 2 are comparable. |
A breakdown of separation between scales 2 and 3 is a much more serious issue. The ratio of mean-free path to hydrodynamic scale lengths is called the Knudsen number, Kn, and the Chapman-Enskog analysis is asymptotic in this quantity. Even for single-component, single-phase fluids, the Navier-Stokes equations may break... |
For the last decade, physicists, chemists, applied mathematicians and engineers faced with the problem of modeling complex fluids have studied lattice models of hydrodynamics. These models consist of particle populations moving about on a lattice and colliding at lattice sites, whose emergent hydrodynamic behavior is t... |
. It was found much easier to introduce effective forces between particle populations on a lattice than to introduce such forces in a continuum setting |
. In this way, the dynamics of immiscible , coexisting , and amphiphilic fluids have all been modeled successfully. It may be argued that the success of lattice models of fluids is purely phenomenological in nature. Attractive or repulsive interparticle potentials are introduced to make immiscible species separate, and... |
. As long as the dimensionless parameters associated with the simulation match the fluid being modeled, the details of the microscopic interactions are deemed unimportant. |
Faith in this phenomenological approach is, to some extent, justified by the observed robustness of hydrodynamic equations to details of the kinetic interactions. If both the interparticle potential range and the mean-free path are on the order of a lattice spacing, loss of separation between scales 1 and 2 is evident.... |
Another reason for faith in the lattice-BGK approach owes to its second-order accuracy. Mathematically, the lattice-BGK equation may be written |
[EQUATION] where [MATH] is the discrete-velocity distribution function corresponding to the [MATH] th velocity [MATH] at spatial position [MATH] and time [MATH] , likewise [MATH] is an equilibrium distribution function that depends only on the conserved densities [MATH] , and [MATH] is a collisional relaxation time. Se... |
has pointed out that we may define the new dependent variable [EQUATION] It is then straightforward to derive the lattice BGK equation for the transformed variable. Using [MATH] to denote the propagation operator , defined by |
[EQUATION] the result may be written [EQUATION] where [MATH] is the identity operator. This form makes manifest the fact that the collision is applied between sites. It also allows a glimpse at the origin of the term [MATH] which will emerge as a factor in the transport coefficient. |
One might hope that the success of lattice BGK models would represent progress toward the end of deriving hydrodynamic equations for complex fluids from first principles. For example, it would be very satisfying if a Chapman-Enskog analysis of an interacting-particle lattice-BGK equation gave rise to a Ginzburg-Landau ... |
. To date, however, a successful derivation of this sort does not exist, most likely due to the aforementioned loss of separation between scales 2 and 3 for such fluids. There is simply no small parameter analogous to the Knudsen number on which to base an asymptotic expansion. |
Part of the problem is that the very first step of the Chapman-Enskog analysis of the lattice-BGK equation is the Taylor expansion of the propagation operator, effectively in powers of the Knudsen number. This has the effect of yielding partial differential equations (PDEs) in space and time. In this work, we argue tha... |
We circumvent the usual need for Taylor expansion by applying the projection operator formalism to the problem of deriving the exact hydrodynamics of the lattice-BGK equation. As an alternative to the more usual Chapman-Enskog expansion, this approach offers many benefits. Most remarkably, it produces absolutely exact,... |
We begin by applying the methodology to a lattice-BGK model for Burgers’ equation. This example is simple enough to display every step of the calculation, but complicated enough to raise most all of the above issues. In particular, spatially varying initial conditions may lead to a shock of characteristic width equal t... |
Burgers’ equation 2.1 Lattice BGK model The method is best illustrated by example, so we begin by applying it to a lattice BGK model for Burgers’ equation in one spatial dimension ( [MATH] ). A number of such models for Burgers’ equation have been developed over the years |
; this one is a variant of an entropic version considered by Boghosian et al. . At each point [MATH] , and at each time [MATH] , we have a two-component distribution function [MATH] , from which it is possible to recover the hydrodynamic density |
[EQUATION] The distribution function obeys the lattice BGK kinetic equation [EQUATION] where we have defined the local equilibrium distribution function |
[EQUATION] Here [MATH] is a parameter that is taken to be of first order in the scaling limit. That is, as the number of lattice points per physical distance is doubled, [MATH] will be halved. |
For the analysis of this model, we shall also find it useful to define the kinetic moment [EQUATION] so that the distribution function may be recovered from knowledge of its hydrodynamic and kinetic moments, |
[EQUATION] 2.2 Chapman-Enskog analysis The simple lattice BGK model described above has pedagogical value as a simple introduction to the Chapman-Enskog asymptotic procedure. To make this presentation self-contained, and to facilitate comparison of the proposed new methodology with the Chapman-Enskog procedure, we outl... |
The usual analysis begins with a Taylor expansion of the kinetic equation ( ) assuming parabolic ordering, wherein spatial derivatives are first order quantities, and time derivatives are second order quantities. The result is |
[EQUATION] where all quantities are understood to be evaluated at [MATH] , and where [MATH] is a formal expansion parameter, introduced for purposes of bookkeeping, that will be set to one at the end of the calculation. Note that we made the substitution [MATH] to reflect the fact that [MATH] is first-order in the scal... |
Taking the sum of the [MATH] components of this equation yields the conservation equation [EQUATION] To close this equation, it will be necessary to express [MATH] in terms of [MATH] . This is done by solving Eq. ( 10 ) perturbatively by taking |
[EQUATION] At order zero in [MATH] , we immediately obtain [EQUATION] This lowest approximation to [MATH] implies [MATH] ; it is therefore insufficient to calculate the kinetic moment, which first appears at order one. |
Proceeding to order one, we obtain [EQUATION] so [EQUATION] where we have used the order zero solution in the last step. This order-one result for the distribution function, |
[EQUATION] yields the kinetic moment [EQUATION] Inserting this result in Eq. ( 11 ) yields [EQUATION] or [EQUATION] Upon the substitution [MATH] , this becomes Burgers’ equation in canonical form, |
[EQUATION] where we have defined the transport coefficient [MATH] This method of simulating Burgers’ equation is simple to implement and remarkably robust. Fig. shows the results of simulating of the model on a periodic lattice of size [MATH] , with [MATH] and [MATH] . The initial conditions used were |
[EQUATION] It is seen that the solution captures the formation and decay of the shock, and that the width of the shock at late times is comparable to the lattice spacing. |
Before going on to the exact analysis, it is worth making some general observations about the Chapman-Enskog analysis for this model: |
We solved the kinetic equation only to first order, but used that solution in the hydrodynamic equation at second order. This interlacing of orders is characteristic of the Chapman-Enskog analysis. |
The second-order accuracy of the lattice BGK equation is not at all evident in the final result obtained. If we were to carry on to the next order – which would involve solving the kinetic equation to second order – it is not at all clear that we would not find corrections to the hydrodynamic equation obtained. |
The transport coefficient is equal to [MATH] . The first term, [MATH] , arose from the gradient correction to the local equilibrium distribution function. The second term, [MATH] , came from the Taylor expansion of the left-hand side of the kinetic equation at order two, and is an artifact of the discrete nature of the... |
2.3 Exact analysis The exact analysis that is the point of this paper comes from projecting the kinetic equation onto hydrodynamic and kinetic subspaces, solving for the kinetic field [MATH] as though the hydrodynamic field [MATH] were a known function of position and time, and then using this solution to obtain a clos... |
. In linear algebra, it is related to the Schur complement . To the best of our knowledge, however, it has not heretofore been applied to the problem of obtaining exact solutions of the lattice BGK equation. |
We begin by using Eqs. ( ) to write coupled evolution equations for [MATH] and [MATH] . After a bit of straightforward algebra, we obtain |
[EQUATION] These two dynamical equations for the hydrodynamic and kinetic fields respectively, taken together, are equivalent to the original kinetic equation. The equation for the kinetic field may be written in the suggestive form |
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