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Lattice-gas crystallization

Lattice-Gas Crystallization

Je?rey Yepez?

Phillips Laboratory,Hanscom Field,Massachusetts01731

yepez@https://www.sodocs.net/doc/7617202375.html,

June20,1994

Abstract

This paper presents a new lattice-gas method for molecular dynamics modeling.A mean?eld treatment is given and is applied to a linear stabil-

ity analysis.Exact numerical simulations of the solid-phase crystallization

is presented,as is a?nite-temperature multiphase liquid-gas system.The

lattice-gas method,a discrete dynamical method,is therefore capable of

representing a variety of collective phenomena in multiple regimes from

the hydrodynamic scale down to a molecular dynamics scale.

KEY WORDS:Lattice-gas automata,discrete hydrodynamics,mul-tiphase?uids,complex?uids,liquid-gas transition,molecular dynamics,

crystallization

1INTRODUCTION

Presented in the paper is a theory of lattice-gas dynamics that includes inter-particle potentials.The microscopic lattice-gas dynamics is a highly discrete form of traditional molecular dynamics.All the usual dynamical quantities ap-pearing in the traditional theory are discrete in the case of a lattice-gas.It is well known that in lattice-gases these discrete quantities include space,time, and momentum.Here the notion of a discrete?eld is introduced and an analyt-ical theory is presented to describe emergent macroscopic dynamics.The use of the?eld concept is shown to be quite useful.In particular,the?eld concept is useful in describing our novel lattice-gas with multiple long-range interactions with di?erent ranges and polarity.This new lattice-gas possesses a liquid-solid transition and can be used as a new general method of simulating molecular dynamics.The theoretical possibilities for such a lattice-gas opens the subject of exactly computable modeling to the areas of dynamical solid-state systems.

It is known that interparticle potentials can be modeled by including a sin-gle anisotropic?xed-range interaction in the lattice-gas dynamics for discrete ?This work supported in part by Phillips Laboratory,and in part by the Air Force O?ce of Scienti?c Research under new initiative No.2304C.

momentum exchange between particles.The simplest theoretical model of this kind is the Kadano?-Swift-Ising model[8].An attractive?xed-range interaction was used in a lattice-gas automaton by Appert and Zaleski[2]in1990to model

a nonthermal liquid-gas phase transition.The use of attractive and repulsive

?xed-range interactions of this sort extended the lattice-gas dynamics to a?-nite temperature liquid-gas transition where a complete pressure,density,and temperature equation of state is modeled,and the complete liquid-gas coexis-tence curve is analytically predicted through a Maxwell construction[15].Our

?nite temperature liquid-gas lattice-gas is presented in the paper for pedagogi-

cal reasons as well as to validate the theoretical method https://www.sodocs.net/doc/7617202375.html,ttice-gas crystallization is introduced as a direct generalization of the?nite temperature liquid-gas lattice-gas model.

This paper is organized in basically two parts.The?rst part of the pa-

per up to and including section§4is well known by the lattice-gas community and is given here as review material for the subject of lattice-gases with purely local collisions.The rest of the paper presents new results,using a new theo-rem referred to here as the lattice multiple theorem presented in appendix12. This theorem is useful for determining the linear response of a lattice-gas with long-range interactions.Appendix12describes in some detail an explicit nu-merical method for implementing the the simplest of long-range interactions:

the bounce-back and clockwise orbits.

2Lattice-Gas Automaton:An Exactly Computable Dynamical System

A boolean formulation of an exactly computable dynamical system,known as

a lattice-gas,may be stated in a way that is consistent with the Boltzmann equation for kinetic transport.In essence the lattice-gas dynamics are a sim-

pli?ed form of molecular transport as we restrict ourselves to a discrete cellular phase space.The macroscopic equations,in particular the continuity equation and the Navier-Stokes equation,are obtained by coarse-graining over a discrete microdynamical transport equation for number boolean variables.The scheme employs the?nite-point group symmetry of a crystallographic spatial lattice.It

is somewhat inevitable that to obtain an exactly computable representation of

?uid dynamics one must perform a statistical treatment over discrete number variables.

Before introducing the basic lattice-gas microdynamical transport equation,

let us give some notational conventions.We consider a spatial lattice with N total sites.The fundamental unit of length is the size of a lattice cell,l,and

the fundamental unit of time,τ,is the time it takes for a speed-one particle to

go from one lattice site to a nearest neighboring site.Particles,with unit mass m,propagate on the lattice.The unit lattice propagation speed is denoted

by c=lτ.Particles occupy this discrete space and can have only a?nite B number of possible momenta.The lattice vectors are denoted by e ai where

a =1,2,...,B .For example,for a single-speed gas on a triangular lattice,a =1,2,...,6.A particle’s state is completely speci?ed at some time,t ,by specifying its position on the lattice,x i ,and its momentum,p i =mce ai ,at that position.The particles obey Pauli exclusion since only one particle can occupy a single momentum state at a time.The total number of con?gurations per site is 2B .The total number of possible single particle momentum states available in the system is N total =BN .With P particles in the system,we denote the

?lling fraction by d =P

N total

.The number variable,denoted by n a (x ,t ),takes the value of one if a particle exists at site x at time t in momentum state mc ?e a ,and takes the value of zero otherwise.The evolution of the lattice-gas can then be written in terms of n a as a two-part process:a collision and streaming part.The collision part reorders the particles locally at each site

n a (x ,t )=n a (x ,t )+?a (

n (x ,t )),(1)

where ?a represents the collision operator and in general depends on all the

particles, n at the site.So as a short-hand we suppress the index on the occu-pation variable when it is an argument of ?a (

n (x ,t ))to represent this general dependence.In the streaming part of the evolution the particle at position x “hops”to its neighboring site at x +l ?e a and then time is incremented by τ

n a (x +l ?

e a ,t +τ)=n a (x ,t )+?a ( n (x ,t )).(2)

Equation (2)is the lattice-gas microdynamical transport equation of motion.The collision operator can only permute the particles locally on the site since we wish the local particle number to be conserved before and after the collision.We construct an n-th rank tensor composed of a product of lattice vectors [13]E

(n )

=E i 1...i n =

(e a )i 1···(e a )i n ,(3)where a =1,...,B .All odd rank E vanish.We wish to express E (2n )in terms

of Kronecker deltas,δij =1for i =j and zero otherwise.We can turn this problem of expressing the E -tensors in terms of products of Kronecker deltas into a problem of combinatoric counting.We use the following tensors

?2ij =δij

(4)?4ijkl

=δij δkl +δik δjl +δil δkj

(5)

and so forth.Then we know that if E is isotropic it must be proportional to ?

E (2n )∝?(2n )

(6)

and that the constant of proportionality may be obtained by counting the num-ber of ways we could write a term comprising a product of n Knonecker deltas.

Consider for example the case n =2.Since the Knonecker delta is symmetric in its indices,the following four products are identical:δij δkl =δij δlk =δji δkl =

δjiδlk.The degeneracy is22.Furthermore,the order of the Kronecker deltas also doesn’t matter since they commute;that is,δijδkl=δklδij.The degen-eracy is2!.For the case where n is arbitrary,there are2n identical ways of writing the product of n Kronecker deltas.For each choice of indices,there are an additional n!number of ways of ordering the products.Therefore,the total number of degeneracies equals2n n!=(2n)!!.The total number of permutations for2n indices equals(2n)!.So from this counting procedure we know that?(2n)

consists of a sum of(2n!)

(2n)!!=(2n?1)!!terms.

In general,the lattice tensors are

E2n+1=0(7)

E2n=

D(D+2)···(D+2n?2)

?2n(8)

3Coarse-Grained Dynamics

To theoretically analysize the lattice-gas dynamics,it is convenient to work in the Boltzmann limit where a?eld point is obtained by an ensemble average over the number variables.That is,we may de?ne a single particle distribution function,f a= n a ,resulting from an ensemble of initial conditions and the neglect of correlations,with the averages taken over the ensemble.

It is essential to determine the macroscopic limit of the microdynamical transport equation(2)and to see how it leads to non-compressible viscous Navier-Stokes hydrodynamics—for a lengthier treatment of this see Frisch et al.[6].

Using the Boltzmann molecular chaos assumption the averaged collision op-erator simpli?es to ?a( n) =?a( n ),and by coarse-graining and Taylor ex-panding(2)we obtain the lattice Boltzmann equation

?t f a+ce ai?i f a=?a( f).(9) We write the particle number density,momentum density,and moment?ux density in terms of the single-particle distribution function as follows

f a=ρ(10)

e ai

f a=ρv i(11)

mc2

e ai e aj

f a=Πij.(12)

Now following Landau and Lifshitz[9]we know that in standard form we must be able to write the momentum?ux density tensor as follows

mc2

e ai e aj

f a=pδij+ρv i v j?σ ij(13)

where in(13)the?rst two terms represent the ideal part of the momentum?ux density tensor andσ ij=η(?i v j??j v i)is the viscous stress tensor.Alternatively the momentum?ux density tensor may be written

Πij=mc2

e ai e aj

f a=?σij+ρv i v j,(14) whereσij is the pressure stress tensor

σij=?pδij+η(?i v j+?i v j).(15) The general form of the single particle distribution function,appropriate for single speed lattice-gases,is a Fermi-Dirac distribution.Fundamentally,this arises because the individual digital bits used to represent particles satisfy a Pauli-exclusion principle.Therefore,the distribution must be written as a func-tion of the sum of scalar collision invariants,α+βe ai v i,implying the following

form

f a=

1+eα+βe ai v i

.(16)

Taylor expanding(16)about v=0to fourth order in the velocity and equating the zeroth,?rst,and second moments of f a to(10),(11),and(12)respectively, the parametersαandβare determined.The inviscid part of the lattice-gas distribution function becomes

(f a eq)ideal

LGA =

e ai v i+g

nD(D+2)

2c2B

?e ai?e aj v i v j

n(D+2)

2c2B

v2,(17)

where

g≡

D+2

1?2d

1?d

.(18)

That is,usingρ=mn for the density and c s=c√

D for the sound speed,the

moments of lattice-gas distribution are

a (f a eq)ideal

LGA

=ρ(19)

a e ai(f a eq)ideal

LGA

=ρv i(20)

mc2

a e ai e aj(f a eq)ideal

LGA

=ρc2s(1?g

)δij

+gρv i v j.(21)

The lattice-gas automaton almost produces the correct form for the momentum ?ux density tensor,except thatΠij appears to have a spurious dependence on the square of the velocity?eld,(1?g v2c2)with a factor g arising as an artifact

of the discreteness of the number variables.Working directly in the Boltzmann limit and using only symmetry arguments,it is possible to?x this problem.

The macroscopic equations of motion are then determined from mass con-servation(continuity equation)and momentum conservation(Euler’s equation)

?tρ+?i(ρv i)=0(22) and

?t(ρv i)+?jΠij=0.(23) Substituting(21)into Euler’s equation(23),gives us the Navier-Stokes equation for a viscous?uid

ρ(?t v i+gv j?j v i)=??i p+η?2v i,(24) given a non-divergent?ow(?i v i=0)appropriate to the incompressible?uid

limit and where the pressure is

p=ρc2s

1+g

.(25)

A general expression for the shear viscosity,η,for a single-speed lattice-gas has been derived by H′e non[7].

In any lattice-gas simulation,one typically obtains a realization of the macro-scopic dynmical variables by block averaging in both space and time over the microscopic variables.In this way,for example,a momentum map can be pro-duced so that the dynamic evolution the the?uid can be monitored.The size of the coarse grain block a?ects the resolution with which one can observe the system but of course does not at all a?ect the underlying dynamics.If too small of a coarse grain block size is used,more?uctuations in the macroscopic variables occurs.

4Lattice BGK Equation

We wish to consider a dynamical transport equation for the particle distribution function given in the previous section.We have a lattice Boltzmann gas de?ned on a discrete spatial lattice.Restricting ourselves to a single speed lattice-gas system,the lattice BGK equation is

?f a ?t +ce ai?i f a=?

(f a?f a eq).(26)

This equation was introduced in1954by D.Bhatnager,E.Gross,and M.Krook [3,10].A way to obtain(26)was introduced by Shiyi Chen et al.[5]by ex-panding the lattice Boltzmann collision term to?rst order about the equilibrium distribution and assuming it diagonal.

It is possible to?x the anomaly in the?uid pressure that occurs in the lattice-gas automaton.Chen et al.[4]have introduced a pressure-corrected equilibrium distribution to have the following Chapman-Enskog expansion

(f a eq)ideal

BGK =

e ai v i+

nD(D+2)

2c2B

?e ai?e aj v i v j?

2c2B

v2(27)

which satis?es

(f a eq )ideal

P C

=ρ(1?v 2

c 2)

(28)mc

e ai (

f a eq )ideal

P C

=ρv i

(29)mc 2

e ai e aj (

f a eq )ideal

P C

=ρc 2s δij +ρv i v j .

(30)

Here the de?nition of the density is modi?ed by the 1?v 2

c 2

factor.

Density Dependent Pressure in the Boltzmann Limit

Here we exploit the analytical facility of the lattice Boltzmann approach and show that the addition of a convective-gradient term in the lattice Boltzmann equation allows one to model a hydrodynamic gaseous ?ow governed by a general equation of state [14].The pressure may have a nonlinear dependence on the local density.It is possible to generalize this to a multi-speed lattice-gas or a to single-speed lattice-gas coupled to a heat bath so that the pressure dependence includes the local temperature as well.

The equation of state for the isothermal gas is

p =c 2s ρ.

(31)

We now wish to consider how we may alter the lattice-Boltzmann equation to

allow for a more general equation of state.Let us add an additional term,P a (x,x +re a ),to the R.H.S.of (9)

?t f a (x,t )+ce ai ?i f a (x,t )

[?a (x,t )+P a (x,x +re a ,t )].

(32)

P a depends on the local con?guration of the system at position x as well as on the local con?guration at a remote position x +re a .We assume that the values of ψat x and x +re a are independent and that therefore P a can be factorized 1

P a (x,x +re a ,t )=ψ(x )ψ(x +re a ).

(33)

1The

e?ect of long-range interactions on f a (x )will actually depend on local con?gurations

at x +re a and at x ?re a .So we could write the full form of the long-range part of the collision

operator as 12[ψ(x )ψ(x +re a )?ψ(x ?re a )ψ(x )],where the factor of 1

2must be included to

avoid double counting when doing any directional sums,P a ,since ψhere does not have any directional dependence.According to Theorem 1in the appendix 12,upon expanding ψ(x )ψ(x ±re a ),both terms would add to remove the 12

factor,so using P a =ψ(x )ψ(x +re a )in the present calculation ultimately gives the same result.In appendix 12where we give the microscopic long-range part of the lattice-gas collision operator,the microscopic ψdoes have directional dependence so we use the full form there.

In a single speed lattice-gas model as we have been considering,P a is a function of the local density.In a single-speed model coupled to a heat bath,P a may depend on the local temperature as well [15].

We wish to constrain the form of P a so as not to violate continuity.We require

P a =0,(34)

and when ?i f a =0,

e ai P a =0.(35)

Constraint (35)is required only under uniform ?lling conditions;i.e.for general

situations a e ai P a is non-zero.In the uniform ?ow limit the lattice-Boltzmann equation reduces to

?t f a (x,t )=

[?a (x,t )+P (x,x +re a ,t )],(36)where we have taken the directional dependence of the long range collisional term to occur only in its argument,P a (x +re a )→P (x +re a ).We assume the probability of a long range collision depends only the density at the spatial location of a momentum transfer event and not on the direction of the momen-tum transfer.That is,we require the interaction distance to be of su?ciently long range that the approximation of local isotropy in the particle distribution is valid.Summing over all lattice directions and using constraint (34)we have maintained the collision property that

?total a

= a

(?a +P a )=0.(37)Thus,for arbitrary ?ows,summing the lattice-Boltzmann equation (32)over all

directions preserves continuity

f a +c a

e ai ?i

f a =0(38)?→?t ρ+?i (ρv i )=0,(39)

where we have used (37).

Multiplying the lattice Boltzmann equation by e ai and then summing over directions gives

e ai

f a (x,t )+c?j

e ai e aj

f a (x,t )=1

τψ(x,t )

e ai ψ(x +re a ,t ).(40)

Using Theorem 1in appendix 12we can expand the R.H.S.

a e ai f a (x,t )+c?j a

e ai e aj

f a (x,t )=+1τψ(x,t )?i rB π2 2(r?)?D

2I D 2(r?)ψ(x,t ) .(41)

Therefore,we again arrive at Euler’s equation

?t(ρu i)+?j(Πij)=0(42) but with an augmented momentum?ux density tensor

Πij(x,t)=mc2

a e ai e aj f a?mc2B

2δij

dx kψ(x,t)?k(r?)?D2I D

(r?)ψ(x,t).

(43)

Since the additional term in the momentum?ux density tensor is diagonal,it can only impart an e?ective density dependent pressure.

De?ning a con?gurational potential energy as

V(x)=mc2B r

dx kψ(x,t)?k(r?)?D2I D

(r?)ψ(x,t)(44)

then Euler’s equation(42)gives us the viscous Navier-Stokes equation for non-ideal?uids

?t(ρv i)+?j(ρv i v j)=??i

c2sρ+V(ρ)

+ρν?2v i.(45)

Therefore,we have arrived at a general equation of state de?ned by the potential energy function V(ρ)where there is an inter-particle force F i(x)=??i V(ρ(x)). The form of the density dependent pressure directly follows

p(ρ)=c2sρ+V(ρ).(46) With this methodology,we can model a system with a general equation of state with completely local dynamics described by the generalization of(26)

f a(x+l?e,t+τ)=f a(x,t)?τ

(f a(x,t)?f a eq(x,t))

+ψ(x,t)ψ(x+re a,t).(47) In the Boltzmann limit,the analysis itself does not indicate the form of V in (46)(or more to the point,does not indicate the form ofψ),but does show it is possible to have a lattice-gas that has Navier-Stokes dynamics as its macroscopic limit with a density dependent pressure(45).This is the motivation needed to develop a more complete microscopic description.With a lattice-gas automaton microscopic description,the interparticle force??i V(ρ)may be caused by long-range momentum exchange between two particles.Calculating the probability of such momentum exchange events should provide a way to determineψ.

Note that in the mesoscopic regime in which the Boltzmann equation is applicable,the lowest order expression for V is proportional toψ2.That is

V(x)=mc2B

dx kψ(x)?kψ(x)(48) 9

V(x)=mc2B

ψ2(x).(49)

A similar calculation has also been done by Shan and Chen[12]who have veri?ed their analysis by comparing with data taken from lattice BGK simulations. They have also presented exact calculations for the liquid-gas interface pro?le and surface tension.In the following section§6we take another view point and write an alternate expression for the potential energy,but one that is also proportional toψ2.This alternate view of the potential energy will help towards developing the lattice-gas automaton microscopic description.

6Interaction Energy

We introduce a potential energy due to non-local2-body interactions

H =1

abmn

(1?f a(x))f b(x)V abmn(1?f m(x ))f n(x ),(50)

where V abmn=VΛabmn andΛabmn is either±1or vanishes for any set{abmn} that violate mass,momentum,or energy conservation.H accounts for the potential energy between particles in coming along lattice directions b and n and outgoing along a and m and is therefore restricted to2-body interactions.

We now try to justify the form of H and in so doing develop a microscopic description of the lattice-gas with long range interactions.We require

˙p i=??i H .(51) H is thought of as the con?gurational potential energy due to momentum trans-fers between two locations.The momentum exchange per unit time between two points x and x in the?uid is

δp i=mv out i?mv in i,(52) where the incoming and outgoing velocity states are quantized:v in i=ce bi and v out i=ce ai.The probabilityψ(x)of there being a local momentum change at some point x depends independently on the probability f b(x)that there is a particle in velocity state ce bi and the probability1?f a(x)there isn’t a particle in velocity state ce ai.So in this factorized approximation that neglects particle-particle correlations,we write

ψ(x)=(1?f a(x))f b(x).(53) As a long range momentum exchange event involves two sites,x and x ,we can de?ne the vector r i=re ai=x i?x i and the therefore parallel and perpendicular components of the local momentum exchange are

δp =δp·?r(54)

δp⊥=|δp×?r|.(55)

The two components of the force mediated by the long range momentum ex-change could be interpreted as created by two separate ?elds

δE =δp (r )c

l ?r (56)δB =

δp ⊥(r )c

δ?p ×?r ,(57)

where δp and δp ⊥are written as a function of r since any kind of functional dependence is allowed provided enough detail is speci?ed for the automaton interaction rules.We have explicitly written the forms of the parallel and per-pendicular components of a lattice-gas force ?eld to stress an analogy with the classical theory of electromagnetism where the electric and magnetic ?elds are expressed,for a di?erential element of charge and current element respectively,by the well known laws of Coulomb and Biot-Savart

d E =δQ

4πr 2?r (58)d B

4πr 2

δl ×?r .(59)

Using (53)we can write the total momentum change as a ?eld itself

δp i (x,x

)=?mc

abmn

(e ai ?e bi +e mi ?e ni )(1?f a (x ))f b (x )Λabmn (1?f m (x ))f n (x ).(60)

Note that the momentum change is zero for a ?uid with uniform density due to the symmetry of the lattice.For central body interparticle momentum ex-changes,in the Boltzmann limit we can then approximate the con?gurational potential energy as

V (x,x

)=?

r i δp i τ=?12mc 2 r l abmn

?r ·(?e a ??e b +?e m ??e n )(1?f a )f b Λabmn (1?f m )f n ,(61)

where r is the range of the interaction.For a system locally isotropic in its particle distributions,letting ψ(x )≡(1?f (x ))f (x ),this may be simpli?ed to

V (x,x

)=?12mc

2 r l

abmn abmn

?r ·(?e a ??e b +?e m ??e n )Λabmn ψ(x )ψ(x ),(62)which is suitable for a bulk description of the ?uid.Now the form of H follows

if we sum over all pairs xx and de?ne

V abmn =mc 2 r l ?r ·(?e a ??e b +?e m ??e n )Λabmn ,(63)so that

H =

V (x,x )=

abmn

ψ(x )V abmn ψ(x ).

(64)

7E Field Construction

It is possible to de?ne a ?eld that exists in a lattice-gas that has long-range momenta exchanges occurring.The notion is to consider each lattice-gas particle as having a delta function type ?eld that exists only at certain ?xed ranges and certain ?xed angles.Therefore,the lattice-gas particle has a highly anisotropic ?eld.However,in the coarse-grained limit obtained by averaging over many particles,a valid description of a continuous ?eld emerges.Of coarse,if there are no gradients in the coarse-grained density of the system,the ?eld must necessarily vanish.

The ?eld at position x due to a particle at position x along the lattice direction a is a delta function

ai (

x ; x

)=mc 2l

α(σ)δ( x ? x ?r σ?e a )e ai (65)

That is,the discrete ?eld must be directed along ?e a and must be a distance r σ

from the source,or x = x

+r σ?e a .The total ?eld is obtained by considering all the possibilities where a particle could contribute.The sum over σis necessary to account for multiple interaction ranges,where α(σ)denotes the strength of the interactions at range r σ.Thus to obtain the total ?eld we must sum over all directions and integrate over all positions

E i (

x )= a

d x ψ( x ) ai ( x ; x )(66)or

E i (

x )=mc 2l

σα(σ) a

e ai ψ( x ?r σ?e a ).(67)

Using Theorem 1we can evaluate the directional sum and express the ?eld as

a gradient of a scalar quantity

E i (

x )=??i mc 2B σμσ π2 2

(r σ?)?D

2I D 2(r σ?)ψ( x )

,(68)where μn is de?ned as

μσ≡?α(σ)

r σ

.(69)

Note that μ>0for attractive interactions and μ<0for repulsive interactions.Since,E i =?i φ,the ?eld’s scalar potential is

φ=?mc 2

μσ π2

(r σ?)?D

2I D 2

(r σ?)ψ( x )

(70)

=?mc 2B

σμσ

1D ψ+12D (D +2)

r 2?2

ψ+···

.(71)

The ?eld E i and the event probability ψ=d (1?d )appear in the Navier-Stokes equations as follows

?t (ρv i )+?j (gρv i v j )=?c 2s ?i ρ 1+g v

c 2

+ψE i +ρν?2v i (72)and according to (43),the term ψE i when due to central-body interactions can modify only the pressure.

8Stability Analysis

In this section we consider the linear response of a lattice-gas with long range central-body interactions.The macroscopic equations of motion are (39),(72),and (68)respectively

?t ρ+?i (ρv i )=0

?t (ρv i )+?j (gρv i v j )=?c 2s ?i ρ 1+g v

+ψE i +ρν?2v i E i ( x )=??i

mc 2B σ

μσ π2 2

(r σ?)?D

2I D 2(r σ?)ψ( x )

We treat the e?ect of the ?eld E i as a perturbation on a resting equilibrium

state where ρis uniform and constant and v =0.Then an ε-expansion of the dynamical variables is

v i =εu i

(73)ρ=ρo +ε (74)ψ

=ψo +ε?.

(75)Using ψ=(1?d )d we have

1?2d o

(76)

where d o =ρo

.Consequently,the linear response equations are ?t +ρo ?i u i =0(77)

ρo ?t u i =

?c 2s ?i

?ψo ?i

mc 2

μσ

π2

(r σ?)?D

2I D 2

(r σ?)?

+ρo νo ?2u i

(78)

Then applying ?t to the continuity equation and ?i to the Navier-Stokes equation

allows us to eliminate u i and to obtain the following second-order equation in

?2t =c 2s ?2 +ψo (1?2d o )c 2?2

μσ π2 2

(r σ?)?D

2I D 2(r σ?) +νo ?2?t .(79)

In an inviscid ?uid (ν=0)with no interparticle potentials (E i =0) would satisfy the wave equation

=ρo e ?iωt +ik i x i .(80)Given a non-zero perturbation, can be Fourier expanded

dωdk i ?ρ

e ?iωt +ik i x i (81)

and we can replace with ? by taking ?t →?iωand ?i →ik i .(79)becomes

?ω2? =?k 2c 2s ? ?ψo (1?2d o )k 2c 2

μσ π2 2

(r σk )?D

2J D 2(r σk )? +iωνo k 2? ,(82)

where we have made use of the identity that relates the hyperbolic Bessel func-tion with imaginary argument to the ordinary Bessel function

(iz )?νI ν(iz )=z ?νJ ν(z ).

(83)

Dividing out ? gives a quadratic equation for ω

c s 2+i νo k 2c s ωc s ?k 2 1+ψo (1?2

d o )D

μσ π2 2

(r σk )?D 2J D 2(r σk ) .(84)

The dispersion relation for ω(k )is then

ωc s =±k 1+ν2o k 24c 2s +ψo (1?2d o )D σ

μσ π2 2

(r σk )?D

2J D 2(r σk )?i νo k 22c s .

(85)

In the long wavelength limit,(85)reduces to linear sound speed dispersion

ω=c s k.

(86)

In the absence of long range interactions,(85)reduces to the dispersion relation for ideal,incompressible,viscous ?ow

ω=c s k

ν2o k 24c 2s ?i νo k 2

2.(87)By choosing di?erent ranges and strengths of the momentum exchanges we

can adjust the dispersion relation (85)and produce a variety of interesting dy-namical behavior.Therefore,in the macroscopic limit,what principally de?nes the linear response of a lattice-gas model with long range interactions is the set of constants α(σ)and r σ.

9Finite Temperature Liquid-Gas Model

A simple two-dimensional example is a lattice-gas on a triangular lattice.The macroscopic equations of motion for the lattice-gas automaton are

?t ρ+?i (ρv i )=0

(88)?t (ρv i )+?j (gρv i v j )=??i p +ρν?2v i

(89)?i p =c 2s ?i ρ 1+g v

?ψE i .(90)

The ?eld to lowest order in the interaction range is

E i (x )=?

mc 2B D r l

?i ψ(x ).(91)

This implies that the local force is

ψE i =??i

α2mc 2s B

r l ψ

(92)

Since ψ=d (1?d ),the pressure is a simple polynomial

p (d,h )=mc 2s B d 1+g v

c 2

+α(h )r 2l

d 2(1?d )2 ,(93)

where we have written the pressure depending on the density and a temperature

control parameter,h ,that modi?es the strength of the interactions,α=α(h ).This will be discussed in more detail below.It is possible to determine the coexistence curve for such a ?nite temperature liquid-gas system.We begin by de?ning the free energy,G ,as

G (d,h )≡

dn n ?p (n,h )

(94)

Using (93)for a ?uid at rest we can carry out the integral to obtain

G (d,h )=mc 2s B

α(h )r

?2d +3d 2

?4d 33+2d o ?3d o 2+

4d o 33

+log

d o

(95)

A Maxwell construction can be performed by making a parametric plot of the free energy versus the pressure and locating the point at which the curve is double valued.That is,there are two densities,corresponding a rare?ed phase and a dense phase,that have the same pressure and minimal free energy.The critical temperature,h c ,can be found by ?nding the isothermal pressure curve that has an in?ection point

?p (d,h c )

?d =0(96)15

α(h c)=

2rd(1?d)(1?2d)

.(97)

To verify our theory,we can perform exact numerical simulations of a?nite temperature multiphase system.We can extend an Appert-type nonthermal model to work in a?nite temperature domain by coupling the long range in-teractions to a heat bath of variable density,denoted by a parameter h,and by allowing repulsive long range interactions in addition to the attractive ones. This is done in such a way that the likelihood of an attractive and repulsive interaction goes as(1?h)2and h2respectively[15].Figure1depicts the long-range interaction called bounce-back and how it is coupled to a heat bath. At zero temperature,when h=0,we recover the minimal model as only at-tractive long range interactions can occur.Fiqure2shows the time evolution of the phase separation process in this case at a density d=0.07and inter-action range r=6l.As h increases,the likelihood of repulsive interactions also increases to the point where at the in?nite temperature limit,h=12,the likelihood of attractive and repulsive interactions becomes equal.The occur-rence of both long range attractive and repulsive interactions is identical to a system with?nite-impact parameter collisions.Therefore,the in?nite temper-ature system behaves as an ideal neutral?uid but with an enhanced mean-free path.The nominal strength of the interaction when coupled to a heat bath is α(h)=?αo(1?h)2+αo h2=?αo(1?2h)given a local momentum change of magnitudeδp=αo mc due to long range interactions of range r.For a two dimensional example D=2,we use the following values m=c=l=1,B=6,αo=2.With the?uid at rest,the pressure is then

p=3d?3rd2(1?d)2(1?2h)(98) and the critical value of h is

h c=1

2rd(1?d)(1?2d)

.(99)

Figure3shows liquid-gas coexistence curves for this lattice-gas model at three di?erent interaction ranges:r=7,9,and11l.Both the mean?eld theory calculation and the exact numerical data are presented.The comparison of the theory to the numerical simulation is in good agreement.In the Boltzmann limit, the probability of a long range interaction goes asψ2=d2(1?d)2.It is expected that this estimate which neglects all particle correlations would su?er the most at low densities where the mean free path between local collisions becomes comparable to the range of the non-local interactions.This may account for the deviations that are observed at low densities.The mean?eld predictions of the critical point is also in quite good agreement with the numerically obtained values.Figure4shows the mean?eld calculation and exact numerical data taken at?ve di?erent interaction ranges:r=7through11l.The calculated value of h c is slightly higher then the measured value for all cases indicating a systematic deviation.

For the liquid-gas system,the dispersion relation(85)reduces to

ωc s =±k

ν2o k2

4c2s

?d o(1?d o)(1?2d o)Dα(h)

2(rσk)?D2J

(rσk)?i

νo k2

2c s

(100)

Figure5shows the real and imaginary parts of the liquid-gas dispersion curve for a two dimensional system with a density of d=0.20and momentum exchanges ofδp=?2mc over a range of r=9l.Also shown for comparison purposes are the dispersion curves for an ideal,viscous?uid.In a long-range lattice-gas, since the kinematic shear viscosity is dependent on the square of the mean-free path which in turn is proportional to the interaction range,the following approximation is made forνo

νo(r)=νr=0

1+0.1r2

.(101)

Numerical con?rmation of the parabolic dependence of the kinematic shear vis-cosity on the interaction range is presented in?gure6.Several lattice-gas models were tested by varying the strength and number of interactions.Viscosity mea-surement were made by the method of a decaying sinusoid and were done for systems at a density of60%?lling.

10Crystallization

Introduced in this section is a lattice-gas automaton with multiple?xed-range interactions that possesses a liquid-solid phase transition.In the previous sec-tion,we have tested our formalism that models interparticle potentials in the coarse-grain limit by using a single anisotropic?xed-range interaction in the lattice-gas dynamics for discrete momentum exchange between particles in the microscopic limit.Here a direct generalization to the?nite temperature liquid-gas model is introduced using long-range repulsive and attractive interactions over multiple ranges.For crystallization to occur,at least two interaction ranges are necessary:an attractive short-range interaction and a longer-range repulsive interaction resulting in a kind of Wigner crystal.

10.1New Way for Molecular Dynamics Modeling

To model a more realistic crystal,that is one that can undergo rigid-body motion such as rotation and that can have well de?ned edges or surfaces,more then two interaction ranges are https://www.sodocs.net/doc/7617202375.html,ually four to eight interaction ranges are used to produce a Leonard-Jones type molecular potential.

The shortest-range interaction creates a potential well that stably traps a group of lattice-gas particles.This group of particles remains in a localized con?guration and behaves as a single collective entity.This persistent collective entity is referred to here as an“atom”.As in the liquid-gas system,each lattice-gas particle possesses a discrete?eld that acts along the lattice directions.But

now as many lattice-gas particles are grouped together,in the coarse-grained limit they act as a single particle with a continuous?eld around it.It can behave like a charged particle and repel other such atoms in the system or can behave like a Leonard-Jones particle and attract other atoms depending on the chosen interactions.Starting from a uniformly random con?guration at d=0.1,the lattice-gas spontaneously crystallizes into arrays of these atoms.The emergent crystalline lattice is hexagonal-close-packed.A two dimensional example,with an underlying512×512lattice,of this time-dependent crystallization process is given in?gure7.The resulting crystal is in a hexagonal-close-pack con?gura-tion since we have strived to make the coarse-grained interatomic potential be radially symmetric.2Three dimensional5123simulations of the crystallization were also carried out,see?gure10.

It is possible to measure the density cross-section for the crystal in its?nal equilibrium state,see?gure8.With a principle crystal direction aligned parallel with the x-axis,average density cross-sectional data was taken for a512×512 system;that is,512samples were averaged.In this case,the lattice-gas model had six interaction ranges:r=?2,?7,19,21,?24,?26.Here the negative sign preceding the range denotes an attractive interaction at that range.The averaged cross-section data very closely produces a Gaussian shaped curve.

In the two dimensional numerical simulation,to obtain isotropy in the macro-scopic limit,12directions are used for long-range momentum exchanges instead of6.This is possible because the underlying triangular lattice has6momen-tum states and the total possible number of central-body momentum exchange directions is always twice the lattice coordination number.With12momentum exchange directions,the crystal is stable under translation along any direction and in fact can undergo free rotation.Therefore,the crystal acts very much like a solid rigid body.This rigid body can also support elastic waves—shear waves and compressional waves have been observed.

The local stability analysis of the equations of motion for the system’s linear response as as carried out in section§8is directly applied to this case.In the short-wavelength limit,the dispersion is identical to that for an ideal,viscous ?uid.However,for small wavelengths,there is a crucial di?erence,see?gure9. The imaginary part of the dispersion relations has a positive peak at about k=0.08.This implies an instability in the lattice-gas system that ultimately gives rise to the crystalline structure characterized by cell size2πk.Therefore, the linear response calculation gives a nearly quantitative prediction about the size of the emergent crystal’s cell size.The interaction ranges used in the lin-ear response calculation are r=7,19,21,26with corresponding interaction strengthsα=?2,2,2,?2,with density d=0.1.The dashed curves are the dispersion relations for an ideal,incompressible,viscous?uid presented here for comparison purposes.

2If the density of the system is increased,one does observe a transition from a hexagonally-ordered bubble phase to ordered and random stripe phases.In the context of lattice-gases, Rothman has shown some pictures similar to?gure7in an two-component immiscible lattice-gas with a short range attractive interaction and a longer range repulsive interaction[11].

10.2The Crystal Recon?guration Process

An expected phenomenon that occurs in the early stages of the crystal formation is the emergence of grain boundaries and defects.Over time,given the inherent ?uctuations of the lattice-gas dynamics,the crystal undergoes an annealing process that removes the defects and eventually produces a prefect crystal.In the two-dimensional case with a radially symmetric coarse-grained potential, the hexagonal-close-pack crystal structure emerges as just mentioned.Defect pairs with?ve and seven neighbors are observed.

An unexpected phenomenon that occurs in the exact simulation is the process by which a defect is removed.To describe this process,consider for example,an atom with?ve neighbors.It may persist in such a frustrated situation for some time.Yet what eventually occurs is that the lattice structure near this defect begins to?uctuate—“tremors”in the crystal structure are observed.That is, the other atoms in the immediate vicinity of the defect begin to vibrate about their metastable positions.The magnitude of the?uctuations increases over time.In fact,the magnitude of the?uctuations appears to grow and one may even say that the temperature of the local crystalline structure appears to rise. When a high enough local temperature of this sort is reached,the microscopic dynamics suddenly recon?gures a cluster of the atoms and the defect vanishes. The recon?guration of the atoms usually entails a small local rotation of a cluster of atoms surrounding the defect site.

One way to characterize the?uctuations that occur in the system leading to the rather sudden recon?guration of a cluster of atoms is to compare the state of the system at one time,t,to the state at some later time,t+T,by computing a hamming length.In discrete models,such as the Ising model or a Hop?eld neutral network,a hamming length is well de?ned.In a lattice-gas,one can also de?ne a hamming length,h l,and in particular we do so in the coarse-grained limit.That is,a block average of the lattice-gas number variables is taken to determine a density?eld,ρ(x).The hamming length is calculated by summing over all points of the density?eld as follows

θ(|ρ(x,t+T)?ρ(x,t)|? ).(102)

h l≡

whereθ(y)is the step function which is zero for negative y and unity for positive y and where is a small threshold value.Fiqure11shows a time series of the hamming length for a long-range lattice-gas of the type described above.The lattice size was1024×1024,block size used was8×8,and the sampling time was T=10.The reason for measuring the hamming length is that it provides a rather direct and simple way of determining the scale of the domain of atoms that participate in a crystal recon?gure event.It is interesting to?nd that the domain sizes of these recon?gures shows power law behavior,see?gure12 where give a log-log plot of the frequency of occurance of a recon?guration versue its hamming length.In?gure12we see a peak at h l 5,which is the most common background?https://www.sodocs.net/doc/7617202375.html,rger scale?uctuations occur but the probability of occurance,p,clearly drops o?according to a power law of the

form p∝1

h?α

l .In this case alpha is approximately6.3.Smaller?uctuations

also occur but these are not responsible for the recon?guration events observed during crystallization.

11Conclusion

We have presented a mean-?eld theory of lattice-gases with long-range interac-tions.We have focused on central body interactions that are mediated by mo-mentum exchange events between remote spatial sites and have used these type of interactions to model two types of physical systems:a)a?nite-temperature liquid-gas dynamical system;and b)a solid-state molecular dynamical system. The latter lattice-gas model is very compelling and is the most important result of this paper.This lattice-gas model of a crystallographic solid-body o?ers an al-ternative to traditional molecular dynamics modeling.The dynamical behavior of the lattice-gas solid is exactly computed,in that there is exact conservation of mass,momentum,and energy.A solid phase is self-consistently produced through the collective and non-linear behavior of billions of lattice-gas particles as they interact via local collisions and long-range interactions.A linear stability analysis is presented that predicts the formation of“molecules”of a character-istic intermolecular spacing,each molecule itself occupying a?nite volume and composed of thousands of lattice-gas particles.Therefore,the molecule,is not a point particle but is distributed over several lattice sites and is stable in a self-consistent way.Each molecule possesses a Lenord-Jones type potential in the coarse-grained limit.The mass of a molecule as well as its?eld are both manifestations of the spatial distribution of lattice-gas particles.A lattice-gas particle is interpreted as an informational token that composes only a small piece of the molecule and contributes to a small piece of its?eld.We have observed an annealing process where defects are removed from the crystal where there is a succession of localized vibrations that continually build to the point where a cluster of molecules around the defect can eventually undergo a recon?guration.

12Acknowledgements

I would like to thank Dr.Bruce Boghosian for his continuing collaboration, particularly those discussions concerning calculations for the linear stability analysis.I thank Dr.Norman Margolus of the MIT Laboratory of Computer Science for his help on the CAM-8implementation that has made this type of calculation practical.Also I thank Dr.Guy Seeley of the Radex Corporation for his useful discussions and help in carrying out the numerical experiments. Thanks expressed also to Donald Grantham and Dr.Robert McClatchey of Phillips Laboratory for their support of the lattice-gas basic research initiative at our laboratory.

Lattice Multipole Theorem

Lattice-gas crystallization

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