a r X i v :c o m p -g a s /9902002v 1 25 F e
b 1999Cellular Automaton Rule184++C .
A Simple Model for the Complex Dynamics of Various Particles
Flow.
Akinori Awazu
Department of Physics Ibaraki University,Mito 310-0056,Japan Abstract A cellular automaton named Rule 184++C is proposed as a meta-model to in-vestigate the flow of various complex particles.In this model,unlike the granular pipe flow and the traffic flow,not only the free-jam phase transition but also the free-intermediate,the intermediate-jam,and the dilute-dense phase transitions ap-pear.Moreover,the freezing phenomena appear if the system contains two types of different particles.Recently,the flow of materials which consist of numerous discrete elements,for example the granular pipe flow,the traffic flow,and so on are investigated analytically,experimentally,
and numerically 2–13.They succeeded to explain phenomena in these systems,e.g,the free-jam phase transition and the 1/f αfluctuation of the local density 4,8,12,16.However,it is fact that the behavior of th
e granular pipe flow depends on materials filled in the pipe 12,16.For example,when the long range interactions like Coulomb force,the inhomogeneity of softness,or the form of materials is taken into account,the system is expected to behave in more complex manner.Now,to discuss such complex flow of various particles,we propose a simple meta-model which we name cellular automaton (CA)named Rule 184++C .The dynamics of Rule 184++C is based on that of Rule 184.The Rule 184is taken as one of the simplest models of the traffic and the granular flow.Here,in addition to the Rule
哀情口令 电影184dynamics,as the+C rules,we employ a set of simple rules for the velocity change of individual particles.The dynamics of each particle is described by the equations; v i n+1=F(v i n,v i+1
n
,d i n)(1) x i n+1=x i n+v i n+1(2) where we number the particles i from the upper part of the traffic stream to the downward. The quantities x i n and v i n are the position and the velocity of the i th particle at time step n,and d i n is the number of empty sites between i th and i+1th particle.The function
F(v i n,v i+1
n
,d i n)and the velocity v i n take0or1,whereas F(∗)obeys the following rules;
I)When d i n>1,F(v i n,v i+1
n ,d i n)=1always holds,and when d i n=0,F(v i n,v i+1
n
,0)=0
always holds.
II)When d i n=1,F(v i n,v i+1
n
,d i n=1)takes0or1which depends on the type of the par-
ticle at i.Since v i n takes0or1,the combination of(v i n,v i+1
n
)takes one of the following four patterns,(0,0),(0,1),(1,0),(1,1).For each combination,F takes the value0or1.Hence,
16types of rules for F(v i n,v i+1
n
,1)(from{F(0,0,1)=0,F(0,1,1)=0,F(1,0,1)=0, F(1,1,1)=0}to{F(0,0,1)=1,F(0,1,1)=1,F(1,0,1)=1,F(1,1,1)=1})are consid-
erd.In other words,we define16types of particles according to the function F(v i n,v i+1
n
,1). Now,we name the types of particles using the rule number C which is defined like Wolfram’s method1,
C=20F C(0,0,1)+21F C(0,1,1)+22F C(1,0,1)+23F C(1,1,1)(3) If the type of all particles is C=15,the dynamics is same as CA Rule184.The boundary condition is set periodic and the positions and the velocities of particles are set random at initial conditions.Hereafter,for thefirst,simulation results for pure systems in which all particles have same C are introduced.Also the statistical and the dynamical properties for each stationary state are discussed.Second,the simulation of mixed systems in which various C particles coexist are discussed.
In Figure1are the typical fundamental diagrams which are the relations between the
particle density of the systemρand theflow f for each C.Here,ρis defined as(the
number of the occupied sites by particles)/(the total number of sites),and f is defined as<
v i n>/(the total number of sites)where<..>means the time average.These fundamental i
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iradondiagrams are classified into following three types;(i)the two phases type(2P-type)where C=0,4,5,6,8,10,12,13,14,or15,(ii)the three phases type(3P-type)where C=2,9,or11, and(iii)the four phases type(4P-type)where C=1,3,or7.The property of the steady state for the2P-type is following;Whenρis low,all particles move at each time step,while slugs appear in the system ifρis higher than a critical value.Here,the term’slug’means an array
=0....Generally,slugs move backward in of particles which do not ,v j n=0v j+1
n
the traffic stream keeping the spatial pattern of them invariant.Each of them is kept by the balance between the incoming free-flow particles from the upward and the outgoing free-flow particles to the downward.As such,withρincreases,the phases transition from the’free-flow state’(without slug)to the’jam-flow state’(with slugs)occurs.This property is qualitatively same as known results of recent traffic and granularflow models2–12.In particular,the dynamics of the pure particle systems with C=12(F12(0,0,1)=0,F12(0,1,1)=0, F12(1,0,1)=1,F12(1,1,1)=1)is equivalent to the dynamics of the deterministic traffic flow model proposed by Takayasu and Takayasu8.Different from such systems,the3P-type systems include the parameter region of the’intermediate-flow state’.At the intermediate ρvalues between those of the free-flow state and of the jam-flow state,the third state which is different from the former two states,takes place as shown in Fig.2(b)and(e).Among the3P-type systems,we discuss the pure systems with C=11and C=9.Figure2(a) (b)and(c)show the space-time evolutions of the stationary states of the pure systems with C=11(F11(0,0,1)=1,F11(0,1,1)=1,F11(1,0,1)=0,F11(1,1,1)=1).They respectively correspond to(a)the free-flow state,(b)the intermediate-flow state and(c)the jam-flow state. Here,dots represent individual
particles where black dots indicate v=0particles and gray dots are v=1particles.The behaviors of particles in(a)and(c)of Fig.2are qualitatively same as the space-time evolutions of the2P-type systems.In the free-flow state,more than
two successive empty sites appear in front of all particles because the particle density is low.
In the jam-flow state,slugs appear and survive stably because the particle density is high. Different from these two kinds of states,in the intermediated-flow state,unstable slugs and more than two successive empty sites coexist(Fig.2(b)).Figure2(d)(e)and(f)show the space-time evolutions of the stationary states of the system with the C=9particles (F9(0,0,1)=1,F9(0,1,1)=0,F9(1,0,1)=0,F9(1,1,1)=1).There are two types of slugs. Isolated slugs consist of only one particle j with v j n=0,whereas large slugs consist of more than two particles with v j n=0,v j+1
=0....Moreover the backward propagation velocity of large slugs is slower than that of isolated ones.In the jam-flow state(Fig.2(f)),some large slugs remain stable in the system.In the intermediate-flow state,however,large slugs are unstable and repeat creation and annihilation irregularly in space
and time(Fig.2(e)).
Next,we discuss the properties of the4P-type systems.As an example,the system with C=3particles(F3(0,0,1)=1,F3(0,1,1)=1,F3(1,0,1)=0,F3(1,1,1)=0)is considered. Infigure3are typical space-time evolutions for several typical densities.When densityρ
is low(0<ρ<1
3
<ρ<2
5
,the system is completelyfilled with dilute slugs.Ifρincreases more
(2
3
),unlike the2P-type systems,different type of slugs from the dilute slugs appear
(Fig.3(c)).In these slugs,the spatial pattern is periodic with the unit v j n=1and v j+1
n
=0 similarly in the dilute slug.However,in this case,the gaps in front of particles j and j+1
repeat(d j n=0,d j+1
n =1)and(d j n+1=1,d j+1
n+1
生育健康网=0)by turn.Moreover,the direction
of movement of these slugs is downward,and these slugs are surrounded by dilute slugs. Now,we name these unusual slugs’advancing slugs’and this state the’advancing jam-flow
state’.Moreover,in the advancing jam-flow state,theflow increases in proportional toρ,in which sense the advancing jam-flow state is similar to the free-flow state.When the density increases more(2
in dilute slugs synchronize each other to oscillate between0and1(Fig.4(a)).Therefore, f oscillates with
large amplitude(Fig.4(b)).iii)The chaoticflow regime(C=6,or9): The space time evolution of particles and the time evolution of f are chaotic.(Examples are shown in Fig.2(e),(f),and Fig.5(a).)In particular,in the C=9particles system,the flow has1/ffluctuation near the critical density(ρ∼0.38)of the phase transition between the intermediate-flow state and the jam-flow state(Fig.5(b)).Both type of C=6and C=9)share the symmetric dynamics F C(0,0,1)=F C(1,1,1)=F C(0,1,1)= F C(1,0,1).In other words,only such systems that contain particles with symmetric rules behave chaotic if the system is pure.
Finally,we focus the mixed systems in which two values of C particles are mixed and compare the behavior of them to those of the pure systems.Figure6are the typical funda-mental diagrams of two types of pure systems and that of mixed systems of these two types of particles.Here,the ratio of two types of particles is1:1.Almost all the cases,the relations like Fig.6(a)are realized.However,the relations like Fig.6(b)also are realized for some cases.Here,theflow of the mixed systems is smaller than that of respective pure particles systems.To discuss them,as an example,we consider mixed systems with C=2and C=4 particles(For C=2:F2(0,0,1)=0,F2(0,1,1)=1,F2(1,0,1)=0,F2(1,1,1)=0.For C=4:F4(0,0,1)=0,F4(0,1,1)=0,F4(1,0,1)=1,F4(1,1,1)=0).Figure7(a)is a fundamental diagram of the almost pure C=2particles system except one C=4particle inside it.Compared with the pure C=2particl
es system,theflow is drastically little, and in particular,noflow forρ≥0.5.The origin of such behavior is following.Whenρis sufficiently large,that is,most of the gaps d i n between successive particles are not larger than
,1).Once the C=4 1,the dynamics of each particle mainly obeys the function F C(v i n,v i+1
n
particle stops,this particle remains stationary if d i n remain not larger than1.Moreover,in such d i n,C=2particle remains stationary when the preceding particle does not move.In other words,a C=4particle works as the coagulant of C=2particles,and change the characters of the whole system.Such relation appears also in different pairs of , C=6and C=4forρ>0.66(Fig.7(b)),and C=2and C=12.The mechanism of the
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