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it is generally surmised that microscopic chaos is a necessary condition for diffusive behavior of a system .
microscopic chaos here means a positive kolmogorov - sinai ( ks ) entropy , or equivalently , at least one positive lyapunov exponent .
while this seemed to be confirmed by an ingenious experiment with a brownian particle  @xcite , we showed that the same behavior on which the microscopic chaoticity of the brownian system was established also held for a manifestly nonchaotic system with zero ks entropy  @xcite , so that both systems exhibited diffusion .
, a one dimensional hard rod model  @xcite and a heavy particle in a lattice of light particles  @xcite . in both cases
there are an infinite number of moving particles , so we expect that the ks entropy is infinite , as in the infinite ideal gas  @xcite . here
we have only one moving particle , and we argued in ref .
@xcite that the ks entropy of our nonchaotic models is zero . ] in these works we used the mean square displacement ( eq .  [ e : msd ] below ) , a two time distribution function , as our definition of diffusion .
an anonymous referee of a ref .
@xcite , alerted us to a definition of diffusion ( eq .  [ e : multi ] ) which involved multitime distribution functions and the question arose whether those could distinguish between chaotic and nonchaotic models , in that all our chaotic models and none of our nonchaotic models would satisfy this definition . in this note
we show that this behavior does not occur : no difference has been found between the diffusive behavior of microscopically chaotic and nonchaotic models even in the multitime distribution functions we studied .    our models , a subset of those of ref .
@xcite , consist of a point particle moving with unit speed in two dimensions , undergoing specular reflections ( collisions ) with fixed non - overlapping scatterers .
the shapes , positions , and orientations of the scatterers in the various models are defined in the next section , and can be classified as lorentz or ( modified ) ehrenfest models . for each fixed configuration of scatterers ,
the particle s position @xmath0 is determined numerically as a function of the time @xmath1 for random initial positions @xmath2 ( which are uniformly distributed over the plane , except that they can not lie inside a scatterer ) and random initial velocity directions ( also uniformly distributed ) of the particle .
the probabilities below are defined with respect to this distribution of initial conditions .
the definition of diffusion proposed by the referee and used in this paper is stated in terms of the probability of the particle initially at the origin visiting @xmath3 regions at @xmath3 different specified times .
explicitly , a system is said to be diffusive , if , for all @xmath4 , the @xmath5 time ( including @xmath6 ) probability distributions @xmath7 satisfy the relation  @xcite : @xmath8 where @xmath9 are arbitrary times and the @xmath10 ( @xmath11 ) are arbitrary subsets of the plane .
@xmath12 is the gaussian green s function of the ( generally anisotropic ) diffusion equation in two dimensions , @xmath13 and determines the probability density of a displacement @xmath14 in time @xmath15 in the limit @xmath16 .
the scaling of the position and time by @xmath17 is based on the fact that for gaussian diffusion the mean square displacement of the particle is proportional to the time , and is a macroscopic limit . here ,
@xmath18 and @xmath19 are distances measured in the eigendirections of the diffusion tensor , while @xmath20 and @xmath21 are the corresponding eigenvalues . in the isotropic case @xmath22 and
the choice of orthogonal directions @xmath23 is arbitrary .
the components of the diffusion tensor in an arbitrary orthonormal basis @xmath24 ( so that @xmath25 ) can be related to the mean square displacement @xmath26 : @xmath27 where @xmath28 .
this expression involves the usual 2-time distribution function , that is , @xmath29 in eq .
( [ e : multi ] ) .
the mean square displacement is discussed in ref .
@xcite , where we found numerically that both chaotic and nonchaotic models exhibited gaussian diffusion .
it is conceivable that a system may have gaussian 2-time distribution functions but non - gaussian multi - time distribution functions , i.e. , for larger values of @xmath3 .
the results of this note show that also for larger @xmath3 , both chaotic and nonchaotic models appear diffusive within the accuracy of the numerical methods .
the models we consider are the same as some of those discussed in detail in ref .
@xcite . , the periodic squares rp4 and the randomly positioned squares r@xmath30
. here , the l stands for the lorentz gas , r for the random orientations of the squares , p for periodic , 4 for the linear dimension of the unit cell ( there are also 4 scatterers per unit cell ) , and @xmath30 for the infinite `` unit cell '' . ] as mentioned above , they all contain a point particle colliding with fixed non - overlapping scatterers in two dimensions .
the density is such that the scatterers cover exactly half of the available area .
the microscopically chaotic models are lorentz models , consisting of circular scatterers distributed in position either periodically or randomly over the plane , such that they do not overlap .
they are microscopically chaotic as they have a positive ks entropy .
see fig .
[ f : d ]    the microscopically nonchaotic models are ehrenfest - like models , consisting of square scatterers distributed in position either periodically or randomly over the plane , such that they do not overlap .
they are microscopically nonchaotic , with zero ks entropy . unlike the original ehrenfest model , the squares in this paper are always oriented at random .
see fig .
[ f : d ] .
( 400,200 ) ( -80,-530 )    in both the lorentz and ehrenfest periodic models , the elementary cell is square and contains four scatterers .
the random models have no preferred directions , so the diffusion is isotropic . in contrast , the periodic models have a directionality defined by the elementary cell , and the diffusion is anisotropic as in eq .
( [ e : gauss ] ) .
we investigated eq .
( [ e : multi ] ) for equally spaced times and two choices of the subsets @xmath10 of the plane :    a : :    the @xmath10 ( @xmath31 ) are all balls of    radius @xmath32 centered on the origin for    @xmath33 .
b : :    @xmath34 , where @xmath35 and    @xmath36 are two balls of radius    @xmath32 centered on randomly chosen points    @xmath37 ; @xmath38 identifies the    different members of our sample of seven configurations of the    @xmath39 , see table  [ t : conf ]
.    .positions defining the centers of the @xmath39 for case * b * , and corresponding symbols in fig .  [ f : b ] below .
[ t : conf ] [ cols="^,>,>,>,>,^",options="header " , ]     ( 400,500 ) ( -50,-300 )    ( 400,500 ) ( -50,-300 )    ( 400,500 ) ( -50,-300 )
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using standard definitions of chaos ( as positive kolmogorov - sinai entropy ) and diffusion ( that multiple time distribution functions are gaussian ) , we show numerically that both chaotic and nonchaotic systems exhibit diffusion , and hence that there is no direct logical connection between the two properties . 
 this extends a previous result for two time distribution functions .