Economics of Queueing
Alistair Tucker∗
Complexity Science DTC, University of Warwick.
Matthew Turner†
Department of Physics, University of Warwick.
(Dated: June 22, 2010)
The N -class totally asymmetric simple exclusion process (TASEP) is a model that has a number of
interesting features, irrespective of any particular physical process it could be said to approximate.
As a nonequilibrium system of many agents, each accorded privilege according to class, it has
something in common with real-world political and economic systems. In that context we examine
the TASEP as implemented on a one-dimensional lattice with periodic boundary conditions. This
is a tractable model and a number of analytic results are already available. We further derive
expressions for each class’s current, with which we identify its productivity, in terms of the overall
class structure. En route we make conjectures as to the analytic form of two-point nearest-neighbour
correlations in the steady state, with excellent support from numerical results. Provable constraints
on the set of two- and three-point nearest-neighbour correlations are also noted.
I.
INTRODUCTION
Some political thinkers have sought the motive forces
of society in its class structure. The work that resulted [1]
has certainly been influential, but is generally perceived,
despite its authors’ hopes, to fall short of a scientific theory.
The questions that provoked that treatment remain,
albeit in a form modified by changed circumstance. And
any modern-day economist also would hope to be able to
respond in a way that is defensible within some scientific
framework.
To what extent do inequalities in society affect its capacity to improve the lives of its members?
Consider the judgment of the school of ‘trickle-down
economics’, providing the justification for policy decisions in Britain (and elsewhere) during the nineteeneighties. Certainly one must admit the theoretical possibility of an unequal society that leaves every member
better off than they would have been under its egalitarian alternative. At that time and in that context, it was
not uncommon to hear the claim that such a possibility
was being made reality.
At the other extreme we have those who defend the
thesis that in an unequal society, every member of society
suffers [2]. They cite, for example, studies suggesting
that even the rich die slightly younger, and lead slightly
worse lives, in the US and the UK than in more equal
Sweden or Japan.
(Policy makers would do well also to note research suggesting that it is inequality itself that causes much misery,
regardless of absolute material wealth [3].)
The truth quite likely lies somewhere between those
two extremes. What sort of models can we dream up
∗
†
agjf.tucker@warwick.ac.uk
m.s.turner@warwick.ac.uk
to give us intuition as to how its value might depend on
parameters of policy?
In the spirit of ‘Complexity Economics’ [4] we examine one model in particular that takes account of the
multiplicity and heterogeneity of agents, and, mirroring
reality, gives rise to a ‘nonequilibrium’ process (one that
is not reversible).
Key to the metaphor is that every individual belongs
to one of an ordered set of classes, and enjoys certain
privileges over members of those classes lower than his
own.
At the microscopic level of individuals, the mechanism
of interaction as specified by our model is far from any
real-world process that we might claim it to parallel. But
that can be regarded as the price of the model’s tractability, which allows us to examine not only macroscopic behaviour, but also the deeper reasons for that behaviour.
In that sense this is an example of the ‘bottom-up’ approach to economics.
In this report we present results relating the classcomposition of our theoretical society to aspects of its
behaviour in the steady state. In particular we derive
exact expressions for the flux of each class, a quantity
with which we identify ‘productivity’. It becomes clear
that, within this model, division of the labour force into
classes of varying proportions has precisely no impact on
overall productivity.
The flux expressions are shown to be a consequence
of particular relations involving two-point nearestneighbour correlations on the one-dimensional lattice.
Those are easily derived; but also presented are expressions for the whole set of two-point correlations, confirmed by simulation data but not so easily derived. An
attempt to do so by relating them to the three-point correlations is ultimately unsuccessful, although it does substantially reduce the dimensionality of the space to which
the set of two-point and three-point correlations may possibly belong.
We also spend some time on diffusion, the second mo-
2
ment of ‘productivity’, in the context of results previously established in the field. The route to an analytical
treatment is pointed out, and some intriguing simulation
results are presented.
The prospects for future work are good. In particular
it is encouraging to find that relations such as those presented herein have such simple forms. This would seem
to suggest that formal proof of their veracity need not be
far off, especially given strong results from previous work
in matrix representations and the like [5]. It also suggests
to me that study of the effects of small perturbations in
microscopic behaviour might be quite generally feasible,
and a route toward linking variation in the model with
variation in the real world.
II.
clusion process (TASEP) as implemented on a large ringshaped lattice with finite densities (ρ1 , ρ2 , . . .) of individuals belonging to different classes.
Individuals try to hop from point to point clockwise
around the ring at random (exponentially distributed)
intervals.
In the single-class case, illustrated by Fig. 1, the only
additional complication is that the progress of one individual may be blocked by another occupying the site
ahead. The small arrows in the figure indicate possible
hops, those that are not blocked. Any such transition
will occur in the next instant dt with probability dt.
THE MODEL
The asymmetric simple exclusion process (ASEP) in
one dimension has been studied extensively within the
mathematical and physical communities.
The earliest paper that we cite [6] was published in
1985, and considers it a model for a ‘lattice gas’. In
1998 an article in which slower second-class individuals appeared [7] found it most natural to speak of the
subject in terms of traffic—fast-moving cars and slowmoving trucks. The same model can be formulated as a
zero-range process [8], or a growing interface in (1 + 1)
dimensions [9]. More recent work has exposed deep connections to queueing theory [10].
It is also noteworthy for its capacity to develop shock
waves (the mass density on the macroscopic level is described by the inviscid Burgers equation). Second-class
individuals appear originally to have been introduced as
a mathematical convenience, a means of studying shock
profiles [11].
In the two-class case, illustrated by Fig. 2, first-class
individuals (blue) may also overtake second-class individuals (red), forcing them backwards.
FIG. 1. Single-class TASEP on a 48-ring
FIG. 3. Three-class TASEP on a 48-ring
Our concern is with the totally asymmetric simple ex-
FIG. 2. Two-class TASEP on a 48-ring
The three-class case, illustrated by Fig. 3, is the ob-
3
vious extension, in which second-class individuals (red)
may also overtake third-class individuals (yellow). Naturally these rules extend to any number of classes without
difficulty.
III.
ANALYSIS AND RESULTS
We have described a Markov chain. By the PerronFrobenius Theorem, it has a unique stationary distribution over configurations (also called its steady state).
We say that a route exists from configuration A to configuration B if the generator matrix has a non-zero entry
so that a jump may occur directly from A to B. It should
be clear that for the TASEP the existence of a route from
A to B does not imply a route from B to A. The chain
does not obey detailed balance (it is not reversible) and
for this reason physicists call its stationary distribution
a nonequilibrium steady state.
We shall be concerned only with behaviour in the
steady state. A key result is that in the steady state
of the single-class model, every configuration (consistent
with the conservation of individuals) is equally likely.
It is not hard to convince ourselves of the truth of this.
In Fig. 1 we count nine routes out of the configuration.
They are the transitions marked with double-headed arrows, and we find one at the front-end of each of the nine
clusters. Conditional on this configuration C , each of
these transitions has probability dt of occurring during
instant of time dt.
Equally one can count nine routes into the configuration, one at the back-end of each of the nine clusters.
Conditional on the associated previous configuration Cj0 ,
each has probability dt of occurring during instant of time
dt.
For every configuration we shall find this same balance between the number of routes in and the number of
routes out, being simply the number of clusters nc in the
configuration.
So, from the Master Equation, we know that the stationary distribution π ∗ satisfies
nc
X
j=1
π ∗ (C ) dt =
nc
X
π ∗ Cj0 dt.
j=1
This is solved by the uniform distribution in which
π ∗ (C ) = π ∗ (C 0 )
for all C and C 0 .
In what follows we adopt a notation such that every
site on the lattice can be labelled with some type. So
let us denote individuals of the first class, second class,
etc. by 1, 2, etc., and holes by ∅. Then the set of types
is the union {1, 2, . . . , ∅}.
Two further intuitions, related to one another, pertain
to the symmetry and grouping of types.
Firstly note, for the single-class (two-type) case,
the symmetry between individuals (type 1) and holes
(type ∅). Holes, moving anticlockwise, have exactly the
same dynamics as do individuals moving clockwise. The
indicative arrows in Fig. 1 are double-headed to express
this symmetry—a clockwise-moving individual merely
exchanges places with an anticlockwise-moving hole.
Secondly note the implications of grouping types in,
say, the two-class (three-type) model. A member of the
first class (type 1) is blind to the difference between a
second-class individual (type 2) and a hole (type ∅). So,
for example, the behaviour of the set of first-class individuals in Fig. 2 will be indistinguishable from the behaviour of the set of first-class individuals in Fig. 1. In
particular the same distribution of first-class individuals,
the uniform distribution with equal weight given to each
configuration, will be found in the steady state.
Equally consider the world according to a hole (type ∅),
a world composed only of holes and not-holes. It is blind
to the difference between a first-class (type 1) and a
second-class (type 2) individual, since neither type causes
it the slightest impediment in its progress anticlockwise.
If a first-class individual were to hop over a second-class
individual at any point (or vice versa, depending on your
point of view), it is of no consequence to the hole; the
world would look the same afterwards as it did before.
Again we have the same distribution of holes found in
the steady state as in the single-class (two-type) model,
the uniform distribution with equal weight given to every
configuration.
The intuition is easily extended to models with more
classes. For a type at one extreme of the class structure—
so either type 1 (first-class individuals) or type ∅
(holes)—we may group all the remaining types together
and consider the resultant system as a simple two-type
model.
(From this logic we already see why it cannot be that
the class-composition of a society will have any effect on
its overall current, or ‘productivity’. This is a quantity
that may be observed without taking any account of the
difference between individuals of the various classes.)
For types in between the extremes (being classes of
individual that are not the first) we may apply a similar, but not identical, line of argument. An nth-class
(n 6= 1) individual is blind to the differences between
higher types (higher classes of individual) and also to
the differences between lower types (lower classes of individual and holes). But it needs to consider its own class
as separate from either group, since members of its own
class, unlike higher types, may not overtake, but unlike
lower types, may not be overtaken.
So the minimal model able to describe the dynamics of
an nth-class individual is the three-type model in which
higher classes are grouped into type 1 while holes and
lower classes are grouped into type ∅.
Without loss of generality then, we focus on the twoclass (three-type) model in what follows.
(Note that we have made a slight departure from the
common notation by adopting ∅ instead of 0 for holes.
Our hope is to avoid the suggestion of an ordering rela-
4
tion 0, 1, 2, . . . when it should be clear that the appropriate ordering is in fact 1, 2, . . . , ∅.)
A.
Correlations
Consider the single-class model with m individuals on
a ring-lattice of size n. Evolution of the chain (from some
initial condition) may be described completely by an aun
tonomous system of m
ordinary differential
equations
n
describing the probabilities of the m
configurations.
(The probabilities must sum to unity so in fact the
n
system of equations has only m
−1 degrees of freedom.)
We adopt the notation [12] of binary variables {τi }
such that τi = 1 if site i is occupied by an individual
and τi = 0 if site i is empty. Angular brackets denote an
average over the ensemble of all possible histories, so, for
example, hτi i is the ‘density’ at site i.
For some configuration in which the m individuals occupy sites j1 , j2 , . . . , jm , we may write its probability as
a correlation,
As we shall see, it is surprising the extent to which twopoint nearest-neighbour correlations do follow a simple
pattern, despite their depth in the hierarchy.
We may simplify our equations by taking our averages
over space as well as over the ensemble of possible histories. Thus we have the following notation for densities,
n
h1i =
1X
hτi i
n i=1
h∅i =
1X
h1 − τi i
n i=1
n
and for nearest-neighbour two-point correlations,
n
h1 1i =
1X
hτi τi+1 i
n i=1
n
1X
h1 ∅i =
hτi (1 − τi+1 )i
n i=1
n
h∅ 1i =
1X
h(1 − τi ) τi+1 i
n i=1
h∅ ∅i =
1X
h(1 − τi ) (1 − τi+1 )i
n i=1
P(C ) = hτj1 τj2 . . . τjm i
n
But correlations such as this are intricate and not particularly helpful. We might like to see simpler and more
informative correlations appear in their place. Two-point
nearest-neighbour correlations, for example, are useful
for working out current (‘productivity’).
In [12] it is explained how there exists a hierarchy of
dependencies in the set of nearest-neighbour correlations.
Noted in particular are the evolution of densities in terms
of two-point nearest-neighbour correlations,
d hτi i
= hτi−1 (1 − τi )i − hτi (1 − τi+1 )i
dt
etc.
For the two-class (three-type) models in which we are
interested, it will be necessary to extend the notation a
little. We introduce binary variables {σi } to indicate the
presence of second-class individuals, just as {τi } indicate
the presence of first-class individuals.
So for the two-class system, we have densities
n
(1)
and of two-point nearest-neighbour correlations in terms
of three-point nearest-neighbour correlations,
On a lattice of finite size, of course, we do not quite
have an infinite hierarchy. Our system
of ODEs need
n
never grow to a size greater than m
.
1X
hτi i
n i=1
h2i =
1X
hσi i
n i=1
n
d hτi τi+1 i
= hτi−1 (1 − τi ) τi+1 i − hτi τi+1 (1 − τi+2 )i .
dt
(2)
Somewhat discouragingly we are told that
Evolution of hτi i requires the knowledge of
hτi τi+1 i which itself requires the knowledge of
hτi−1 τi+1 i and hτi−1 τi τi+1 i so that the problem is intrinsically an N -body in the sense
that the calculation of any correlation function requires the knowledge of all the others. This is a situation quite common in equilibrium statistical mechanics where, although
one can write relationships between different
correlation functions, there is an infinite hierarchy of equations which in general makes
the problem intractable.
h1i =
n
1X
h∅i =
h1 − τi − σi i
n i=1
1.
Provable Correlations
From
hτi i = hτi [τi+1 + σi+1 + (1 − τi+1 − σi+1 )]i
= hτi τi+1 i + hτi σi+1 i + hτi (1 − τi+1 − σi+1 )i
we derive
h1i = h1 1i + h1 2i + h1 ∅i
(3)
Other relationships of the same type may be derived
in the same way, and should be obvious.
Earlier we presented arguments designed to convince
that the distribution of first-class individuals is uniform
5
in the steady state, with each configuration having equal
probability. On a suitably large ring, it is essentially
Bernoulli, so we can write,
h1i = ρ1
h1 1i = ρ1 ρ1
In conjunction with Eq. (3), these imply that
The two sets of numerical results agree in supporting
the conjectures here presented in Table I regarding the
values of nearest-neighbour two-point correlations.
Tables III and IV display the evidence.
The arguments given thus far are sufficient to derive
the expressions for h1 1i and h0 0i. However it does not
appear be entirely trivial to explain the other correlations
in Table I.
3.
h1 2i + h1 ∅i = ρ1 (1 − ρ1 ) = ρ1 (ρ2 + ρ∅ )
while a parallel line of reasoning gives us that
h2 1i + h∅ 1i = ρ1 (1 − ρ1 ) = ρ1 (ρ2 + ρ∅ )
(5)
Similarly we may write,
h∅i = ρ∅
h∅ ∅i = ρ∅ ρ∅
giving us
h1 ∅i + h2 ∅i = ρ∅ (1 − ρ∅ ) = ρ∅ (ρ1 + ρ2 )
(6)
h∅ 1i + h∅ 2i = ρ∅ (1 − ρ∅ ) = ρ∅ (ρ1 + ρ2 )
(7)
and
Further, Eqs. (4) and (6) together give us an expression
that will prove useful in calculating currents,
h2 ∅i − h1 2i = ρ2 (ρ∅ − ρ1 )
(8)
while Eqs. (5) and (7) can be combined into
h∅ 2i − h∅ 1i = ρ2 (ρ∅ − ρ1 )
Higher-Order Correlations
(4)
(9)
We have already seen in Eq. (2) an indication of how
we might relate the evolution of two-point correlations to
three-point correlations.
Naturally in the two-class (three-type) case the equations become more intricate. Nonetheless it is quite possible to derive expressions for the rate of change of each of
the nine (spatially averaged) two-point nearest-neighbour
correlations in terms of the twenty-seven (spatially averaged) three-point nearest-neighbour correlations.
For example the fact that
d hσi τi+1 i
= hτi σi+1 i + hσi−1 (1 − τi − σi )τi+1 i
dt
− hτi−1 σi τi+1 i − hσi τi+1 (1 − τi+1 )i
allows us to write
d h2 1i
= h1 2i + h2 ∅ 1i − h1 2 1i − h2 1 2i − h2 1 ∅i
dt
We further expand the two-point correlation h1 2i in
that equation in terms of three point correlations, in a
similar way to Eq. (3), giving us
d h2 1i
= h1 1 2i + h∅ 1 2i + h2 ∅ 1i − h1 2 1i − h2 1 ∅i
dt
or alternatively
2.
d h2 1i
= h1 2 2i + h1 2 ∅i + h2 ∅ 1i − h2 1 2i − h2 1 ∅i
dt
Observed Correlations
We created computer code to sample from the steady
state of an arbitrarily-composed system, using the algorithm described in [5]. We also wrote code to simulate
the dynamics directly.
Table II shows expressions derived for the whole of the
set of nearest-neighbour two-point correlations.
TABLE II. Evolution of two-point correlations in terms of
three-point correlations.
TABLE I. Observed values of the nearest-neighbour two-point
correlations on a large ring in its steady state
Correlation
h1 1i
h1 2i
h1 ∅i
h2 1i
h2 2i
h2 ∅i
h∅ 1i
h∅ 2i
h∅ ∅i
Value
ρ1 ρ1
ρ1 ρ2 (1 − ρ∅ )
ρ1 ρ∅ (1 + ρ2 )
ρ1 ρ2
ρ2 (ρ2 + ρ1 ρ∅ )
ρ2 ρ∅ (1 − ρ1 )
ρ1 ρ∅
ρ2 ρ∅
ρ∅ ρ∅
Correlation
h1 1i
h1 2i
h1 ∅i
h2 1i
h2 2i
h2 ∅i
h∅ 1i
h∅ 2i
h∅ ∅i
Rate of Change
h1 2 1i + h1 ∅ 1i − h1 1 2i − h1 1 ∅i
h1 ∅ 2i + h1 1 2i − h1 2 1i − 2 h1 2 ∅i
h1 1 ∅i + 2 h1 2 ∅i − h1 ∅ 1i − h1 ∅ 2i
h1 1 2i + h∅ 1 2i + h2 ∅ 1i − h1 2 1i − h2 1 ∅i
h2 ∅ 2i + h2 1 2i − h1 2 2i − h2 2 ∅i
h2 ∅ ∅i + h2 1 ∅i − 2 h1 2 ∅i − h∅ 2 ∅i
h1 1 ∅i + h2 1 ∅i − h1 ∅ 1i − h2 ∅ 1i − h∅ 1 2i
h∅ 1 2i + h2 ∅ 1i + h2 ∅ ∅i − h1 ∅ 2i − h∅ 2 ∅i
h∅ 1 ∅i + h∅ 2 ∅i − h1 ∅ ∅i − h2 ∅ ∅i
We may write these in matrix form, as below.
6
TABLE III. Twenty density vectors (ρ1 , ρ2 , ρ∅ ) were drawn at random from the valid region of the plane ρ1 + ρ2 + ρ∅ = 1. For
each we used the sampling procedure described in [5] to generate N = 20 independent samples of the 100 000-ring in its steady
state. Using these we made estimates of the underlying mean µ̂ and associated error bars √1N σ̂ for each of the nine two-point
nearest-neighbour correlations. We have extremely good agreement with the proposed analytic expressions. Coloured in red
are the cases that lie outside a 90% confidence interval—as one might expect this happens for about one in ten.
ρ1
0.00
0.02
0.04
0.08
0.12
0.13
0.19
0.22
0.25
0.27
0.32
0.32
0.46
0.55
0.56
0.62
0.69
0.74
0.79
0.80
ρ2
0.06
0.89
0.05
0.54
0.25
0.48
0.28
0.16
0.65
0.41
0.30
0.39
0.00
0.20
0.35
0.23
0.22
0.01
0.17
0.05
ρ∅
0.94
0.09
0.91
0.38
0.63
0.39
0.53
0.62
0.10
0.32
0.38
0.29
0.54
0.25
0.09
0.15
0.09
0.25
0.04
0.15
ρ1 ρ1
0.0000
0.0004
0.0016
0.0064
0.0144
0.0169
0.0361
0.0484
0.0625
0.0729
0.1024
0.1024
0.2116
0.3025
0.3136
0.3844
0.4761
0.5476
0.6241
0.6400
µ̂h1 1i
0.0000
0.0004
0.0016
0.0064
0.0144
0.0169
0.0361
0.0484
0.0625
0.0729
0.1025
0.1023
0.2114
0.3023
0.3137
0.3845
0.4761
0.5473
0.6240
0.6402
√1 σ̂h1 1i
N
0.00 × 10−4
0.15 × 10−4
0.29 × 10−4
0.61 × 10−4
0.72 × 10−4
0.79 × 10−4
1.07 × 10−4
1.10 × 10−4
0.84 × 10−4
1.22 × 10−4
1.40 × 10−4
1.54 × 10−4
1.77 × 10−4
1.42 × 10−4
2.01 × 10−4
1.95 × 10−4
1.18 × 10−4
1.01 × 10−4
1.50 × 10−4
0.98 × 10−4
ρ1 ρ2 (1 − ρ∅ )
0.0000
0.0162
0.0002
0.0268
0.0111
0.0381
0.0250
0.0134
0.1463
0.0753
0.0595
0.0886
0.0000
0.0825
0.1784
0.1212
0.1381
0.0056
0.1289
0.0340
µ̂h1 2i
0.0000
0.0162
0.0002
0.0268
0.0111
0.0381
0.0250
0.0133
0.1462
0.0755
0.0595
0.0887
0.0000
0.0826
0.1784
0.1212
0.1381
0.0055
0.1290
0.0338
√1 σ̂h1 2i
N
0.00 × 10−4
0.42 × 10−4
0.09 × 10−4
0.75 × 10−4
0.77 × 10−4
1.07 × 10−4
1.28 × 10−4
0.92 × 10−4
1.29 × 10−4
1.46 × 10−4
1.37 × 10−4
1.39 × 10−4
0.00 × 10−4
0.89 × 10−4
2.00 × 10−4
1.51 × 10−4
1.05 × 10−4
0.28 × 10−4
1.55 × 10−4
0.66 × 10−4
ρ1 ρ∅ (1 + ρ2 )
0.0000
0.0034
0.0382
0.0468
0.0945
0.0750
0.1289
0.1582
0.0412
0.1218
0.1581
0.1290
0.2484
0.1650
0.0680
0.1144
0.0758
0.1868
0.0370
0.1260
µ̂h1 ∅i
0.0000
0.0034
0.0383
0.0468
0.0945
0.0750
0.1289
0.1583
0.0413
0.1216
0.1580
0.1290
0.2486
0.1651
0.0680
0.1143
0.0758
0.1872
0.0370
0.1259
√1 σ̂h1 ∅i
N
0.00 × 10−4
0.34 × 10−4
0.34 × 10−4
0.83 × 10−4
1.12 × 10−4
1.05 × 10−4
1.55 × 10−4
1.15 × 10−4
0.95 × 10−4
1.45 × 10−4
1.64 × 10−4
1.84 × 10−4
1.77 × 10−4
1.21 × 10−4
0.84 × 10−4
0.88 × 10−4
0.99 × 10−4
1.00 × 10−4
0.35 × 10−4
1.00 × 10−4
ρ1
0.00
0.02
0.04
0.08
0.12
0.13
0.19
0.22
0.25
0.27
0.32
0.32
0.46
0.55
0.56
0.62
0.69
0.74
0.79
0.80
ρ2
0.06
0.89
0.05
0.54
0.25
0.48
0.28
0.16
0.65
0.41
0.30
0.39
0.00
0.20
0.35
0.23
0.22
0.01
0.17
0.05
ρ∅
0.94
0.09
0.91
0.38
0.63
0.39
0.53
0.62
0.10
0.32
0.38
0.29
0.54
0.25
0.09
0.15
0.09
0.25
0.04
0.15
ρ1 ρ2
0.0000
0.0178
0.0020
0.0432
0.0300
0.0624
0.0532
0.0352
0.1625
0.1107
0.0960
0.1248
0.0000
0.1100
0.1960
0.1426
0.1518
0.0074
0.1343
0.0400
µ̂h2 1i
0.0000
0.0178
0.0020
0.0432
0.0301
0.0623
0.0532
0.0352
0.1625
0.1104
0.0958
0.1249
0.0000
0.1101
0.1957
0.1425
0.1518
0.0074
0.1343
0.0400
√1
N
σ̂h2 1i
0.00 × 10−4
0.35 × 10−4
0.22 × 10−4
0.93 × 10−4
0.84 × 10−4
1.18 × 10−4
1.18 × 10−4
1.08 × 10−4
0.83 × 10−4
1.54 × 10−4
1.52 × 10−4
1.67 × 10−4
0.00 × 10−4
1.40 × 10−4
1.86 × 10−4
1.30 × 10−4
1.10 × 10−4
0.24 × 10−4
1.49 × 10−4
0.67 × 10−4
ρ2 (ρ2 + ρ1 ρ∅ )
0.0036
0.7937
0.0043
0.3080
0.0814
0.2547
0.1066
0.0474
0.4388
0.2035
0.1265
0.1883
0.0000
0.0675
0.1401
0.0743
0.0621
0.0020
0.0343
0.0085
µ̂h2 2i
0.0036
0.7938
0.0043
0.3080
0.0810
0.2546
0.1068
0.0474
0.4388
0.2036
0.1268
0.1882
0.0000
0.0672
0.1402
0.0743
0.0621
0.0020
0.0343
0.0085
√1
N
σ̂h2 2i
0.36 × 10−4
0.83 × 10−4
0.53 × 10−4
1.68 × 10−4
1.62 × 10−4
2.45 × 10−4
1.60 × 10−4
1.01 × 10−4
1.32 × 10−4
1.48 × 10−4
1.74 × 10−4
1.69 × 10−4
0.00 × 10−4
1.11 × 10−4
1.92 × 10−4
1.12 × 10−4
0.95 × 10−4
0.24 × 10−4
1.51 × 10−4
0.68 × 10−4
ρ2 ρ∅ (1 − ρ1 )
0.0564
0.0785
0.0437
0.1888
0.1386
0.1629
0.1202
0.0774
0.0488
0.0958
0.0775
0.0769
0.0000
0.0225
0.0139
0.0131
0.0061
0.0007
0.0014
0.0015
µ̂h2 ∅i
0.0564
0.0784
0.0437
0.1888
0.1389
0.1631
0.1200
0.0774
0.0487
0.0960
0.0774
0.0770
0.0000
0.0226
0.0141
0.0131
0.0061
0.0006
0.0014
0.0015
√1
N
σ̂h2 ∅i
0.36 × 10−4
0.62 × 10−4
0.56 × 10−4
1.72 × 10−4
1.41 × 10−4
2.11 × 10−4
1.28 × 10−4
1.09 × 10−4
0.92 × 10−4
1.24 × 10−4
1.29 × 10−4
1.68 × 10−4
0.00 × 10−4
1.00 × 10−4
0.85 × 10−4
0.64 × 10−4
0.76 × 10−4
0.14 × 10−4
0.23 × 10−4
0.27 × 10−4
ρ1
0.00
0.02
0.04
0.08
0.12
0.13
0.19
0.22
0.25
0.27
0.32
0.32
0.46
0.55
0.56
0.62
0.69
0.74
0.79
0.80
ρ2
0.06
0.89
0.05
0.54
0.25
0.48
0.28
0.16
0.65
0.41
0.30
0.39
0.00
0.20
0.35
0.23
0.22
0.01
0.17
0.05
ρ∅
0.94
0.09
0.91
0.38
0.63
0.39
0.53
0.62
0.10
0.32
0.38
0.29
0.54
0.25
0.09
0.15
0.09
0.25
0.04
0.15
ρ1 ρ∅
0.0000
0.0018
0.0364
0.0304
0.0756
0.0507
0.1007
0.1364
0.0250
0.0864
0.1216
0.0928
0.2484
0.1375
0.0504
0.0930
0.0621
0.1850
0.0316
0.1200
µ̂h∅ 1i
0.0000
0.0018
0.0364
0.0304
0.0755
0.0508
0.1007
0.1363
0.0250
0.0867
0.1216
0.0929
0.2486
0.1375
0.0506
0.0930
0.0621
0.1853
0.0317
0.1198
√1 σ̂h∅ 1i
N
0.00 × 10−4
0.30 × 10−4
0.40 × 10−4
0.83 × 10−4
1.01 × 10−4
1.05 × 10−4
1.43 × 10−4
1.33 × 10−4
1.17 × 10−4
2.04 × 10−4
1.57 × 10−4
1.44 × 10−4
1.77 × 10−4
1.54 × 10−4
0.69 × 10−4
1.45 × 10−4
0.86 × 10−4
0.98 × 10−4
0.63 × 10−4
0.95 × 10−4
ρ2 ρ∅
0.0564
0.0801
0.0455
0.2052
0.1575
0.1872
0.1484
0.0992
0.0650
0.1312
0.1140
0.1131
0.0000
0.0500
0.0315
0.0345
0.0198
0.0025
0.0068
0.0075
µ̂h∅ 2i
0.0564
0.0800
0.0456
0.2052
0.1578
0.1872
0.1482
0.0993
0.0650
0.1309
0.1137
0.1131
0.0000
0.0502
0.0314
0.0344
0.0198
0.0025
0.0068
0.0076
√1 σ̂h∅ 2i
N
0.36 × 10−4
0.64 × 10−4
0.53 × 10−4
1.62 × 10−4
1.55 × 10−4
2.23 × 10−4
1.69 × 10−4
0.92 × 10−4
1.21 × 10−4
1.62 × 10−4
1.72 × 10−4
1.35 × 10−4
0.00 × 10−4
1.11 × 10−4
0.66 × 10−4
1.07 × 10−4
0.73 × 10−4
0.31 × 10−4
0.57 × 10−4
0.52 × 10−4
ρ∅ ρ∅
0.8836
0.0081
0.8281
0.1444
0.3969
0.1521
0.2809
0.3844
0.0100
0.1024
0.1444
0.0841
0.2916
0.0625
0.0081
0.0225
0.0081
0.0625
0.0016
0.0225
µ̂h∅ ∅i
0.8836
0.0082
0.8280
0.1444
0.3966
0.1519
0.2811
0.3844
0.0100
0.1024
0.1446
0.0840
0.2914
0.0623
0.0080
0.0226
0.0081
0.0622
0.0016
0.0226
√1 σ̂h∅ ∅i
N
0.36 × 10−4
0.48 × 10−4
0.72 × 10−4
1.61 × 10−4
1.78 × 10−4
2.02 × 10−4
1.66 × 10−4
1.48 × 10−4
0.61 × 10−4
1.66 × 10−4
1.55 × 10−4
1.57 × 10−4
1.77 × 10−4
1.30 × 10−4
0.22 × 10−4
0.84 × 10−4
0.53 × 10−4
0.98 × 10−4
0.31 × 10−4
0.94 × 10−4
7
TABLE IV. For the same twenty density vectors we simulated the dynamics on the 100 000-ring directly, generating another
N = 20 samples for each. Again our results show very good agreement, thus we also have some confirmation that our various
sampling and simulation procedures are accurate in theory and correctly programmed in practice.
ρ1
0.00
0.02
0.04
0.08
0.12
0.13
0.19
0.22
0.25
0.27
0.32
0.32
0.46
0.55
0.56
0.62
0.69
0.74
0.79
0.80
ρ2
0.06
0.89
0.05
0.54
0.25
0.48
0.28
0.16
0.65
0.41
0.30
0.39
0.00
0.20
0.35
0.23
0.22
0.01
0.17
0.05
ρ∅
0.94
0.09
0.91
0.38
0.63
0.39
0.53
0.62
0.10
0.32
0.38
0.29
0.54
0.25
0.09
0.15
0.09
0.25
0.04
0.15
ρ1 ρ1
0.0000
0.0004
0.0016
0.0064
0.0144
0.0169
0.0361
0.0484
0.0625
0.0729
0.1024
0.1024
0.2116
0.3025
0.3136
0.3844
0.4761
0.5476
0.6241
0.6400
µ̂h1 1i
0.0000
0.0004
0.0016
0.0065
0.0145
0.0169
0.0361
0.0485
0.0628
0.0729
0.1023
0.1022
0.2116
0.3022
0.3137
0.3843
0.4760
0.5477
0.6240
0.6397
√1 σ̂h1 1i
N
0.00 × 10−4
0.15 × 10−4
0.18 × 10−4
0.45 × 10−4
0.59 × 10−4
0.79 × 10−4
0.84 × 10−4
1.27 × 10−4
1.46 × 10−4
1.03 × 10−4
1.28 × 10−4
1.52 × 10−4
1.37 × 10−4
1.59 × 10−4
1.36 × 10−4
1.34 × 10−4
1.72 × 10−4
1.46 × 10−4
1.30 × 10−4
1.16 × 10−4
ρ1 ρ2 (1 − ρ∅ )
0.0000
0.0162
0.0002
0.0268
0.0111
0.0381
0.0250
0.0134
0.1463
0.0753
0.0595
0.0886
0.0000
0.0825
0.1784
0.1212
0.1381
0.0056
0.1289
0.0340
µ̂h1 2i
0.0000
0.0162
0.0002
0.0267
0.0111
0.0379
0.0250
0.0134
0.1460
0.0754
0.0596
0.0888
0.0000
0.0827
0.1782
0.1210
0.1382
0.0056
0.1290
0.0342
√1 σ̂h1 2i
N
0.00 × 10−4
0.37 × 10−4
0.08 × 10−4
0.86 × 10−4
0.59 × 10−4
0.94 × 10−4
0.87 × 10−4
0.81 × 10−4
1.39 × 10−4
1.46 × 10−4
1.11 × 10−4
1.87 × 10−4
0.00 × 10−4
1.67 × 10−4
1.15 × 10−4
1.65 × 10−4
1.32 × 10−4
0.32 × 10−4
1.26 × 10−4
0.90 × 10−4
ρ1 ρ∅ (1 + ρ2 )
0.0000
0.0034
0.0382
0.0468
0.0945
0.0750
0.1289
0.1582
0.0412
0.1218
0.1581
0.1290
0.2484
0.1650
0.0680
0.1144
0.0758
0.1868
0.0370
0.1260
µ̂h1 ∅i
0.0000
0.0034
0.0382
0.0468
0.0944
0.0752
0.1289
0.1581
0.0412
0.1218
0.1581
0.1290
0.2484
0.1650
0.0681
0.1146
0.0758
0.1867
0.0369
0.1261
√1 σ̂h1 ∅i
N
0.00 × 10−4
0.34 × 10−4
0.20 × 10−4
0.88 × 10−4
0.73 × 10−4
1.03 × 10−4
1.28 × 10−4
1.49 × 10−4
0.90 × 10−4
1.43 × 10−4
1.33 × 10−4
1.65 × 10−4
1.37 × 10−4
1.29 × 10−4
0.97 × 10−4
0.98 × 10−4
0.82 × 10−4
1.38 × 10−4
0.39 × 10−4
0.89 × 10−4
ρ1
0.00
0.02
0.04
0.08
0.12
0.13
0.19
0.22
0.25
0.27
0.32
0.32
0.46
0.55
0.56
0.62
0.69
0.74
0.79
0.80
ρ2
0.06
0.89
0.05
0.54
0.25
0.48
0.28
0.16
0.65
0.41
0.30
0.39
0.00
0.20
0.35
0.23
0.22
0.01
0.17
0.05
ρ∅
0.94
0.09
0.91
0.38
0.63
0.39
0.53
0.62
0.10
0.32
0.38
0.29
0.54
0.25
0.09
0.15
0.09
0.25
0.04
0.15
ρ1 ρ2
0.0000
0.0178
0.0020
0.0432
0.0300
0.0624
0.0532
0.0352
0.1625
0.1107
0.0960
0.1248
0.0000
0.1100
0.1960
0.1426
0.1518
0.0074
0.1343
0.0400
µ̂h2 1i
0.0000
0.0178
0.0020
0.0431
0.0300
0.0623
0.0533
0.0352
0.1624
0.1108
0.0959
0.1248
0.0000
0.1102
0.1959
0.1426
0.1519
0.0074
0.1344
0.0402
√1 σ̂h2 1i
N
0.00 × 10−4
0.33 × 10−4
0.27 × 10−4
0.94 × 10−4
0.81 × 10−4
1.14 × 10−4
1.28 × 10−4
0.81 × 10−4
1.37 × 10−4
1.42 × 10−4
1.62 × 10−4
1.78 × 10−4
0.00 × 10−4
1.50 × 10−4
1.42 × 10−4
1.40 × 10−4
1.38 × 10−4
0.26 × 10−4
1.14 × 10−4
0.68 × 10−4
ρ2 (ρ2 + ρ1 ρ∅ )
0.0036
0.7937
0.0043
0.3080
0.0814
0.2547
0.1066
0.0474
0.4388
0.2035
0.1265
0.1883
0.0000
0.0675
0.1401
0.0743
0.0621
0.0020
0.0343
0.0085
µ̂h2 2i
0.0036
0.7936
0.0043
0.3085
0.0815
0.2550
0.1066
0.0475
0.4389
0.2033
0.1266
0.1884
0.0000
0.0673
0.1404
0.0745
0.0619
0.0019
0.0342
0.0084
√1 σ̂h2 2i
N
0.42 × 10−4
0.61 × 10−4
0.35 × 10−4
1.82 × 10−4
1.17 × 10−4
1.53 × 10−4
1.52 × 10−4
1.53 × 10−4
1.69 × 10−4
2.17 × 10−4
1.66 × 10−4
2.17 × 10−4
0.00 × 10−4
1.22 × 10−4
1.36 × 10−4
1.44 × 10−4
1.38 × 10−4
0.20 × 10−4
1.14 × 10−4
0.61 × 10−4
ρ2 ρ∅ (1 − ρ1 )
0.0564
0.0785
0.0437
0.1888
0.1386
0.1629
0.1202
0.0774
0.0488
0.0958
0.0775
0.0769
0.0000
0.0225
0.0139
0.0131
0.0061
0.0007
0.0014
0.0015
µ̂h2 ∅i
0.0564
0.0785
0.0436
0.1884
0.1385
0.1626
0.1201
0.0773
0.0487
0.0959
0.0776
0.0768
0.0000
0.0224
0.0138
0.0129
0.0062
0.0007
0.0015
0.0015
√1 σ̂h2 ∅i
N
0.42 × 10−4
0.55 × 10−4
0.47 × 10−4
1.64 × 10−4
1.47 × 10−4
1.82 × 10−4
1.84 × 10−4
1.63 × 10−4
1.15 × 10−4
1.23 × 10−4
1.46 × 10−4
1.41 × 10−4
0.00 × 10−4
0.95 × 10−4
0.70 × 10−4
0.66 × 10−4
0.55 × 10−4
0.20 × 10−4
0.25 × 10−4
0.26 × 10−4
ρ1
0.00
0.02
0.04
0.08
0.12
0.13
0.19
0.22
0.25
0.27
0.32
0.32
0.46
0.55
0.56
0.62
0.69
0.74
0.79
0.80
ρ2
0.06
0.89
0.05
0.54
0.25
0.48
0.28
0.16
0.65
0.41
0.30
0.39
0.00
0.20
0.35
0.23
0.22
0.01
0.17
0.05
ρ∅
0.94
0.09
0.91
0.38
0.63
0.39
0.53
0.62
0.10
0.32
0.38
0.29
0.54
0.25
0.09
0.15
0.09
0.25
0.04
0.15
ρ1 ρ∅
0.0000
0.0018
0.0364
0.0304
0.0756
0.0507
0.1007
0.1364
0.0250
0.0864
0.1216
0.0928
0.2484
0.1375
0.0504
0.0930
0.0621
0.1850
0.0316
0.1200
µ̂h∅ 1i
0.0000
0.0018
0.0364
0.0304
0.0755
0.0507
0.1006
0.1364
0.0248
0.0863
0.1218
0.0930
0.2484
0.1376
0.0505
0.0930
0.0621
0.1849
0.0316
0.1201
√1 σ̂h∅ 1i
N
0.00 × 10−4
0.28 × 10−4
0.21 × 10−4
1.01 × 10−4
1.15 × 10−4
1.00 × 10−4
1.28 × 10−4
1.39 × 10−4
0.90 × 10−4
1.46 × 10−4
1.74 × 10−4
1.26 × 10−4
1.37 × 10−4
1.37 × 10−4
1.04 × 10−4
1.15 × 10−4
0.88 × 10−4
1.32 × 10−4
0.62 × 10−4
1.11 × 10−4
ρ2 ρ∅
0.0564
0.0801
0.0455
0.2052
0.1575
0.1872
0.1484
0.0992
0.0650
0.1312
0.1140
0.1131
0.0000
0.0500
0.0315
0.0345
0.0198
0.0025
0.0068
0.0075
µ̂h∅ 2i
0.0564
0.0801
0.0455
0.2048
0.1574
0.1871
0.1484
0.0991
0.0651
0.1313
0.1138
0.1128
0.0000
0.0499
0.0314
0.0345
0.0199
0.0025
0.0068
0.0075
√1 σ̂h∅ 2i
N
0.42 × 10−4
0.56 × 10−4
0.34 × 10−4
1.85 × 10−4
1.03 × 10−4
1.34 × 10−4
1.27 × 10−4
1.49 × 10−4
1.15 × 10−4
1.26 × 10−4
1.96 × 10−4
1.23 × 10−4
0.00 × 10−4
1.40 × 10−4
1.22 × 10−4
1.04 × 10−4
0.69 × 10−4
0.24 × 10−4
0.59 × 10−4
0.56 × 10−4
ρ∅ ρ∅
0.8836
0.0081
0.8281
0.1444
0.3969
0.1521
0.2809
0.3844
0.0100
0.1024
0.1444
0.0841
0.2916
0.0625
0.0081
0.0225
0.0081
0.0625
0.0016
0.0225
µ̂h∅ ∅i
0.8836
0.0081
0.8282
0.1448
0.3970
0.1522
0.2810
0.3846
0.0101
0.1024
0.1443
0.0842
0.2916
0.0625
0.0081
0.0224
0.0080
0.0626
0.0016
0.0224
√1 σ̂h∅ ∅i
N
0.42 × 10−4
0.54 × 10−4
0.37 × 10−4
1.68 × 10−4
1.49 × 10−4
1.61 × 10−4
1.77 × 10−4
1.76 × 10−4
0.68 × 10−4
1.08 × 10−4
1.47 × 10−4
1.20 × 10−4
1.37 × 10−4
1.29 × 10−4
0.58 × 10−4
0.98 × 10−4
0.56 × 10−4
1.29 × 10−4
0.20 × 10−4
0.93 × 10−4
8
 h1 1 1i 
h1 1 2i

 h1
 1 ∅i 
 h1 2 1i 
 h1 2 2i 


 h1 2 ∅i 
 h1 ∅ 1i 
 h1 ∅ 2i 


 h1 ∅ ∅i 
 h2 1 1i 
 h1 1i 

 0 −1 −1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  
 h2 1 2i 
h1 2i
0 1 0 −1 0 −2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 h2 1 ∅i 
 h1 ∅i 

  0 0 1 0 0 2 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  

h2 1i 
 0 1 0 −1 0 0 0 0 0 0 0 −1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0   h2 2 1i 
d 
 h2 2i  =  0 0 0 0 −1 0 0 0 0 0 1 0 0 0 −1 0 1 0 0 0 0 0 0 0 0 0 0   h2 2 2i 

 
0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 −1 0 0 0  
dt 
 h2 2 ∅i 
 h2 ∅i   00 00 01 00 00 −2
0 −1 0 0 0 0 1 0 0 0 −1 0 0 0 −1 0 0 0 0 0 0 0   h2 ∅ 1i 
 h∅ 1i 
0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 −1 0 0 0
 h2 ∅ 2i 
h∅ 2i
0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 −1 0 0 1 0 0 1 0 0 0


h∅ ∅i
 h2 ∅ ∅i 
 h∅ 1 1i 
 h∅ 1 2i 


 h∅ 1 ∅i 
 h∅ 2 1i 


 h∅ 2 2i 
 h∅ 2 ∅i 


 h∅ ∅ 1i 
h∅ ∅ 2i
h∅ ∅ ∅i
We know that in the steady state, the rate of change
for each of these two-point correlations is zero. With
that fact incorporated, our matrix expression constitutes
a constraint on the space of values that the three-point
correlations may possibly take, in the system’s steady
state.
Other summing relationships (just as Eq. (3)) must be
satisfied by the three-point correlations. These should
also be included to further constrain the space of values
in a way that ensures self-consistency.
However experimentation with the actual matrices in
MATLAB has demonstrated that we have not been lucky
enough to find constraints on the set of three-point correlations sufficient to determine any of the set of two-point
correlations (other than h1 1i and h∅ ∅i whose values we
know a priori). Our expressions for the two-point correlations therefore remain conjectures.
Nonetheless, we now have strong constraints on the
set of three-point correlations. If we now also include
our conjectures as to two-point correlations in the matrix
expression, we come to
 h1 1 1i 
h1 1 2i
 h1

 1 ∅i 

 h1 2 1i  
1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0
 h1 2 2i 
0 1 0 −1 0 −2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0

 0 0 1 0 0 2 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  
h1 2 ∅i 
0
 0 1 0 −1 0 0 0 0 0 0 0 −1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0  


0
h1 ∅ 1i 
 0 0 0 0 −1 0 0 0 0 0 1 0 0 0 −1 0 1 0 0 0 0 0 0 0 0 0 0  


0


 0 0 0 0 0 −2 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 −1 0 0 0   h1 ∅ 2i  

0
 0 0 1 0 0 0 −1 0 0 0 0 1 0 0 0 −1 0 0 0 −1 0 0 0 0 0 0 0   h1 ∅ ∅i  

0
 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 −1 0 0 0  


0
 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 −1 0 0 1 0 0 1 0 0 0   h2 1 1i 


0

  h2 1 2i 


 
0
 0 1 1 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0  

0
 0 −1 0 1 1 1 0 0 0 0 −1 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0   h2 1 ∅i 


0
 0 0 −1 0 0 0 1 1 1 0 0 −1 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0   h2 2 1i 


0
 0 0 0 −1 0 0 0 0 0 1 1 1 −1 0 0 0 0 0 0 0 0 −1 0 0 0 0 0   h2 2 2i 



0
=
 0 0 0 0 −1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 −1 0 0 0 0  


0
 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 −1 1 1 1 0 0 0 0 0 −1 0 0 0   h2 2 ∅i 



0
 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 −1 0 0 1 1 1 0 0 0 −1 0 0   h2 ∅ 1i  

0





0
 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 −1 0 0 0 0 1 1 1 0 −1 0   h2 ∅ 2i  

ρ1
 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 1 1 0   h2 ∅ ∅i  

ρ
2
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0


 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0   h∅ 1 1i 
 ρρ1 ∅ρ1 

 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1   h∅ 1 2i   ρ∅ ρ∅ 
 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   h∅ 1 ∅i   ρ1 ρ∅ 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1


ρ1 ρ∅ (1+ρ2 )
 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0   h∅ 2 1i 

ρ
ρ
ρ
0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1
1
1
 h∅ 2 2i 
ρ∅ ρ∅ ρ∅
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 h∅ 2 ∅i 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1


h∅
∅
1i


 0 −1 −1
h∅ ∅ 2i
h∅ ∅ ∅i
9
MATLAB tells us that this is a matrix with rank 21.
Since our system is in 27 variables, we have eliminated all
but 6 degrees of freedom from the space of possible values
taken by the three-point correlations. But again this is
not enough to fix any one of the three-point correlations.
(Except that h1 1 1i and h∅ ∅ ∅i we have fixed a priori
at ρ1 ρ1 ρ1 and ρ∅ ρ∅ ρ∅ respectively, and included those
fixings in the system of equations.)
B.
Currents
We work within the two-class (three-type) model. In
the steady state, the current will be the same at every
site on the periodic lattice. To work out the forward
current we need simply subtract the density of situations
in which a backward hop might occur from the density
of situations in which a forward hop might occur.
The current of first-class individuals is
J1 = h1 ∅i + h1 2i = ρ1 (ρ2 + ρ∅ )
So the ‘productivity’ of a single first-class worker is
given by
v1 =
J1
= ρ2 + ρ∅
ρ1
For second-class individuals, recall that Eq. (8) is a
rigorously derived result.
J2 = h2 ∅i − h1 2i = ρ2 (ρ∅ − ρ1 ) = ρ2 (1 − 2ρ1 − ρ2 )
Therefore
v2 =
J2
= 1 − 2ρ1 − ρ2
ρ2
In general we may apply the logic of grouping types,
as discussed on page 3. Thus
J1 = ρ1 (1 − ρ1 )
J2 = ρ2 (1 − 2ρ1 − ρ2 )
..
.
!
k−1
X
Jk = ρk 1 − 2
ρi − ρk
i=1
and
v1 = 1 − ρ 1
v2 = 1 − 2ρ1 − ρ2
..
.
k−1
X
vk = 1 − 2
ρi − ρk
i=1
C.
Diffusion
Let Yt be the total forward distance travelled between
time 0 and time t by a single individual, as the system
evolves within its steady state. We may study the distribution of Yt .
The first moment, its mean, is a simple consequence
of the velocity expressions we have already deduced in
section III B. To discover those, it was necessary only to
examine a few relationships between correlations. There
was no need to differentiate between members of the same
class.
This is in contrast to the search for higher moments.
In either simulation or in analysis, we are now required
to perform an explicit ‘tagging’ of the individual in order
to proceed.
The diffusion constant is defined [6] by
1 2 2
Yt − hYt i
t→∞ t
∆ = lim
(10)
where that quantity exists.
While much has been written on the subject of the diffusion constant in the single-class model, we have not
come across an analysis for the model with multiple
classes.
The earliest analyses appear to have taken place on
the infinite line rather than on a ring [6]. Through an
argument that relates diffusion to autocovariance, it is
shown that the diffusion constant is identical to velocity,
just as would be the case for a random walker without
interactions.
Later work [13] performed the analysis on the ring. Different arguments were made, appealing either to a matrix
formalism or to an application of the Bethe Ansatz [7].
For an idea of how this sort of analysis might be extended
to the many-class case, please see the attached notes [14].
The very existence of the diffusion constant depends
in effect on the order in which the limits are taken. We
have so far talked about large rings as if considering the
limit of large n. But in Eq. (10) we are also considering
a limit of infinite time. So which comes first?
If the first limit to be taken is in the ring size n, we
are effectively dealing with the infinite line, and so with a
well-defined diffusion constant as described above. However if the time limit is taken first, one finds a diffusion
1
constant that scales as n− 2 , and that therefore vanishes
entirely in the large n limit; i.e. the process becomes
subdiffusive.
The reason for this is given succinctly in [7].
Because on a ring the particles cannot overtake each other, they all cover the same distance Yt up to fluctuations which remain
bounded when t becomes large.
Simulation on a large ring suggests that the effects of
both regimes may be observed in practice. At timescales
of O(T ) where 1 T n, the behaviour as described
10
for the infinite line dominates, and the process is diffusive. However on timescales
where T n, the lower (ul
The parameters (fugacities) we would have to adjust
are explicitly solved for in [11],
1
timately vanishing) O n− 2 level of diffusion dominates.
In between the two regimes interesting oscillations are
seen.
IV.
DISCUSSION AND FURTHER WORK
If our aim was to understand the relationship between
class structure and productivity in this model, then we
have enjoyed some success. The results of section III B
are straightforward and beyond dispute.
Perhaps the most surprising discovery we have made
is that of exact expressions for the two-point nearestneighbour correlations. Those expressions might not be
too hard to prove formally, e.g. via the matrix representation [5].
Higher moments, the existence of diffusive behaviour
etc., are more difficult to analyse, but there does exist a
body of successful work on the subject [7].
In an economic context, class mobility might be an
interesting aspect to model. It is worth noting that there
already exists a natural way in which class mobility could
be defined. On a large ring, one might periodically use
Gibbs sampling from the grand canonical distribution to
change the class of an individual, without ever leaving
the steady state.
[1] F. Engels, Anti-Dühring (Vorwärts, 1877–1878).
[2] K. Pickett and R. Wilkinson, The Spirit Level: Why
More Equal Societies Almost Always Do Better (Penguin,
2009).
[3] “Chasing the dream: Why don’t rising incomes make everybody happier?”, The Economist, 368(8336), 68 (Aug
9, 2003).
[4] E. D. Beinhocker, The Origin of Wealth: Evolution,
Complexity and the Radical Remaking of Economics
(Random House Business Books, 2006).
[5] M. R. Evans, P. A. Ferrari, and K. Mallick, J. Stat.
Phys., 135, 217 (2009).
[6] A. De Masi and P. A. Ferrari, J. Stat. Phys., 38, 603
(1985).
[7] B. Derrida, Phys. Rep., 301, 65 (1998).
[8] M. R. Evans and T. Hanney, J. Phys. A: Math. Gen., 38,
x
y+x
y 2 − xz
ρ2 =
(y + x)(y + z)
z
ρ∅ =
y+z
ρ1 =
An economist might appreciate having more parameters to play with, more knobs to turn. . .
Results also exist for the model where hop rates vary
according to class [7] and for models in which hops
are only partially asymmetric [12]. There are presumably various trade-offs involved in the quest to maximise
productivity—it might be interesting to model these in
some game.
Another variation is where individuals are allowed to
overtake members of their own class. This does not affect
the dynamics of the class as a whole, nor current, but it
does affect diffusion for example.
There is perhaps some interest in exploring the limit
where the number of classes is equal to the number of
individuals, so that every individual belongs to its own
class. The situation as described may actually turn out
to have simpler properties than the one where there are
a relatively small number of classes.
R195 (2005).
[9] P. Meakin, P. Ramanlal, L. M. Sander, and R. C. Ball,
Phys. Rev. A, 34, 5091 (1986).
[10] P. A. Ferrari and J. B. Martin, Ann. Probab., 35, 807
(2007).
[11] B. Derrida, S. A. Janowsky, J. L. Lebowitz, and E. R.
Speer, J. Stat. Phys., 73, 813 (1993).
[12] M. R. Evans and R. A. Blythe, Physica A, 313, 110
(2002).
[13] B. Derrida, M. R. Evans, and D. Mukamel, J. Phys. A:
Math. Gen., 26, 4911 (1993).
[14] See EPAPS Document No. 12345 for details of our suggested approach. For more information on EPAPS, see
http://www.aip.org/pubservs/epaps.html.
EPAPS: Route to higher moments
We follow Y the distance travelled by a tagged individual.
Let points on the ring be labelled
0, . . . , N − 1
and keep the tagged individual at 0, shifting the rest of the configuration C
when necessary.
K0 transitions imply no change in Y , but K1 transitions correspond to an
increase of 1 and K−1 transitions to a decrease of 1.
The Master Equation is
X d
Pt (C , Y ) =
K0 (C , C 0 )Pt (C 0 , Y )
C0
dt
+ K1 (C , C 0 )Pt (C 0 , Y − 1) + K−1 (C , C 0 )Pt (C 0 , Y + 1)
X −
K0 (C 0 , C ) + K1 (C 0 , C ) + K−1 (C 0 , C ) Pt (C , Y )
0
C
If we group the terms in matrices thus
M0 (C , C 0 ) = K0 (C , C 0 ) for C 6= C 0
X M0 (C , C ) = −
K0 (C 0 , C ) + K1 (C 0 , C ) + K−1 (C 0 , C )
0
C
M1 (C , C 0 ) = K1 (C , C 0 )
M−1 (C , C 0 ) = K−1 (C , C 0 )
we may write
d
P t (Y ) = M0 P t (Y ) + M1 P t (Y − 1) + M−1 P t (Y + 1)
dt
where the vector P t (Y ) runs over the set of configurations.
We now turn to generating functions. Note that
∞
X
Y =−∞
f (Y )
X
M (C , C 0 )P (C 0 , Y ) =
C0
X
M (C , C 0 )
C0
So for
Gt (s) =
∞
X
Y =−∞
1
sY P t (Y )
∞
X
Y =−∞
f (Y )P (C 0 , Y )
we have
∞
X
d
Gt (s) =
sY M0 P t (Y ) + M1 P t (Y − 1) + M−1 P t (Y + 1)
dt
Y =−∞
!
∞
∞
X
X
Y
Y −1
= M0
s P t (Y ) + M1 s
s
P t (Y − 1)
Y =−∞
Y =−∞
+ M−1
−1
s
∞
X
!
Y +1
s
P t (Y + 1)
Y =−∞
= M0 + sM1 + s−1 M−1 Gt (s)
But also for
∞
X
G0t (s) =
Y sY −1 P t (Y )
Y =−∞
we have
∞
X
d 0
Y sY −1 M0 P t (Y ) + M1 P t (Y − 1) + M−1 P t (Y + 1)
Gt (s) =
dt
Y =−∞
∞
X
Y sY −1 P t (Y )
= M0
Y =−∞
+ M1
s
∞
X
Y −2
(Y − 1)s
P t (Y − 1) +
Y =−∞
+ M−1
s−1
∞
X
∞
X
!
Y −1
s
P t (Y − 1)
Y =−∞
(Y + 1)sY P t (Y + 1)
Y =−∞
−2
−s
= M0 + sM1 + s−1 M−1 G0t (s) + M1 −
And for
G00t (s) =
∞
X
∞
X
2
s
P t (Y + 1)
Y =−∞
−2
s M−1 Gt (s)
Y (Y − 1)sY −2 P t (Y )
Y =−∞
!
Y +1
we have
∞
X
d 00
Gt (s) =
Y (Y − 1)sY −2
dt
Y =−∞
· M0 P t (Y ) + M1 P t (Y − 1) + M−1 P t (Y + 1)
∞
X
= M0
Y (Y − 1)sY −2 P t (Y )
Y =−∞
+ M1
∞
X
s
(Y − 1)(Y − 2)sY −3 P t (Y − 1)
Y =−∞
∞
X
+2
!
Y −2
(Y − 1)s
P t (Y − 1)
Y =−∞
+ M−1
−1
s
∞
X
(Y + 1)Y sY −1 P t (Y + 1)
Y =−∞
− 2s−2
∞
X
(Y + 1)sY P t (Y + 1)
Y =−∞
−3
+2s
∞
X
!
Y +1
s
P t (Y + 1)
Y =−∞
= M0 + sM1 + s−1 M−1 G00t (s)
+ 2 M1 − s−2 M−1 G0t (s) + 2s−3 Gt (s)
In terms of the moments of Y
(1 · · · 1) Gt (1) = 1
(1 · · · 1) G0t (1) = hYt i
(1 · · · 1) G00t (1) = Yt2 − hYt i
and noting that
(1 · · · 1) (M0 + M1 + M−1 ) = (0 · · · 0)
we have
d
hYt i = (1 · · · 1) (M0 + M1 + M−1 ) G0t (1) + (1 · · · 1) (M1 − M−1 ) Gt (1)
dt
= (1 · · · 1) (M1 − M−1 ) Gt (1)
d 2
Y = 2 (1 · · · 1) (M1 − M−1 ) G0t (1) + 2 (1 · · · 1) Gt (1)
dt t
+ (1 · · · 1) (M1 − M−1 ) Gt (1)
= 2 (1 · · · 1) (M1 − M−1 ) G0t (1) + 2 + (1 · · · 1) (M1 − M−1 ) Gt (1)
Continue by seeking the largest eigenvalue of
M0 + sM1 + s−1 M−1
3