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arXiv:1201.2855v1 [cond-mat.stat-mech] 13 Jan 2012
Multiple phase transitions in a system of exclusion
processes with limited reservoirs of particles and
fuel carriers
Chris A Brackley1, Luca Ciandrini1and M Carmen
Romano1,2
1Institute for Complex Systems and Mathematical Biology, SUPA, University
of Aberdeen, Aberdeen, AB24 3UE, United Kingdom
2Institute of Medical Sciences, Foresterhill, University of Aberdeen, Aberdeen,
AB25 2ZD, United Kingdom
E-mail: cab@chrisbrackley.co.uk, l.ciandrini@abdn.ac.uk
PACS numbers: 05.60.-k,05.40.-a,02.50.Ey
Abstract.
physics, which describes particles hopping along a lattice of discrete sites. The
TASEP is applicable to a broad range of different transport systems, but does not
consider the fact that in many such systems the availability of resources required
for the transport is limited.In this paper we extend the TASEP to include
the effect of a limited number of two different fundamental transport resources:
the hopping particles, and the “fuel carriers”, which provide the energy required
to drive the system away from equilibrium.
dynamics are substantially affected: a “limited resources” regime emerges, where
the current is limited by the rate of refuelling, and the usual coexistence line
between low and high particle density opens into a broad region on the phase
plane. Due to the combination of a limited amount of both resources, multiple
phase transitions are possible when increasing the exit rate β for a fixed entry
rate α. This is a new feature that can only be obtained by the inclusion of both
kinds of limited resources. We also show that the fluctuations in particle density
in the LD and HD phases are unaffected by fluctuations in the number of loaded
fuel carriers, except by the fact that when these fuel resources become limited,
the particle hopping rate is severely reduced.
The TASEP is a paradigmatic model from non-equilibrium statistical
As as consequence, the system’s
Keywords: driven diffusive systems (theory), stochastic processes (theory)
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Multiple phase transitions in a TASEP with limited particles and fuel carriers2
1. Introduction
The totally asymmetric simple exclusion process (TASEP) is one of the fundamental
models of non-equilibrium statistical mechanics [1, 2, 3]. Essentially a driven diffusion
model, it has many applications in physics and beyond, including traffic models [4],
the movement of molecular motors in biological systems [5], and protein synthesis in
messenger RNA (mRNA) translation [6, 7]. It also belongs to the same universality
class as some surface growth models [8]. In this paper we study a constrained TASEP
where finite resources are shared among several lattices. By finite resources we mean
a constrained number of both particles and “fuel carriers”, whose role is to provide
the energy needed to the movement of the particles. Molecular motors requiring ATP
or GTP molecules are an example of such systems occurring in nature. In this paper
we introduce a new model which includes the finite availability of both resources, in
contrast to previous works where the effect of having a finite number of a single type
of resource was studied in isolation [9, 10, 11, 12, 13, 14]. As a result, multiple phase
transitions can occur when varying one of the fundamental parameters of the model
–the exit rate β– while keeping the rest of the parameters constant: the system can
go from a high density regime, to a shock phase, then to a high density phase again,
visit the shock phase once more, and finally reach a low density phase. This is a novel
effect that emerges only by combining both limited resources. We use a mean-field
approach and verify our results by means of Monte Carlo simulations.
In its most simple form, the TASEP consists of a 1D lattice of L sites upon which
particles can sit, see figure 1(a). Each site can be occupied only by one particle at
a time, and particles move from site to site in one direction (say rightward) with
a hopping rate k. Since particles cannot pass each other, movement requires that
the downstream site is vacant. A system with open boundaries, as we shall consider
here, can display rich dynamics with multiple boundary induced phases [15]. Particles
are allowed to hop onto the lattice with rate α at one end, and off of the lattice
with rate β at the other. For a system with constant internal hopping rate k it is
possible to solve the steady-state of the system exactly [16, 2, 17, 18], whilst the
full relaxation dynamics have been solved using matrix methods [19, 20]. Mean-field
methods have also been extensively used [16], since they are easily tractable and yield
a good approximation in many cases. There has been much extension of this simple
model, for example variable hopping rates (site or particle dependent) [21, 22, 23],
extended particles which cover more than one site [6, 24], branching lattices [25, 26],
particles which have multiple internal states [27, 28, 29], as well as a TASEP with a
constrained reservoir of particles [9, 10, 11, 12].
For the standard TASEP we denote the occupation of the ith lattice site ni= 1 if
the site contains a particle and ni= 0 otherwise. The system is characterised by the
steady-state particle current J (the rate at which particles pass any given point on
the lattice), and the mean site occupancy (often called density) ρi= ?ni?, where ?···?
denotes average over realisations of the system (which we assume is ergodic, so this is
equivalent to a time average). The average density is therefore given by ρ = L−1?
or low density (LD) phase, the exit limited or high density (HD) phase, a maximal
current (MC) phase where the current depends only on the internal hopping rate, and
a mixed LD-HD or shock phase (SP). A mean-field approach [16] (which turns out to
be exact in the L → ∞ limit) can be used to calculate J and ρifor given α and β.
iρi.
There are four possible phases depending on the values of α and β: the entry limited
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Multiple phase transitions in a TASEP with limited particles and fuel carriers3
Figure 1. Schematic diagrams describing the various systems. (a) The TASEP
with open boundaries its most simple form. Particles enter at fixed rate α, hop
at fixed rate k and leave with rate β. (b) A finite number of fuel carriers can be
introduced. When a particle hops, fuel from one carrier is used. (c) Complete
model with finite fuel carrier and finite particles. The entry rate depends on the
number of particles in the reservoir, and the hopping rate depends on the number
of loaded fuel carriers.
The density in the bulk (far from the ends of the system) is given by
(LD)
(HD)
(MC)
ρLD= α/k
ρHD= 1 − β/k
ρMC= 1/2
for α < β,α < k/2 ,
for β < α,β < k/2 ,
for β,α ≥ k/2 ,
(1)
and the current is always given by J = kρ(1 − ρ).
α = β < k/2, presents an HD region on the right of the lattice and an LD region
on the left, separated by a boundary which diffuses freely through the lattice. This
has often been described using a domain wall (DW) theory [30]. Due to the free
diffusion of the DW a time average of the density in SP gives ρSP = 1/2, but the
current depends on the density in either the LD or HD regions of the lattice, i.e.
JSP= α(1 − α/k) = β(1 − β/k).
In this paper we consider several TASEPs which share a common finite pool of
both particles and fuel carriers. The entry rate, which is the same for each TASEP,
depends on the availability of particles in a common pool (i.e., particles which are not
involved with any lattice). A model describing several TASEPs sharing a common
pool of particles has been introduced and thoroughly studied in [9, 10, 11], where
the authors use the DW theory along with known exact results. In this paper we
use an alternative recent mean-field (MF) approach that allows us to simplify the
calculations [12]. Importantly, we combine this with a model for a finite pool of fuel
carriers [13, 14] which, as noted above, can be viewed as carriers that provide the
The SP, which occurs for
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Multiple phase transitions in a TASEP with limited particles and fuel carriers4
energy which drives the motion, i.e. allowing the particles to hop. Although we
consider a fixed number of fuel carriers, we suppose that it takes a finite time to
“refuel” them with their cargo once it has been used (figure 1(b) shows a schematic
representation of this model). We show that novel effects arise when both types of
limited resource are considered, e.g. multiple phase transitions can occur when varying
the exit rate β. The outline of the paper is as follows: in section 2 we summarise the
previous results for the two models separately, before describing in section 3 a mean-
field model for a system with both a finite pool of particles and fuel carriers which are
refuelled at a finite rate (figure 1(c)). We then interpret the mean-field model results
and compare them with results from Monte Carlo simulations. Finally, in section 4
we analyse the effect of both limited resources on the fluctuations in the number of
particles on the lattice.
2. Finite resources - Review of previous results
We first introduce and describe a system containing multiple TASEPs in which each
lattice shares the same reservoir of particles; then we present the concept of fuel-
carriers and the effect of a finite rate of refuelling on the exclusion process dynamics.
2.1. Finite number of particles
In this work we analyse a system of M identical lattices of length L. The total number
of available particles is N, while the number of free particles in the reservoir is Nr.
Since the lattices are identical and experience the same injection and depletion rates,
we observe the same phase for each. We can write the total number of particles as
N = Nr+ LMρ,(2)
where ρ is the density on each lattice. The entry rate of the M lattices depends on
the number of free particles via a saturating function
α = α0tanh
?Nr
?N − LMρ
N∗
?
= α0tanh
N∗
?
, (3)
where the constant α0gives the entry rate in the limit Nr→ ∞ and is an intrinsic
property of the lattices ‡ . Without loss of generality, we fix the normalisation factor
N∗to be LM/2, i.e. the total number of particles used if all the lattices were in the
MC phase.
Throughout this paper we define the different phases according to the values of α
and β and the resulting density ρ, following [12]. With this choice of nomenclature we
solve equations for α in terms of α0and N. Since the densities in each phase are the
same as those in the standard TASEP (equations (1)), for a given set of parameters
(α0, β, N) we find the resulting α which determines the phase; e.g. if α < β and
α < k/2 the system will be in the LD phase. By substituting equation (3) into these
inequalities, we get a representation of the different phases on the α0–β plane.
As a consequence of having a finite number of particles, we encounter different
regimes for small, mid-range and large values of N. We show typical phase diagrams
‡ Equation (3) is consistent with the function used in [9, 10, 11], and is relevant, e.g.
application to protein synthesis.
for the
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Multiple phase transitions in a TASEP with limited particles and fuel carriers5
Figure 2. Phase diagrams for a TASEP of length L = 500 with a finite number
of (a) N = 225, (b) N = 475, and (c) N = 600 particles (infinite amount of loaded
fuel carriers). For small N there are not enough particles to support the HD and
MC phases, and the SP (coexistence) line opens into a region. For N > LM/2 all
four phases can be obtained. If N is increased further the HD phase grows at the
expense of a shrinking SP.
for these regimes in figure 2. If N < LM/2, then the HD and MC phases no longer
exist – there are too few particles to support the high density or maximal current
phases. Instead, there are only two phases: the LD phase and the SP (figure 2(a)). As
described in the previous section, the latter occurs when the entry and exit rates are
equal, i.e., α = β, and there is coexistence between an LD region and an HD region;
since α depends on both α0and β (through its dependence on the bulk density), the
line opens into a region on the α0-β phase plane. That is to say, the condition α = β
is fulfilled for a certain range of α0[12]. If N = LM/2, then the lattices can support
an MC phase, and for N > LM/2 there are enough particles for an HD phase to exist
(figure 2(b)). As N is increased, the size of the HD phase on the α0-β plane increases,
at the cost of reducing the size of the SP phase (figure 2(c)). For N ≫ LM the SP
phase reduces to a line and we recover the original unconstrained TASEP.
In the unconstrained TASEP within the SP, the LD and HD regions of the system
are separated by a domain wall (DW) which can diffuse freely across the lattice.
However, if there is a finite number of particles, in the case of a single lattice the DW
is pinned to one position [9, 10] (actually the DW fluctuates about its mean position
like a noisy damped oscillator). This is because if the DW were to move to the right
this would increase the number of free particles, increasing the entry rate and therefore
driving the DW leftwards. Similarly if the DW moves to the left the number of free
particles decreases, decreasing the entry rate and driving the DW rightwards. The
opening of the SP line into a region on the α0-β plane is possible because a different
mean position for the DW corresponds to a different value of α0, while keeping α = β.
Hence, the system can maintain α = β for different values of α0. As detailed in [11], if
more than one TASEP is in contact with the same pool of particles, the DW on each
lattice once again performs a random walk; there is however a pinning of the total
number of particles on all lattices.
2.2. Finite reloading time for fuel carriers
In many of the systems that can be described by a driven lattice gas, the
energy required for the advancement of the particles is obtained from some kind
of finite resource.For instance, molecular motors consume ATP molecules, and
ribosome movement on mRNAs during protein synthesis requires aminoacylated tRNA
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Multiple phase transitions in a TASEP with limited particles and fuel carriers6
complexes and GTP. The latter case has recently been described in [13, 14, 31], and
here we briefly review those results, before in the next section combining this with the
a finite pool of particles model.
We consider a finite number¯T of fuel carriers, T of which are carrying fuel. Every
time a particle moves, the fuel from one of the loaded carriers is used, and hence, T
is reduced by one. The hopping rate of particles depends on the availability of loaded
fuel carriers, and the empty carriers are refuelled at a rate V . For simplicity, the
hopping rate is taken to be directly proportional to the loaded fuel carriers, i.e.
k = aT,(4)
where a is a constant. Moreover, the rate of refuelling is taken to depend on the
number¯T − T of unloaded fuel carriers as
V =V0(¯T − T)
b +¯T − T,
which has the form of the well known Michaelis-Menten equation in biochemistry.
The recharging rate is therefore a saturating function of the number of empty carriers
(¯T − T) with maximum value V0and saturation determined by the constant b. Any
saturating function will give the same qualitative results, but the above formulation
allows for a straightforward analytical treatment [13, 14] §.
In this section we describe a collection of M identical TASEPs with a finite
number of fuel carriers (but no constraints on the number of particles); following the
common mean-field treatment [16] the particle density on the ith site of each lattice
is given by
dρ1
dt
dρi
dt
dρL
dt
=
=
=
α(1 − ρ1) − kρ1(1 − ρ2),
kρi−1(1 − ρi) − kρi(1 − ρi+1),
kρL−1(1 − ρL) − βρL.
i = 2,...L − 1,
(5)
The inclusion of a finite pool of fuel carriers leads to the additional equation
dT
dt=V0(¯T − T)
b +¯T − T
−
(L−1)M
?
j=1
kρj(1 − ρj), (6)
where the sum is over all of the L − 1 sites which use fuel carriers on each of the M
lattices. We assume that the particles do not require a fuel carrier to leave the Lth
site, i.e. the exit rate β is constant. In the steady-state we identify the term under
the sum in (6) as the particle current, and using equation (4) we find
k = a¯T −
abJ(L − 1)M
V0− J(L − 1)M,
(7)
i.e., the hopping rate is now itself a function of the current. Following [13, 14], upon
solving equations (5) in the steady-state we find the four phases as in the original
TASEP, but now the current and density are given as follows
(LD)
(HD)
(MC)
(SP)
JLD= J(α),
JHD= J(β),
JMC= α∗/2,
JSP= J(α),
ρLD= D(α)
ρHD= 1 − D(β)
ρMC= 1/2
ρSP= 1/2
for α < α∗and α < β,
for β < α∗and β < α,
for α,β ≥ α∗,
for α = β < α∗,
(8)
§ An alternative model would be to have a hopping rate which is a saturating function of¯ T, and
then have a constant refuelling rate. This would give qualitatively similar behaviour to the present
definitions. Our choice is most applicable to protein synthesis, i.e. refuelling due to an enzymatic
reaction.
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Multiple phase transitions in a TASEP with limited particles and fuel carriers7
where
J(α) =1
2
?
α
?
1 −
α
a(¯T + b)
??
?
+
¯T
¯T + b
V0
L′M
−
¯T
¯T + b
V0
L′M+ α
?
1 −
α
a(¯T + b)
??2
−4α(a¯T − α)
a(¯T + b)
V0
L′M
,
D(α) =
1
2α
?
α
?
1 +
α
a(¯T + b)
??
?
V0
L′M+ α
−
¯T
¯T + b
V0
L′M
+
¯T
¯T + b
?
1 −
α
a(¯T + b)
??2
−4α(a¯T − α)
a(¯T + b)
V0
L′M
,
and
α∗=a
4(¯T + b) +
V0
L′M−
??a
4(¯T + b) +
V0
L′M
?2
− a¯T
V0
L′M,
with L′= L − 1. The behaviour of these functions as α and β are varied depends
on the parameters a, b and V0. By considering the steady-state of equation (6) and
noticing that the maximal value that the recharging rate can possibly have is equal to
V0, we note that the particle current is limited from above by V0/L′M. Hence, there
are substantially two different cases: (i) if V0/L′M ≫ 1, the recharging rate is very fast
and the particle current is not influenced by it; we recover the results of the original
TASEP; (ii) if in contrast V0/L′M ≪ 1, the recharging of the fuel carriers can limit
the value of the particle current. Figure 3 shows the current for sets of parameters
corresponding to each case. In case (ii) (figure 3(b)), J(α) shows a sharp change from
increasing with α, to almost independent of α (though we note that the derivative of
J(α) remains continuous). The value of J(α) is severely reduced compared to the
one obtained for case (i) (see figure 3(a)). We refer to the regime where the current
appears independent of α as a limited resources (LR) regime, since the rate at which
fuel is used by the particles approaches the rate at which fuel carriers are reloaded.
Thus the pool of loaded carriers becomes depleted and the hopping rate k reduces. For
some choices of a and b, the LR regime exists within each of the phases (LD, HD and
MC). In the LR regime within the LD phase, the sensitivity of the current to changes
in α or β is greatly reduced, whilst the sensitivity of the density is greatly increased.
In the MC phase, the current is greatly reduced in the LR regime compared to that
in case (i). For further details see [13, 14].
The onset of the LR regime depends particularly on the value of the three
quantities a¯T, V0/LM and b/¯T, the former two controlling at what value of α or β the
onset will occur, and the latter controlling the sharpness of the change in behaviour.
In the rest of this paper we choose b such that there is a sharp onset of LR, and take
V0 as the control parameter for the fuel carriers, fixing the other parameters. This
choice not only gives the most interesting dynamics, but it has also been shown to
be the biological relevant regime in the context of protein synthesis [13, 14]. When
the onset of LR is sharp we can estimate the value of α or β at which this occurs
by equating the rate of fuel carrier use (approximately αL′M for small α in LD and
βL′M for small β in HD) and the maximum recharging rate. This gives
¯TV0
(¯T + b)L′M.
αLR= βLR≈
(9)
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Multiple phase transitions in a TASEP with limited particles and fuel carriers8
Figure 3. Plots showing the current as a function of α for different parameters.
In (a) V0 = 300 s−1, and in (b) V0 = 20 s−1. In both cases a = 2 × 10−4,
¯T = 5000, and b = 50. Solid lines show the current in the LD phase, JLD= J(α),
and dashed lines the current in the MC phase, JMC= α∗/2. The dotted line is at
α∗, where there is a transition from LD→MC. The dot-dashed line in (b) shows
the value of αLDas estimated in equation (9).
Figure 4. Steady-state fuelling level T/¯T as a function of rate V0 for a model
with no constraints on the number of particles (section 2.2). Other parameters are
L = 500, α = 0.5 s−1, β = 0.1 s−1, a = 2 × 10−4s−1, b = 50, and¯ T = 5000. For
small V0the number of loaded fuel carriers is depleted; for large V0the carriers are
practically always fully loaded, and we recover the original unconstrained TASEP.
This value is represented in figure 3(b) by a dot-dashed vertical line. As it is shown
there, the estimation predicts quite accurately the onset of the LR regime.
3. Constrained reservoir of particles and finite refuelling rate
A much more realistic model for natural processes such as biological transport has to
include the finite availability of both particles and fuel carriers. As we show later
in this section, it is only when combining the two schemes discussed above that
we can see emerging novel effects, such as multiple phase transitions. Analogous
to the dependence of the entry rate α on the number of particles N, the steady-
state proportion of loaded fuel carriers is a saturating function of V0 (see figure 4).
Therefore, by regarding N and V0as control parameters, we can vary the number of
available particles and loaded carriers respectively; in both cases a saturating function
of the resource determines the dynamics.
The quantities of interest are the particle current J, the number of loaded fuel
carriers T and the particle density ρ, which is linked to the number of free particles
by equation (2). The expressions for J and ρ given in equations (8) still hold; however
α is no longer a control parameter, and it can be eliminated using equation (3). This
is the case in all phases except in the SP: since the DW cannot move freely on the
lattice, the mean density now depends on the size of the LD and HD regions. We
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Multiple phase transitions in a TASEP with limited particles and fuel carriers9
calculate the average value ρSPusing equation (3); using the fact that in this phase
α = β leads to
ρSP=
N
LM
?
1 −N∗
N
tanh−1
?β
α0
??
. (10)
The hopping rate k (and therefore T) in each phase can be found by substituting the
appropriate equation for the current J in (7).
We now turn to the problem of finding the boundaries between the different phases
as functions of α0, β, N and V0— i.e., eliminating α. Our aim is to draw the α0-β
phase plane for any given values of V0and N. As noted in previous sections, due to
finite particles, the SP line opens into a region, and the HD and MC phases do not
exist if N < LM/2. Each phase boundary can be written in terms of either α0as a
function of β, or vice versa. We now consider the boundaries between each phase in
turn, consulting figure 2 as an ansatz for the arrangement of the phases.
(i) MC/LD phase boundary. The MC phase can exist if N > LM/2. If we consider
starting in the MC phase with large β and reducing α0, we will cross into the LD
phase when α = α∗. Using (3) this gives an equation for the boundary in the
α0-β plane, where
?N
Note that α∗depends on a, b,¯T, V0, L and M. Hence the MC/LD boundary is
a vertical line on the α0-β phase plane.
(ii) MC/HD phase boundary. If we now consider starting in the MC phase with large
α0 and reducing β, then we cross into the HD phase when β = α∗, i.e., in the
α0-β plane this boundary is given by the horizontal line
α0= α∗coth
N∗−LM
2N∗
?
for β ≥ α∗and N > LM/2 .
β = α∗for α0≥ α∗coth
?N
N∗−LM
2N∗
?
and N > LM/2 .
(iii) HD/SP phase boundary. Here we consider moving from the HD to the SP. For
the system to be in the HD phase requires β < α∗, N > LM/2 and β < α. Using
equation (3) in the latter inequality gives β < α0tanh(Nr/N∗); using equation
(2) and the density in HD phase gives the following equation for the boundary
α0= β coth
?N
N∗−LM
N∗(1 − D(β))
?
for β < α∗and N > LM/2 .(11)
Hence, the HD/SP phase boundary is a curved line on the α0-β phase plane.
(iv) LD/SP phase boundary. Finally we consider starting the the LD phase and
moving to the SP. In LD we require α < β and α < α∗. Finding an expression
for these inequalities poses some difficulty, since by using equation (3) and the
density in the LD phase we obtain
α = α0tanh
?N
N∗−LM
N∗D(α)
?
,
an equation which cannot be solved analytically to find α as a function of α0.
Instead we solve this numerically, setting α = β (which occurs at the SP) to find
β as a function of α0. This gives another curved line on the α0-β phase plane.
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Multiple phase transitions in a TASEP with limited particles and fuel carriers 10
Figure 5. Colour on-line. Phase diagrams in the α0-β plane at different values
of V0and N. White lines show the phase boundaries as determined by the mean-
field model of section 3. Colour maps show the steady-state particle density ρ
from Monte Carlo simulations. Parameters used are V0 = 20 or 300 s−1, and
N = 200, 450, or 600. Other parameters are L = 500, M = 1, a = 2 × 10−4s−1,
b = 50, and¯T = 5000. The dotted line in (f) shows the line of constant α0 used
in figure 8.
We can then construct the phase plane by plotting the phase boundaries (β as a
function of α0) for any given values of N and V0. Unless otherwise stated, throughout
the rest of this paper we use parameters¯T = 5000 so that¯T ≫ LM and hence, it is
always the refuelling which is the limiting process, and not the total number of fuel
carriers. This represents realistic scenarios in biological transport processes, such as
protein synthesis. We set the time scale of the system by choosing a = 2 × 10−4s−1,
such that the maximum hopping rate is k = 1 s−1. A value of b = 50 then gives a
sharp onset of LR as shown in figure 3(b). The phase diagram boundaries calculated
using the mean-field approach are shown in figure 5 using white lines.
To test the validity of the mean-field results derived above we perform simulations
using a continuous time Monte Carlo method [32]. The length of the Monte Carlo time
step is chosen from an exponential distribution, such that the events occur according
to a Poisson process, with a single event occurring at each step. Possible events are
the movement of a particle (either on to, along, or off of the lattice) or the refuelling of
a fuel carrier. The event which occurs is chosen stochastically from the set of particles
which have a vacancy to their right and the set of empty fuel carriers. Particles are
chosen with a probability such that they move with a rate k, and empty fuel carriers
are chosen with a probability such that they are recharged with rate V ; after each
event T is updated accordingly. To remove any transient effects associated with the
initial condition we disregard the first 5 × 106time steps. Assuming that the system
is ergodic we average currents and densities over at least a further 4 × 107steps.
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Multiple phase transitions in a TASEP with limited particles and fuel carriers 11
Figure 6.
the steady-state fuelling level T/¯ T at different values of α0 and β for small
V0 = 20 s−1. White lines show phase boundaries as predicted by the mean-
field model. (a) Small N = 200; (b) mid-range N = 450; (c) large N = 600. For
these parameters we have αLR= βLR≈ 0.040 from (9). Other parameters are as
figure 5.
Colour on-line.Colour maps showing Monte Carlo results for
In figure 5 we present a series of phase planes for different values of N and V0.
We show the particle density ρ obtained from Monte Carlo simulations as a colour
map with the phase boundaries from the mean-field model overlaid (white lines). Note
that the mean-field model very closely predicts the boundaries. Here we show data
for M = 1 lattices, but the plots look the same for M > 1 with appropriately scaled
parameters. Introducing more lattices does not change the macroscopic behaviour,
but changes the microscopic behaviour for the SP (see section 3.1).
In figure 5 we note that for small V0 the phase diagrams look broadly similar
to the large V0case, but as we expect from [13, 14], the phase transitions occur at
much smaller values of α0 and β. We also obtain a limited resources (LR) regime
within each of the phases. This can seen in figure 6, which shows colour plots for the
proportion of charged fuel carriers (T/¯T) for different values of N, with small V0. The
onset of the LR regime can be clearly seen as T/¯T decreases dramatically over a small
range of α0or β. For large V0the refuelling is so quick that T/¯T is constant through
all phases, i.e. we recover the results for a TASEP with a finite pool of particles, but
no constraints on the fuel carriers, since these are refuelled almost as soon as they are
used (data not shown). For clarity, throughout the rest of the paper when we refer
to LR or limited resources, we specifically mean the regime where the pool of loaded
fuel carriers has become depleted.
Crucially, the presence of the LR regime also alters the shape of the phase
boundaries; a noticeable “kink” can be seen in the LD/SP and HD/SP phase
boundaries at the point of LR onset. Particularly striking is the shape of the HD/SP
boundary for the large N small V0case, and we examine this in detail in section 3.2.
3.1. The Shock Phase
As noted above, due to the finite number of particles, the coexistence line present
in the original TASEP – the shock phase – opens out into a region on the phase
plane. We examine the behaviour in this phase by locating the domain wall (DW)
which separates the regions of LD and HD, and examining how the position of this
is affected by the finite number of fuel carriers and how it changes at different values
of α0 and β. The introduction of a finite number of particles also gives a change
from a DW which can wander freely along the lattice, to one where the wandering is
constrained by the presence of the reservoir. This is most easily explained in the case
Page 12
Multiple phase transitions in a TASEP with limited particles and fuel carriers 12
of M = 1 lattices, where the DW is on average fixed in position. A fluctuation which
leads to movement of the DW away from its mean position will change the number of
particles in the reservoir; this in turn changes the entry rate α, acting like a restoring
force on the DW. In actual fact the DW executes Gaussian fluctuations about its
mean value, and we discuss fluctuations further in section 4. If multiple lattice are
introduced (M > 1), then as in [11] the DW is not pinned, but rather it is the sum of
the DW position on all lattices which executes Gaussian fluctuations about a mean.
We focus on the M = 1 case for the rest of this section.
The relative mean position x ∈ [0,1] of the pinned DW (where xL gives the site
at which it is located) can be estimated from the mean-field model by approximating
the density in the SP as follows
ρSP= xρLD+ (1 − x)ρHD.
Since x is always selected such as to maintain the condition α = β, we can use equation
(10) and the densities from (8) to show that
1
1 − 2D(β)
We also note that the difference between ρLDand ρHDdecreases as β increases, i.e.,
the “height” of the wall decreases.
As it is the most interesting case, we focus on parameters where the SP has the
largest area on the phase diagram, namely the mid-range N cases, i.e, figures 5(b)
and (e). In figure 7 we show plots for the mean position of the DW as a function of
β, for large and small values of V0. Also shown for each case is the density in the HD
region of the lattice (to the right of the DW) as a function of β, which will aid in the
following discussion.
For large V0(figures 7 (a) and (c)) there is a monotonic increase in x with β. A
larger value of β requires that more particles be present in the reservoir in order to
achieve α = β. At small β, x is approximately constant with β; this is because the
decrease in density on the HD side of the lattice is a sufficient release of particles to
maintain α = β. Due to the saturating form of the function α(Nr) (equation (3)), for
larger values of Nr a greater increase in Nr is required to give the same increase in
α. So for larger values of β, the change in the HD density as β increases no-longer
releases sufficient particles to keep α = β; the DW also must move towards the right
such that there is a steep increase of x with β.
In the small V0 case there is an LR regime within the SP which results in an
interesting dependence of x on β (figure. 7(b)); in contrast to the large V0value case,
x does not increase monotonically with β. We can understand this behaviour by again
considering the density in the HD region of the lattice. We note that for the small V0
case, ρHDchanges differently with β depending on whether the system is in the LR
regime or not, and that the maximum in x at β ≈ 0.04 s−1corresponds to the onset of
the LR regime. For β < 0.04 s−1we see from figure 7(d) that, increasing β results in
a decrease in the density in the HD region – and therefore a release of particles to the
reservoir and an increase in α. However the decrease of ρHDwith β is not enough to
maintain α = β. The DW must also move rightwards, i.e. there is an initial increase
of x with β. After the onset of LR, β > 0.04 s−1, figure 7(d) shows that ρHDdecreases
much more quickly with increasing β. So now the density on the HD side of the DW
decreases much more rapidly as β increases. The resulting release of particles would
be too great to maintain α = β if the wall did not also move leftwards – x decreases
again.
x =
?
1 − D(β) −
N
LM
?
1 −N∗
N
tanh−1
?β
α0
???
.
Page 13
Multiple phase transitions in a TASEP with limited particles and fuel carriers 13
Figure 7. Plots (a) and (b) show the relative mean DW position for systems in
the SP with V0= 300 s−1and V0= 20 s−1respectively. In both cases N = 450;
in (a) α = 0.6 s−1, and in (b) α = 0.1125 s−1. In (b) the onset of the LR
regime is at β ≈ 0.04 s−1. Plots (c) and (d) show how the density in the HD
phase ρHD= 1 − D(β) varies with β, again for V0 = 300 s−1and V0 = 20 s−1
respectively. In the SP this is the density to the right of the DW. In (d), initially
the density decreases slowly with increasing β; at the onset of LR the rate of
density decrease becomes more severe - a change in the behaviour not seen in a
model with an infinite number of fuel carries.
Deeper within the LR regime figure 7(d) we have the same situation as before:
due to the saturating function α(Nr), for large Nrwe need a greater increase in Nrto
give the same increase in α. The wall has to move rightward as β increases in order
to release enough particles to maintain α = β.
3.2. Multiple Phase Transitions
By combining the effects of both types of limited resources, we obtain a novel phase
diagram (figure 5(f) and figure 6(c)). There is an unusual kink shape in the phase
boundary between the HD and SP regimes, given by equation (11). As can be seen in
figure 6(c), this is at the point where the system enters the limited resources regime.
It is possible to draw a vertical line at constant α0through the phase diagram (dashed
line in figure 5(f)), which cuts through the phases HD→SP→HD→SP→LD as β
increases, i.e., by varying only one parameter we can go from the HD phase through
a transition to SP, and then a transition back to HD, etc. Figure 8 shows how the
quantities ρ, T/¯T, J, x (where applicable) and α, vary along this line of constant
α0; we show both Monte Carlo results and the prediction of the mean-field model.
The mean-field model performs well deep within each phase, but begins to show some
discrepancy near the phase boundaries; we discuss this further below.
We label the phases shown in figure 8 with roman numerals I–V and explain each
in turn. These are that phases which are crossed by the dashed line in figure 5(f).
Phase I At very small values of β we have α ≫ β, so the system is in the HD
phase. Again considering how ρHD varies with β, from figure 7(d) we see that
for small β the slope is small, dρHD/dβ ∼ −1. A decreasing density means an
Page 14
Multiple phase transitions in a TASEP with limited particles and fuel carriers 14
increasing number of free particles, i.e., dNr/dβ ∼ 1; the hyperbolic tangent form
of equation (3) means that α increases with β but at a very low rate (dα/dβ ≪ 1,
see figure figure 8(d)) ?.
Phase II We arrive at phase II as follows: in phase I we started with very small
values of β such that α ≫ β. By increasing β, particles are freed and therefore,
α also increases. However, dα/dβ ≪ 1 in phase I, and hence, we eventually reach
α = β, and there is a transition to an SP – phase II. Here we have coexistence of
both LD and HD separated by a DW.
The current — and therefore the fuel carrier use rate — increases with β through
phase I, and initially in phase II (figure 8(b)). About half way through phase II
(β ≈ 0.04 s−1) the system enters the LR regime (see crosses in figure 8(a)).
From figure 7(d) we know that in the first half of phase II, the density in the HD
part of the system decreases slowly with β — dρHD/dβ ∼ −1. The corresponding
increase in the number of free particles would not be enough to keep α = β, so
the DW also moves rightward, i.e. there is an initial increase in x in phase II.
At the onset of LR the slope of the curve in figure 7(d) gets steeper, i.e.,
dρHD/dβ ≪ −1. As β is increased further (in the second half of phase II) the DW
must move leftwards again in order to keep α = β; i.e. after initially increasing
with β, x then decreases as LR onsets, as shown in figure 8(c) (see inset).
Phase III Once the DW reaches the leftmost side of the lattice the system can no
longer maintain the condition α = β, so there is a phase transition and we re-enter
the HD phase. Further increase of β increases the number of free particles Nr,
however since α is a saturating function of Nr, dα/dβ begins to decrease. That
is, as β increases through phase III the slope of α(β) gets shallower (figure 8(d)).
Phase IV If we keep increasing β, we reach β = α, and then the system enters the SP
for a second time. From figure 7(d) we see that deep within the LR regime ρHD
again varies slowly with β, and the difference between the LD and HD densities
is small; therefore in this second SP, changes in the density in the two regions
of the lattice would not significantly change the number of free particles. Thus
rapid variation of the DW position is required as β increases in order to maintain
α = β.
Phase V Once the DW reaches the rightmost edge of the lattice, the system can no
longer maintain the condition α = β by moving the wall, and there is a transition
to the LD phase.
The above description accounts for the changes in DW position, current and
density predicted by the mean-field theory, but as we noted previously there is some
discrepancy with the Monte Carlo results, particularly near the transitions. This is due
to the fact that our mean-field model assumes that the density is constant throughout
the lattice, when in fact there is some change near the edges [7]. Also the mean-field
treatment ignores correlations in the density which occur near the DW. These edge
effects become less significant as L is increased, and therefore the discrepancy between
the mean-field and simulation results reduces (data not shown).
We also note that there is some difficulty in determining the existence and
position of the domain wall. In figure 8(c) we define DW position from simulations by
considering the mean particle density at each lattice site ρi; we define the existence
of a DW if for any pair of adjacent lattice sites i,i + 1 the density cuts through 0.5.
? In the model with finite resources α depends on β, in contrast to the standard TASEP.
Page 15
Multiple phase transitions in a TASEP with limited particles and fuel carriers 15
Figure 8.
fuelling level T/¯ T (crosses), (b) the current J, (c) the mean DW position x, and
(d) the entry rate α as given by equation (3), vary with β for small V0and large N
at a fixed value of α0= 0.08 s−1. Points show Monte Carlo results and lines the
prediction from the mean-field model. The system passes through several phases
and we indicate with dotted lines the positions of the boundaries as predicted by
the mean-field model. We label each phase with roman numerals I-V. The inset
in plot (c) shows a zoom around the SP phase II. The dashed line in (d) shows
α = β.
Colour on-line. Plot showing (a) how the density ρ (points) and
Then, the position of the DW is given by lattice site i. The difficulty arises in the fact
that this can also occur near the edges of the system when it is not in the SP. This
explains why it appears that there are DWs when the system is not in the SP – we
are actually detecting the decrease in the density at the edge of the system.
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