# Multiple phase transitions in a system of exclusion processes with limited reservoirs of particles and fuel carriers

**Abstract**

The TASEP is a paradigmatic model from non-equilibrium statistical 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. As as consequence,
the system's 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 beta for a fixed entry rate alpha. 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.

# Figures

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 Brackley

1

, Luca Ciandrini

1

and M Carmen

Romano

1,2

1

Institute for Complex Systems and Mathematical Biology, SUPA, University

of Aberdeen, Aberdeen, AB24 3UE, United Kingdom

2

Institute of Medical Sciences, Foresterhill, University of Aberdeen, Aberdeen,

AB25 2ZD, United Ki ngdom

E-mail: cab@chrisbrackley.co.uk, l.ciandrini@abdn.ac.uk

PACS numbers : 05.60.-k, 05.40.-a,02.50.Ey

Abstract. The TASEP is a paradigmatic model from non-equilibrium statistical

physics, which describes particles hopping along a lattice of discrete si tes. 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. As as consequence, the system’s

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 limi ted,

the particle hopping rate is severely reduced.

Keywords: driven diﬀusive systems (theory), stochastic processes (theory)

Multiple phase transitions in a TASEP with limited particles and fuel carriers 2

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) trans lation [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 numbe r 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 s uch systems occurring in nature. In this paper

we introduce a new model which includes the ﬁnite availability of both re sources, 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 sys tem can

go from a high density r e gime, to a shock phase, then to a high density phase again,

visit the shock phase once mor e, 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 re sults by means of Monte Carlo s imulations.

In its most simple form, the TASEP consists of a 1D lattice of L sites upon which

particles can sit, se e 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 tha t

the downstream site is vacant. A s ystem with open boundaries, as we shall conside r

here, can display rich dy namics with multiple boundary induced phases [15]. Particles

are allowed to hop onto the lattice with rate α at one e nd, 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 be en solved using matrix methods [19, 2 0]. Mean-ﬁeld

methods have also been extensively used [16], since they are easily tractable a nd yield

a good appr oximation in many cases. There has been much extension of this simple

model, for example varia ble hopping rates (site or pa rticle dependent) [21, 22, 23],

extended particles which cover more than one site [6, 2 4], 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 o ccupation of the ith lattice site n

i

= 1 if

the site contains a particle and n

i

= 0 otherw ise. The system is characterised by the

steady-state particle current J (the r ate at which particles pass any given point on

the lattice), and the mean site occupancy (often called density) ρ

i

= hn

i

i, where h· · ·i

denotes average over realisations of the system (which we assume is ergodic, so this is

equivale nt to a time average). The average density is therefore given by ρ = L

−1

P

i

ρ

i

.

There are four possible phases depending on the values of α and β: the entry limited

or low density (LD) phase, the exit limited or high density (HD) phase, a maximal

current (MC) phase wher e 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 ρ

i

for given α and β.

Multiple phase transitions in a TASEP with limited particles and fuel carriers 3

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 wi th 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 o f the system) is given by

(LD) ρ

LD

= α/k for α < β, α < k/2 ,

(HD) ρ

HD

= 1 − β/k for β < α, β < k/2 ,

(MC) ρ

MC

= 1/2 for β, α ≥ k/2 ,

(1)

and the current is always given by J = kρ(1 − ρ). The SP, which occurs for

α = β < k/2, presents an HD region on the right of the lattice a nd an LD region

on the left, separated by a boundary which diﬀuses fr eely through the lattice. This

has often been described using a domain wall (DW) theor y [30]. Due to the fr ee

diﬀusion of the DW a time average of the density in SP gives ρ

SP

= 1/2, but the

current depends on the density in either the LD or HD regions o f the lattice, i.e.

J

SP

= α(1 − α/k) = β(1 − β/k).

In this paper we consider several TASEPs which share a c ommon ﬁnite pool of

both particles and fuel carriers. The entry rate, which is the sa me for each TASEP,

depe nds 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 alterna tive 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

Multiple phase transitions in a TASEP with limited particles and fuel carriers 4

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 w ith their cargo once it has been used (ﬁgure 1 (b) shows a s chematic

representation of this model). We show that novel eﬀects arise when both types of

limited resource are considered, e.g. multiple phase transitions ca n occur when varying

the exit rate β. The outline of the paper is as follows: in sec tion 2 we summarise the

previous results for the two models s e parately, before describing in section 3 a mean-

ﬁeld mo de l for a system with b oth 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. Fina lly, 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 s hares the s ame r eservoir of particles; then we present the concept of fuel-

carriers and the eﬀect of a ﬁnite rate of refuelling on the exclusion pr ocess dynamics.

2.1. Finite number of particles

In this work we analyse a s ystem 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 N

r

.

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 = N

r

+ LMρ, (2)

where ρ is the density on each lattice. The entry rate of the M lattices depends on

the numb e r of free particles via a saturating function

α = α

0

tanh

N

r

N

∗

= α

0

tanh

N − LMρ

N

∗

, (3)

where the constant α

0

gives the e ntry rate in the limit N

r

→ ∞ and is an intrinsic

property of the lattices ‡ . Without loss of ge ne rality, we ﬁx the normalisation factor

N

∗

to be LM/2, i.e. the tota l 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 e quations for α in terms of α

0

and N. Since the densities in each phase are the

same as those in the sta ndard TASEP (equations (1)), for a given set of parameters

(α

0

, β, N ) we ﬁnd the resulting α which determines the phas e ; e.g. if α < β and

α < k/2 the system will be in the LD pha se. By substituting equation (3) into these

inequalities, we get a re presentation of the diﬀerent phases on the α

0

–β plane.

As a c onsequence 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 wi th the function used in [9, 10, 11], and is relevant, e.g. for the

application to protein synthesis .

Multiple phase transitions in a TASEP with limited particles and fuel carriers 5

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 l oaded

fuel carriers). For s mall 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 reg imes 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, ther e 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 regio n;

since α depends on both α

0

and β (through its depe ndenc e on the bulk density), the

line opens into a region on the α

0

-β phase plane. That is to say, the condition α = β

is fulﬁlled for a certain range of α

0

[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 o f the HD phase on the α

0

-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 unconstra ine d 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 be c ause 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. Similar ly if the DW moves to the left the number o f free

particles decreases, decreasing the entry rate and driving the DW rightwards. The

opening of the SP line into a region on the α

0

-β pla ne is possible because a diﬀerent

mean p osition for the DW corr esponds to a diﬀerent value of α

0

, while keeping α = β.

Hence, the system can maintain α = β for diﬀerent values of α

0

. 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 ga s, the

energy required fo r the advancement of the particles is obtained from some kind

of ﬁnite resourc e. For instance, molecular motors consume ATP molecules, and

ribosome movement on mRNAs during protein synthesis requires aminoacylated tRNA

Multiple phase transitions in a TASEP with limited particles and fuel carriers 6

complexes and GTP. The latter case has re cently been described in [13, 14, 31], and

here we brieﬂy review those results, before in the next sectio n combining this with the

a ﬁnite pool of particles model.

We consider a ﬁnite numb e r

¯

T of fuel c arriers, T of which are carrying fuel. Every

time a pa rticle moves, the fuel from one of the loaded ca rriers is used, and hence, T

is reduced by one. The hopping rate of pa rticles depends on the availability of loa de d

fuel carrie rs, and the empty carriers are refuelled at a ra te 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 =

V

0

(

¯

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 V

0

and saturation determined by the constant b. Any

saturating function will give the same qualitative res ults, but the above fo rmulation

allows for a straightforward analytical treatment [13, 14] §.

In this section we describe a c ollection of M identical TASEPs with a ﬁnite

number of fuel carrie rs (but no constraints on the number of particles); following the

common mean-ﬁeld trea tment [16] the particle density on the ith site of each lattice

is given by

dρ

1

dt

= α(1 − ρ

1

) − kρ

1

(1 − ρ

2

),

dρ

i

dt

= kρ

i−1

(1 − ρ

i

) − kρ

i

(1 − ρ

i+1

), i = 2, ...L − 1,

dρ

L

dt

= kρ

L−1

(1 − ρ

L

) − βρ

L

.

(5)

The inclusion of a ﬁnite pool of fuel carriers leads to the additional equation

dT

dt

=

V

0

(

¯

T − T )

b +

¯

T − T

−

(L−1)M

X

j=1

kρ

j

(1 − ρ

j

), (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

V

0

− J(L − 1)M

, (7)

i.e., the hopping rate is now itself a function of the current. Following [13, 14], upon

solving eq uations (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) J

LD

= J (α), ρ

LD

= D(α) for α < α

∗

and α < β,

(HD) J

HD

= J (β), ρ

HD

= 1 − D(β) for β < α

∗

and β < α,

(MC) J

MC

= α

∗

/2, ρ

MC

= 1/2 for α, β ≥ α

∗

,

(SP) J

SP

= J (α), ρ

SP

= 1/2 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.

Multiple phase transitions in a TASEP with limited particles and fuel carriers 7

where

J (α) =

1

2

α

1 −

α

a(

¯

T + b)

+

¯

T

¯

T + b

V

0

L

′

M

−

s

¯

T

¯

T + b

V

0

L

′

M

+ α

1 −

α

a(

¯

T + b)

2

−

4α(a

¯

T − α)

a(

¯

T + b)

V

0

L

′

M

,

D(α) =

1

2α

α

1 +

α

a(

¯

T + b)

−

¯

T

¯

T + b

V

0

L

′

M

+

s

¯

T

¯

T + b

V

0

L

′

M

+ α

1 −

α

a(

¯

T + b)

2

−

4α(a

¯

T − α)

a(

¯

T + b)

V

0

L

′

M

,

and

α

∗

=

a

4

(

¯

T + b) +

V

0

L

′

M

−

s

a

4

(

¯

T + b) +

V

0

L

′

M

2

− a

¯

T

V

0

L

′

M

,

with L

′

= L − 1. The behaviour of these functions as α and β are varied dep ends

on the parameters a, b and V

0

. By considering the steady-state of equatio n (6) and

noticing that the maximal value that the recharging rate can possibly have is equal to

V

0

, we note that the particle current is limited from above by V

0

/L

′

M. Hence , ther e

are substantially two diﬀerent cases: (i) if V

0

/L

′

M ≫ 1, the recharging rate is very fast

and the particle current is not inﬂuenced by it; we recover the r esults of the original

TASEP; (ii) if in co ntrast V

0

/L

′

M ≪ 1, the recharging o f the fuel carriers can limit

the value of the particle current. Figure 3 shows the current for sets of parameters

corres ponding to each case. In ca se (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 r efer to the regime w he re the current

appears independent of α as a limited resources (LR) regime, s inc e the rate at which

fuel is used by the particles approaches the rate at which fuel c arriers ar e re loaded.

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 re gime within the LD phase, the sensitivity of the current to changes

in α or β is greatly r educed, whilst the sensitivity of the density is greatly increased.

In the MC phas e, 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 , V

0

/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 pa per we choose b such that there is a sharp onset of LR, a nd take

V

0

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 oc curs

by equating the rate of fuel carrier use (appr oximately αL

′

M for small α in LD and

βL

′

M for small β in HD) and the maximum recharging rate. This gives

α

LR

= β

LR

≈

¯

T V

0

(

¯

T + b)L

′

M

. (9)

Multiple phase transitions in a TASEP with limited particles and fuel carriers 8

Figure 3. Plots showing the current as a function of α for diﬀerent parameters.

In (a) V

0

= 300 s

−1

, and in (b) V

0

= 20 s

−1

. In both cases a = 2 × 10

−4

,

¯

T = 5000, and b = 50. Solid lines show the current in the LD phase, J

LD

= J (α),

and dashed lines the current in the MC phase, J

MC

= α

∗

/2. The dotted line is at

α

∗

, where there is a transi tion from LD→MC. The dot-dashed line in (b) shows

the value of α

LD

as estimated in equation (9).

Figure 4. Steady-state fuelling level T /

¯

T as a function of rate V

0

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

−4

s

−1

, b = 50, and

¯

T = 5000. For

small V

0

the number of loaded fuel carriers is depleted; for large V

0

the carriers are

practically always f ully loaded, and we recover the original unconstrained TASEP.

This value is repres ented in ﬁgure 3(b) by a dot-dashed vertical line. As it is shown

there, the estimation predicts q uite 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 par ticle s and fuel carriers. As we show later

in this section, it is only when combining the two schemes discussed above that

we can se e emerging novel eﬀects, such as multiple phase transitions. Analogous

to the de pendence of the entry rate α on the number of pa rticles N , the s teady-

state proportion of loaded fuel carriers is a saturating function of V

0

(see ﬁgure 4).

Therefore, by regarding N and V

0

as control parameters, we can va ry 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 a nd 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 exce pt 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

Multiple phase transitions in a TASEP with limited particles and fuel carriers 9

calculate the average value ρ

SP

using equation (3); using the fact that in this pha se

α = β leads to

ρ

SP

=

N

LM

1 −

N

∗

N

tanh

−1

β

α

0

. (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 pro ble m of ﬁnding the boundaries between the diﬀerent phases

as functions of α

0

, β, N and V

0

— i.e., eliminating α. Our aim is to draw the α

0

-β

phase plane for any given values of V

0

and N . As noted in previous sections, due to

ﬁnite particles, the SP line o pens 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 α

0

as a

function of β, or vice versa. We now consider the boundaries between e ach phase in

turn, cons ulting ﬁgure 2 as an ansatz for the arrangement of the phases .

(i) MC/LD phase boundary. The MC phas e can exist if N > LM/2. If we consider

starting in the MC phase with large β and reducing α

0

, we will cros s into the LD

phase when α = α

∗

. Using (3) this gives an equation for the boundary in the

α

0

-β plane, where

α

0

= α

∗

coth

N

N

∗

−

LM

2N

∗

for β ≥ α

∗

and N > LM/2 .

Note that α

∗

depe nds on a, b,

¯

T , V

0

, L and M. Hence the MC/LD boundary is

a vertical line on the α

0

-β phase plane.

(ii) MC/HD phase boundary. If we now consider starting in the MC phase with large

α

0

and r educing β, then we cross into the HD phase when β = α

∗

, i.e., in the

α

0

-β plane this boundary is given by the horizontal line

β = α

∗

for α

0

≥ 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 < α

0

tanh(N

r

/N

∗

); using equation

(2) and the density in HD phase gives the following equation for the b oundary

α

0

= 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 α

0

-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 thes e inequa lities poses some diﬃculty, since by using equation (3) and the

density in the LD phase we obtain

α = α

0

tanh

N

N

∗

−

LM

N

∗

D(α)

,

an equation which cannot be so lved analytically to ﬁnd α as a function of α

0

.

Instead we solve this numerically, setting α = β (which o ccurs at the SP) to ﬁnd

β as a function of α

0

. This gives ano ther curved line on the α

0

-β phase plane.

Multiple phase transitions in a TASEP with limited particles and fuel carriers 10

Figure 5. Colour on-line. Phase diagrams in the α

0

-β plane at diﬀerent values

of V

0

and 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 M onte Carlo simulations. Parameters used are V

0

= 20 or 300 s

−1

, and

N = 200, 450, or 600. Other parameters are L = 500, M = 1, a = 2 × 10

−4

s

−1

,

b = 50, and

¯

T = 5000. The dotted line in (f) shows the line of constant α

0

used

in ﬁgure 8.

We can then construct the phase plane by plotting the phase boundaries (β as a

function of α

0

) for any given values of N and V

0

. 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 re alistic scenarios in biological transport processes, such as

protein synthesis. We set the time scale of the system by choosing a = 2 × 10

−4

s

−1

,

such that the maximum hopping rate is k = 1 s

−1

. A value of b = 50 then gives a

sharp onse t of LR as shown in ﬁgure 3(b). The phase diagram boundaries calculated

using the mea n-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 le ngth of the Monte Carlo time

step is chosen from an exponential distributio n, 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 chos e n stochastically from the set of particles

which have a vacancy to their rig ht and the set of empty fuel carriers. Particles are

chosen with a probability such that they move with a rate k, and empty fuel car riers

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 × 1 0

6

time s teps. Assuming that the system

is ergodic we average currents and densities over at least a further 4 × 10

7

steps.

Multiple phase transitions in a TASEP with limited particles and fuel carriers 11

Figure 6. Col our on-line. Colour maps showing Monte Carlo results for

the steady-state fuelling level T /

¯

T at diﬀerent values of α

0

and β for sm all

V

0

= 20 s

−1

. White lines show phase boundaries as predicted by the mean-

ﬁeld model. (a) Small N = 200; (b) mi d-range N = 450; (c) large N = 600. For

these parameters we have α

LR

= β

LR

≈ 0.040 from (9). Other parameters are as

ﬁgure 5.

In ﬁgure 5 we prese nt a ser ies of phase planes for diﬀerent values of N and V

0

.

We show the pa rticle density ρ o btained from Monte Carlo simulations as a colour

map with the phas e boundaries from the mean-ﬁeld model overlaid (white lines ). Note

that the mean-ﬁeld model very closely predic ts 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 doe s not change the macroscopic behaviour,

but changes the microscopic behaviour for the SP (see section 3.1).

In ﬁgur e 5 we note that for small V

0

the phase diagrams look broadly similar

to the larg e V

0

case, but as we expect from [13, 14], the phase transitions occur at

much smaller values of α

0

and β. We also obtain a limited r esources (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 V

0

. The

onset of the LR regime can b e c le arly seen as T/

¯

T decreases dramatically over a small

range of α

0

or β. For large V

0

the refuelling is so quick that T/

¯

T is constant through

all phases, i.e. we recover the res ults for a TASEP with a ﬁnite pool of particles, but

no constraints on the fuel carriers, since thes e are refuelled almost as soon as they are

used (data not shown). For clarity, throughout the rest of the pa per when we refer

to LR or limited resources, we spec iﬁc ally 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 see n in the LD/SP and HD/SP phase

boundaries at the p oint of LR onset. Particularly strik ing is the s hape of the HD/SP

boundary for the large N small V

0

case, and we exa mine this in deta il 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 phas e

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 α

0

and β. The introduction of a ﬁnite numbe r 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

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 mea n position will change the number of

particles in the reservoir; this in turn changes the entry rate α, a cting like a restoring

force on the DW. In actual fact the DW executes Gaussian ﬂuctuations about its

mean value, and we discuss ﬂuctua tions 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 ca se 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 fro m the mean-ﬁeld model by approximating

the density in the SP as follows

ρ

SP

= xρ

LD

+ (1 − x)ρ

HD

.

Since x is always selected such as to maintain the condition α = β, we can use equation

(10) a nd the densities from (8) to show that

x =

1

1 − 2D(β)

1 − D(β) −

N

LM

1 −

N

∗

N

tanh

−1

β

α

0

.

We also note that the diﬀerence between ρ

LD

and ρ

HD

decreases as β increases, i.e.,

the “height” of the wall decreases.

As it is the most interesting ca se, 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 V

0

. Also s hown 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 dis cussion.

For lar ge V

0

(ﬁgures 7 (a) and (c)) there is a monotonic increase in x with β. A

larger va lue of β req uires that more particles be present in the reser voir in order to

achieve α = β. At sma ll β, x is approximately constant with β; this is because the

decrease in density on the HD side of the la ttice is a suﬃcient release of particles to

maintain α = β. Due to the saturating form of the function α(N

r

) (equation (3)), for

larger va lue s of N

r

a greater increase in N

r

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-longe r

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 V

0

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 V

0

value case,

x does not increase monotonically with β. We c an understand this behaviour by again

considering the density in the HD region of the lattice. We note that for the sma ll V

0

case, ρ

HD

changes 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

−1

corres ponds to the ons e t of

the LR regime. For β < 0.04 s

−1

we see from ﬁgure 7(d) that, increasing β results in

a decrease in the density in the HD region – and ther e fore a release of particles to the

reservoir and an increase in α. However the decrease of ρ

HD

with β is not eno ugh to

maintain α = β. The DW must also move rightwards, i.e. there is a n initial increas e

of x with β. After the onset of LR, β > 0 .04 s

−1

, ﬁgure 7(d) shows that ρ

HD

decreases

much more quickly with increasing β. So now the density on the HD side of the DW

decreases much more r apidly as β incre ases. The resulting release of particles would

be too gre at to mainta in α = β if the wall did not also move leftwards – x decreases

again.

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 V

0

= 300 s

−1

and V

0

= 20 s

−1

resp ectively. 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 ρ

HD

= 1 − D(β) varies with β, again for V

0

= 300 s

−1

and V

0

= 20 s

−1

resp ectively. In the SP this is the density to the right of the DW. In (d), initiall y

the density decreases s lowly w ith 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 wi th 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(N

r

), for lar ge N

r

we need a gre ater increase in N

r

to

give the same increase in α. The wall has to move rightward as β increa ses in order

to release enough pa rticles to mainta in 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 be tween the HD and SP regimes, given by equation (11). As can be seen in

ﬁgure 6(c), this is at the point where the sys tem enters the limited resources regime.

It is possible to dr aw a vertical line at constant α

0

through the phase diagram (dashed

line in ﬁgure 5(f)), which cuts throug h the phases HD→SP→HD→SP→LD a s 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

α

0

; we show both Monte Carlo results and the prediction of the mean-ﬁeld model.

The mean-ﬁeld mo del p e rforms well deep within each phase, but begins to show s ome

discrepancy near the phas e 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 consider ing how ρ

HD

varies with β, from ﬁgure 7(d) we see that

for small β the s lope is small, dρ

HD

/dβ ∼ −1. A decreasing density means an

Multiple phase transitions in a TASEP with limited particles and fuel carriers 14

increasing number of free particles, i.e., dN

r

/dβ ∼ 1; the hyperbolic tangent form

of equation (3) means that α incre ases with β but at a very low rate (dα/dβ ≪ 1,

see ﬁgure ﬁgure 8(d)) k.

Phase II We arrive at phase II as follows: in phase I we started with very sma ll

values of β such that α ≫ β. By increasing β, particles are freed and therefo re,

α also incre ases. However, dα/dβ ≪ 1 in phase I, and hence, we eventua lly reach

α = β, and there is a transition to an SP – phase I I. Here we have coexistence of

both LD and HD separated by a DW.

The current — and therefore the fuel carrie r use rate — increases with β through

phase I, and initially in phas e II (ﬁgure 8(b)). About half way through phase II

(β ≈ 0.04 s

−1

) the s ystem 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 — dρ

HD

/dβ ∼ −1. The cor responding

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 c urve in ﬁgure 7(d) gets steeper, i.e.,

dρ

HD

/dβ ≪ −1. As β is increased further (in the second half of phase II) the DW

must move leftwards again in o rder to keep α = β; i.e. after initially increasing

with β, x then de c reases as LR onsets, as shown in ﬁgure 8(c) (see inset).

Phase III Once the DW reaches the le ftmos t 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 incr eases the numb er of free particles N

r

,

however since α is a saturating function of N

r

, dα/dβ begins to decrease. That

is, as β increases through phase III the slope of α(β) gets sha llower (ﬁgure 8(d)).

Phase IV If we keep increasing β, we reach β = α, and then the system enter s the SP

for a second time. From ﬁgure 7(d) we see that deep within the LR regime ρ

HD

again varies slowly with β, and the diﬀerence between the LD and HD densities

is small; therefore in this s e cond 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 ca n 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 a s we noted prev iously 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 thr oughout

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 (da ta 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 de nsity at each lattice site ρ

i

; we deﬁne the existence

of a DW if for any pair of adjacent lattice sites i, i + 1 the density cuts through 0.5.

k In the model with ﬁnite resources α depends on β, in contrast to the standard TASEP.

Multiple phase transitions in a TASEP with limited particles and fuel carriers 15

Figure 8. Colour on-line. Plot s howing (a) how the density ρ (points) and

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 smal l V

0

and large N

at a ﬁxed value of α

0

= 0.08 s

−1

. Points show Monte Car lo 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 l ine in (d) shows

α = β.

Then, the position of the DW is g iven by lattice site i. The diﬃculty arises in the fact

that this c an 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 actua lly detecting the decrease in the density at the edge of the system.

Multiple phase transitions in a TASEP with limited particles and fuel carriers 16

4. Fluctuations

In this section we examine ﬂuctuations in the density of particles on the la ttice,

focusing on the case of M = 1 lattice. We obtain power spectra for the ﬂuctuations

in density by taking the average o f the Fourier transform of several diﬀerent time

series. These time series are ge ne rated by recording the densities at time intervals

of ∼ 125 s. Since our simulation method does not advance time in regular steps,

the time intervals are not exactly uniform; therefore we use c ubic spline ﬁtting to

obtain a time series at regular intervals. Deﬁning ρ(t) as the instantaneous density

at time t, the power spectrum is given by I(ω) = h|FT [ρ(t) − ρ] |

2

i, where the angled

brackets denote average over diﬀerent simulation runs, and FT[f(t)](ω) is the discrete

Fourier transform of the time series f(t). As before ρ denotes the time average density.

From now on, we use the fo llowing notation: if we mean the instantaneous value of a

quantity at time t, then explicit time dependence is indicated; symbols without time

depe ndence denote the time average of the quantity.

4.1. LD and HD Phase

A good approximation to the power spectrum of the density ﬂuctuations in the LD

phase for the original TASEP (no particle or fuel carriers constraints) can be found

via a continuum description, with ﬂuctuations travelling through the lattice with a

velocity v and an eﬀective diﬀusion coeﬃcient D [33].¶ At low frequencies (ω ≪ v

2

/D)

the power spectrum shows oscillations, or dips in power, at unit multiple of 2πv/L,

i.e., at frequencies co rresponding to the length of time it takes a ﬂuctuation to traverse

the entire lattice. These oscillations are damped for ω > v

3

/DL, after which I ∝ ω

−2

;

for large frequencies I ∝ ω

−3/2

. For the case of ﬁnite re-fuelling, but no particle

constraints (i.e. the N → ∞ limit), we obtain a similar power spectrum . As shown in

ﬁgure 9(a), we have oscillations with dips at multiples of 2πv/L (with v ≡ k(1 − 2 ρ ),

see [33]); at larger values of α, when the system is in the LR regime, the oscillations

are severely damped, ﬁgure 9(b). This is because the hopping rate k is highly reduced

in the LR regime, meaning that v is reduced, and hence, the damping occurs at

much lower freq uenc ies. What remains o f the dips is still clearly visible in the power

spectrum at low multiples of 2πv/L, implying that the density ﬂuctuations are not

largely a ﬀected by the ﬂuctuations in the hopping rate in this regime. The I ∝ ω

−2

for mid- ranged ω and I ∝ ω

−2/3

for large ω re lationships are maintained even for

parameters such that the mean number of loaded fuel carriers T is small, c onditions

under which one might have though that ﬂuctuations in T would become important

(data not shown). We ﬁnd that actually ﬂuctuations in loaded carrier levels become

small when the mean value is small.

+

In summary, just as in the original TASEP, when there are constrained fuel

carriers the oscillations in the power spectrum of the number of particles on the lattice

are damped for ω > v

3

/DL. In the original TASEP this is most noticeable at larger

values of α, which give large values of ρ, and therefore small v. With constrained fuel

carriers small v is obtained in the LR regime due to the reduction in the value of k,

i.e., at much smaller values of α.

¶ Due to particle-hole symmetry (which is maintained in the present model), our understanding of

ﬂuctuations in the LD phase can also be applied to the HD phase.

+

In fact we ﬁnd that T ≈ h(T (t) − T )

2

i for small T in the LR. The probability of ﬁnding T loaded

fuel carriers appears to be very close to a Poisson distribution in this regime, but further discussion

of this is beyond the scope of the present work.

Multiple phase transitions in a TASEP with limited particles and fuel carriers 17

Figure 9. Plots showing power spectra of the total density from simulations of

a system with ﬁnite fuel carriers with V

0

= 20 s

−1

, but no constraints on the

number of particles (N → ∞). Plot (a) is for α = 0.01 s

−1

, (b) for α = 0.04 s

−1

,

and (c) α = 0.06 s

−1

; in the latter two plots the system is in the LR regime, and

we note that the oscillations have been damped out (see text). The dashed line

has a slope ω

−2

, and the dotted lines in each plot are at ω = 2πv/L and 4πv/L.

These spectra were obtained by averaging the Fourier tr ansforms of 500 diﬀerent

time series.

Figure 10. Colour on-line. Pl ots comparing power spectra of density ﬂuctuations

for systems with a l arge number of particles (black) and a small number of particles

(red). (a) Systems not in the LR regime, with V

0

= 300 s

−1

. Black lines show

results for a large number of particles: N = 600 and α

0

= 0.4 s

−1

. Red l ines show

results for a small number of particles: N = 220 and α

0

= 2.83 s

−1

. (b) Systems

in the LR regime, with V

0

= 20 s

−1

. Again black lines show results for a large

number of particles: N = 600 and α

0

= 0.07 s

−1

. Red lines show results for a

small number of particles: N = 220 and α

0

= 0.65 s

−1

. The diﬀerent values of

α

0

used in each case are chosen such that the systems we compare have the same

mean density and m ean hopping rate.

Density ﬂuctuations have also previously been studied for the TASEP with

constrained particles [34]. The eﬀect in the LD re gime is to suppress the ﬂuctuations.

This is due to the feedback eﬀect of the particle pool which stabilises the de nsity

(an increase in density leads to a decrease in entry rate). The eﬀect is reduced fo r

higher frequencies, as these correspond to short time scales over which the feedback

from the particle pool has less inﬂuence, i.e. for time scales shorter than the

time it takes a ﬂuctuation to traverse the entire lattice. Turning to the present

model with constrained par ticle s and fuel carriers, ﬁgure 10 shows power spectra

for systems with diﬀerent total numb e rs of particles, both when there are no limited

fuel resources (ﬁgure 10(a)) and when there are (ﬁgure 10(b)). No te that to fairly

compare ﬂuctuations from two simulations the mean densities and hopping rates (and

therefore positions of the “dips” in the power spectrum) must be the same; we therefo re

choose diﬀerent va lue s of α

0

to give the same value of v in each case. Again we see

little qualitative diﬀerence between the non-LR and LR ca ses, i.e., ﬂuctuations in the

number of loaded fuel carriers have little eﬀect on the ﬂuctuations in the density.

Multiple phase transitions in a TASEP with limited particles and fuel carriers 18

4.2. Shock Phase

We now consider ﬂuctua tions in the density, and in the position of the DW, when the

system is in the SP. We ag ain consider a single lattice (M = 1), and focus on the case

of small V

0

, and a mid-range value of N, corresponding to ﬁgures 5(e) and 6(b), for

which the SP has the largest area on the α

0

-β plane.

To locate the DW we average the occupation of each site over a short time

window of length τ, so as to average over short time microscopic ﬂuctuations in the

occupation [25]. The value of τ is chosen short enough so as to probe the movement

of the DW on mesoscopic time scales rather than probing only the mean-ﬁeld de nsity

proﬁle. Following the approach of [25] we choose for τ the smallest value which gives

exactly one micro-domain wall (µDW) in the averaged proﬁle. A µDW is deﬁned as

any point where the density crosses 0.5 (from above or below) b etween one lattice site

and the next.

We show in ﬁgure 11 the behaviour of the DW at diﬀerent values of α

0

and

β. Figures 11(a) and (b) show the long time mean density proﬁle for parameters

for outside and just within the LR re gime respectively. Also shown is the mean

over the mesoscopic time τ , i.e. a snapshot of the density proﬁle from which

the “instanta ne ous” DW position can be found. We note that in the LR regime

(ﬁgure 11(b)) there are larger ﬂuctuations in the τ averaged proﬁle than for non-LR,

and s hape of the DW in the long time average proﬁle is wider. The τ averaging

method works well for small values of β; however for larger β when the sys tem is in

the LR regime, determination of the position of the shock is much more diﬃcult. This

is because, due to the limited ava ilability of the fuel carriers, the particles move more

slowly – the density ﬂuctuations in the regions to the left and rig ht of the DW e xist

on time scales similar to tha t of the movement of the wall. Any value of τ which will

average out the microscopic ﬂuctuations, will also average out the movement of the

wall. This problem is co mpounded by the fact that the diﬀerence between the mean

density on either side of the DW decreases with increasing β. Therefore we can only

accurately measure the time course of the DW position just inside the LR regime;

deep within that r egime, we can only measure the mean wall position.

Figures 11(c)-(d), (e) a nd (f) show respectively typical time courses, normalised

histograms of the wall positions, and the correlation function of the time course deﬁned

as

C(t

′

) =

hδx(t)δx(t

′

)i

t

hδx(t)

2

i

t

, (12)

where δx(t) = x(t) − x, and h· · ·i

t

denotes average over time. The correlation

functions are approximately exponential, and the examples shown have correlation

time 2.3 × 10

3

s (β = 0.007 s

−1

) and 3.6 × 10

3

s (β = 0.049 s

−1

- the LR regime). We

ﬁnd that whilst the time scale over which the DW moves is comparable in both cases,

the width of the distribution is much wider in the LR case.

For the SP, previous studies [33, 34] have treated dens ity ﬂuctuatio ns analytically

by making the approximation that any ﬂuctuations travel quickly along the lattice and

are absorbed by the DW. That is to say, any ﬂuctuation can be treated as a movement

of the DW, and so only ﬂuctuations in the rate at which particles move onto the i = 1

and oﬀ o f the i = L sites need be c onsidered. The ﬂuctuations can then be described

using a simple Langevin equation, leading to a power spectra I ∝ (ω

2

+ γ

2

)

−1

, where

the constant γ represents the restoring force which localises the DW.

Multiple phase transitions in a TASEP with limited particles and fuel carriers 19

Figure 11. Colour on-line. Simulation data showing the behaviour of the domain

wall in the SP for smal l V

0

and mid-range N corresponding to ﬁgure 5(e), for

α

0

= 0.1125 s

−1

and various values of β. Plots (a)-(b) show density proﬁles, and

(c)-(d) the DW position time courses for β = 0.007 s

−1

(black) and β = 0.049 s

−1

(red). The latter corresponds to a system which is in the LR regime. For the

density proﬁles long time averages ar e shown with heavy lines, and mesoscopic

time averages (fr om which we determine the instantaneous wall positions) with

light l ines. The mesoscopic averaging time was τ = 500 s. Plot (e) shows DW

posi tion histograms (normalised) for the same two values of β, and plot (f) shows

the time correlation function from (12).

Figure 12. Plot showing power spectra of density ﬂuctuations from simulations of

systems in the SP with α

0

= 0.1 s

−1

and values of β = 0.01, 0.02, 0.03, 0.04, 0.05

and 0.06 s

−1

. The onset of the LR regime is at β ∼ 0.04 s

−1

, so the top 2 curves

are for systems with LR. Other parameters are L = 500, V

0

= 20 s

−1

and N = 450.

The dashed line shows the slope ω

−2

.

Multiple phase transitions in a TASEP with limited particles and fuel carriers 20

Figure 13. Plot showing<