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The dynamics of the system is given by the rates at which a particle leaves a site m one can think of it as the topmost particle-see Fig. As our first example we assume it moves to the left nearest neighbour site m The hopping rates u n are a function of n the number of particles at the site of departure. The important attribute of the zero-range process is that it yields a steady state described by a product measure.

Here the normalisation Z M , L is introduced so that the sum of the probabilities for all configurations, with the correct number of particles L , is one. We shall explore later in Section the interesting possibilities afforded by the form 5. Note that f n is defined only up to a multiplicative constant and we could have included a factor z n in 6.

Later this factor reappears as a fugacity in Section.

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The proof of 5, 6 is, happily, straightforward. One simply considers the stationarity condition on the probability of a configuration probability current out of the configuration due to hops is equal to probability current into the configuration due to hops :. There exists an exact mapping from a zero-range process to an asymmetric exclusion process.

This is illustrated in Fig. The idea is to consider the particles of the zero-range process as the holes empty sites of the exclusion process. Then the sites of the zero-range process become the moving particles of the exclusion process. This is possible because of the preservation of the order of particles under the simple exclusion dynamics.

An interesting feature of the mapping is that it converts a model where the local degree of freedom can take unbounded values particle number in the zero-range process to a model where the local site variable is restricted to two values. On the other hand, a hopping rate u m which is dependent on m corresponds to a hopping rate in the exclusion process which depends on the gap to the particle in front.

So in principle the particles can feel each other's presence and it is possible to have a long-range interaction. We now show how the totally asymmetric, homogeneous zero-range process we have considered so far may be generalised yet retain steady states of a similar form to 5,6. First we consider an inhomogeneous system by which we mean the hopping rates are site dependent: the hopping rate out of site m when it contains n m particles is u m n m.

It is easy to check that the steady state is simply modified to. The proof is identical to that given above for the homogeneous case, with the trivial replacement of u n m by u m n m. A further generalisation is to the case of discrete time dynamics. This has been studied in [59] in the context of a disordered asymmetric exclusion process. Here we translate the results into the zero-range process. Rather than processes occurring with a rate, time is counted in discrete steps and at each time step events occur with certain probabilities.

In the case of Parallel Dynamics , at each time-step all sites are updated. One particle from each site m is moved to the left, each with probability p m n m where n m is the number of particles at the site before the update. Note that the particles move simultaneously and particles do not move more than one site.

In this way one can interpolate between discrete time, parallel dynamics and continuous time dynamics. In [59] ordered sequential updating schemes were also considered. These are discrete time updating schemes were one site is updated at a time, but the sites are updated in a fixed order.

The steady states for the forwards and backwards updating sequences were derived and it turns out they too have the form 11 with f m taking an expression related to the parallel case In the original paper paper of Spitzer [10] some more general versions of the zero range process were considered. Here we discuss one interesting case which serves to generalise the totally asymmetric zero range process defined above to a process on a more general lattice or for any finite collection of points with a prescribed transition matrix for the dynamics of a single particle [11].

Thus the probability that in time dt a particle at m moves to n is. We assume that the transition matrix is irreducible i.

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We now show that the steady state for the many-particle problem defined above is given by 11 where now f m n is given by. The proof is again a straightforward generalisation of that of Section. Equation 7 is modified to. Equating the terms m on each side of 18 , assuming 11 and cancelling common factors yields. Let us also discuss an example where the single particle problem has an inhomogeneous steady state. We consider a one-dimensional lattice where hops to the left and right neighbours are allowed but with probabilities that depend on the site.

Thus, we may write. The steady state of the single particle problem random walker on a disordered one dimensional lattice [60]. This network is relevant to the disordered one-dimensional exclusion process studied in [61, 62, 63]. The sites in the present model correspond to the particles in the exclusion process which each have their own forward and backward hopping rates.

Another, particular instance of this network occurs in [52], where a repton model of gel electrophoresis is studied in the case of periodic boundary conditions see Section. An interesting consequence of the form of the steady state 17 is that it allows one to relate an arbitrary zero-range process to a model obeying detailed balance.

The idea is that if detailed balance doesn't hold, we can always define a new zero-range process to be denoted by a prime with the same steady state, but with a different dynamics obeying detailed balance. To do this, we solve the single particle problem 16 for the original model to obtain s m. It is easy to check from 17 that the new model has the same steady state as the original. Thus, within the realm of zero-range processes, to the steady state of any nonequilibrium model we can always identify a model satisfying detailed balance and therefore an energy function.

Of course, although the steady states are the same, there is no reason for the dynamical behaviour of the two systems to be related. To clarify this point we will discuss a simple example in Section. The marginals 17 have the interesting structure of being a product of a term s m n that depends on the nature of the network and a term involving the product of u n m which reflects the interactions at the site. The network can represent an arbitrary dimensional lattice or the effects of disorder, the only difficulty to surmount in obtaining the steady state is the solution of the single particle problem.

We now proceed to analyse the steady states of form 11 and the condensation transition that may occur. The important quantity to consider is the normalisation Z M , L as it plays the role of the partition sum.

Phase Transitions of Simple Systems

The normalisation is defined through the condition. The normalisation may be considered as the analogue of a canonical partition function of a thermodynamic system. Note that 30 tells us that the speed is independent of site and thus may be considered a conserved quantity in the steady state of the system. In the totally asymmetric system considered in Section III. More generally, however, the speed is not equal to the current and the fact that the speed is a conserved quantity is not a priori obvious. For example, in a system obeying detailed balance the net current is zero, but the speed as defined above remains finite.

The speed is a ratio of partition functions of different system sizes 30 and corresponds to a fugacity, as we shall see below. We will consider also the probability distribution of the number of particles at a site, taken here to be site 1. In general the probability distribution is site dependent but for a homogeneous system f m independent of m it will be the same for all sites. For large M , L 32 is dominated by the saddle point of the integral and the value of z at the saddle point is the fugacity.

The equation for the saddle point reduces to. We expect that for low f the saddle point is valid but, as we shall discuss, there exists a maximum value of z and if at this maximum value the rhs of 35 is finite, then for large f 35 cannot be satisfied. We now consider separately, and in more detail, how condensation may occur in the inhomogeneous and the homogeneous case.

In general, the inhomogeneous case i. Here we would just like to give an idea of how a condensation transition may occur by discussing a simple example.

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  • We then go on to analyse perhaps the simplest example of a condensation transition: a single inhomogeneous site [62]. First we take the general model discussed in Section IV. For the moment we do not specify further the transition matrix; later we will discuss two specific examples one obeying detailed balance and one not.

    Under these conditions f m is given by. Thus the ground state corresponds to the site with the lowest hopping rate. The normalisation Z M , L is equivalent to the canonical partition function of the Bose gas. Interpreting u as a density of states, equation 39 corresponds to the condition that in the grand canonical ensemble of an ideal Bose gas the number Bosons per state is f. The theory of Bose condensation [64] tells us that when certain conditions on the density of low energy states pertain we can have a condensation transition.

    Then 35 can no longer be satisfied and we have a condensation of particles into the ground state, which is here the site with the slowest hopping rate. This case is discussed further, in the context of an asymmetric exclusion process on an infinite system, by J. Krug in this volume [65]. It is easy to see that 11 simplifies to. In this case the normalisation Z M , L is easy to calculate combinatorially:. In the low density phase 42 the particles are evenly spread between all sites and we will refer to it as the homogeneous phase. We now discuss two models which both have this steady state: a driven system and a system obeying detailed balance.

    This provides an illustration of the idea discussed in Section IV. First we take the totally asymmetric model so that particles move to the site to the left: the transition matrix is. So this model is similar to that discussed in Section III. The equivalent exclusion process is illustrated in Fig. We see that the equivalent exclusion process is system of hard-core particles hopping to the right, one particle being slower than the rest. The interpretation of the two phases within the context of the exclusion process is that in the condensed phase for the exclusion process a low density of particles a 'traffic jam' forms behind the slow particle and the slow particle has a finite fraction of the lattice as 'empty road' ahead.

    Whereas in the homogeneous phase a high density of particle for the exclusion process the particles are roughly evenly spaced. On the other hand one may consider the case where the one particle problem is a symmetric random walk so that the system obeys detailed balance. The transition matrix is given by. When we map this system to a simple exclusion process we see from Fig.

    In the condensed phase the gap between these particles diverges. Previously a single asymmetric particle in a sea of symmetric particles has been studied [66] but in that case there is no phase transition. At first it seems that we have found a counterexample to the received wisdom that no phase transition should occur in an equilibrium system, since we have a condensation transition in a model with local dynamics obeying detailed balance.

    Inferring an energy function from the steady state 40 through the following equation. Therefore the energy is 'unphysical' in that it has very long range interactions. Thus the phase transition can be rationalised within the categories of exceptions discussed in Section II.

    We have seen that this simplest example of a condensation transition a single inhomogeneous site in the zero range process is exhibited both in a driven model and also in a model obeying detailed balance but with long-range energy function. Again it should be stressed that although the steady states of these two models are equivalent, the dynamical properties should be very different. However in the homogeneous phase of the model obeying detailed balance we expect symmetric exclusion like behaviour and the dynamic exponent to be 2 implying relaxation times of M 2 [67].

    A similar analysis has been carried out in the context of balls-in-boxes and branched polymer models [69]. The fugacity z must be chosen so that F converges or else we could not have performed From 33 we see that b is the limiting value value of the u m i. We interpret 35 as giving a relation between the density of holes number of holes per site and the fugacity z.

    The saddle point condition 35 becomes. Given that the rhs of 45 is a monotonically increasing function of z which is not difficult to prove we deduce that density of particle increases with fugacity. Physically, the condensation would correspond to a spontaneous symmetry breaking where one of the sites is spontaneously selected to hold a finite fraction of the particles.

    Thus, for condensation to occur i. We now assume that u n decreases uniformly to b in the large n limit as. Analysis of the series. Therefore the 'no phase transition rule' does not apply. One also gains insight by translating the results into the language of the simple exclusion process. Having discussed the case where a true phase transition occurs we now consider a homogeneous example where, although there is no strict condensation transition, some interesting crossover phenomena occur [68]. One can interpret these hop rates as meaning that a site only distinguishes whether it contains greater than r particles.

    When we use the mapping of Section III. In fact these phases correspond exactly to those of the single defect problem discussed in the previous subsection 42, 43 with b playing the role of p. For finite r , z is actually a smoothly increasing function of f but we see from 55, 57 that the curve sharpens as r increases. One sees a dramatic sharpening as r increases leading to a sharp crossover between a low density and high density regime. In order to see the effects of this sharp crossover it is interesting to consider the particle number probability distribution 31 which for this system is site independent and given by.

    Therefore to simulate the model one requires a number of particles very much larger than this! If care is not taken to do this, and the total number of particles in the system is comparable to the typical occupancy, one would have an apparent condensate on a finite system. An example of this phenomenon was studied recently within the context of a 'bus route model' [70].

    There the underlying motivation was to consider how a non-conserved quantity could mediate an effective long-range interaction amongst a conserved quantity in a driven system with a strictly local dynamical rule. The model considered was defined on a 1 d lattice. Each site bus-stop is either empty, contains a bus a conserved particle or contains a passenger non-conserved quantity.

    The dynamical processes are that passengers arrive at an empty site with rate l ; a bus moves forward to the next stop with rate 1 if that stop is empty; if the next stop contains passengers the bus moves forward with rate b and removes the passengers. Since the buses are conserved, there is a well defined steady state average speed v. This fact can be used to integrate out the non-conserved quantity passengers within a mean-field approximation.

    From this probability an effective hopping rate for a bus into a gap of size n is obtained by averaging the two possible hop rates 1, b :. We can now see that this mean-field approximation to the bus-route model is equivalent to a homogeneous zero-range process as discussed earlier in this section.

    It is reasonable to believe that the system behaves in a similar way to the system with a finite 'range' r discussed in Section V. In the bus route problem this corresponds to the universally irritating situation of all the buses on the route arriving at once.

    As mentioned earlier the zero-range process and related models have appeared several times in the modelling of nonequilibrium phenomena. Here we briefly discuss a few of these instances to illustrate the ubiquity of the basic model. In [53] models of sandpile dynamics are considered. A zero range process is used to model the toppling of sand on a one-dimensional lattice; specifically the system is homogeneous and the occupation number of a site becomes the height of sand h at that site. This limit means that a particle grain of sand keeps moving until it finds an unoccupied site, thus a hopping event may play the role of an avalanche.

    Although in terms of sandpiles and self-organised criticality this model is rather trivial, it did serve to investigate the idea of a diverging diffusion constant. In a different context Barma and Ramaswamy [55] introduced the 'drop-push' model of activated flow involving transport through a series of traps. Each trap can only hold a finite number of particles. For the trap depth set equal to one this model is essentially the same as the sandpile model of [53] discussed above i. A generalisation to inhomogeneous traps, and partially asymmetric hopping rates dependent on the occupancy of the trap was made in [72] and a steady state similar to 11, 17 demonstrated.

    The interface can be visualised as an ascending staircase of terraces. Adatoms land on the terraces and diffuse until they bind to the ascending step. If the ratio of deposition rates over diffusion rates tends to zero then the resulting dynamics is that a terrace shrinks by one unit and the adjacent higher terrace grows by one unit with a rate proportional to the size of the terrace. Thus the terrace lengths are equivalent to the site occupancies of an asymmetric zero-range process that was discussed in Section III.

    The equivalence of zero range processes to a general class of step flow models is discussed in [57]. Finally we note that the repton model of gel-electrophoresis [73] studied in the case of periodic boundary conditions by [52] is equivalent to an inhomogeneous zero-range process. In this case, the particles of the zero-range process represent the excess stored length of a polymer which diffuses along the tube of the polymer. The sites in the zero-range process represent the segments of the polymer tube and the inhomogeneities in site hopping rates reflect the shape of the polymer tube.

    In this work the aims were to give an overview of the area of phase transitions and ordering in one-dimensional systems and also to analyse in some detail a particularly simple model, the zero-range process. In Section II several features were identified which could lead to the anomalous behaviour of ordering and phase transitions in equilibrium systems: long-range interactions; zero temperature; unbounded local variable. For nonequilibrium systems some concepts which may be important emerged: conserved order parameter; drive; forbidden microscopic transitions.

    The simplicity of the zero-range process allowed us to analyse the steady state of the model in detail. We then analysed the condensation transitions that can occur. On an inhomogeneous system the condensation is very reminiscent of Bose-Einstein condensation. For it to occur requires certain conditions to hold on the distribution of hopping rates. In the homogeneous system the condensation corresponds to a spontaneous symmetry breaking, since an arbitrary site is selected to hold the condensate.

    The condition for it to occur is that the hopping rate dependence on the site occupancy decays sufficiently slowly. It was also shown that when the condition for condensation does not hold, one can still observe very sharp crossover behaviour and apparent condensation on a finite system. An interesting possibility that was explored was that of the existence of an effective energy function. Dirk V. Despite using different formalizations and investigating very different kinds of systems, the same unimodal dependence between disorder and complexity has been found in several independently conducted studies.

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