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    Fisher, and J. We shall now review recent progress in applying the Lee-Yang theory to nonequilibrium phase transitions. We consider first in section IV. An appealing feature, discussed in more detail below, is that some one-dimensional cases have been solved exactly, the normalisation 23 calculated, and nonequilibrium phase transitions analysed. The existence of one-dimensional phase transitions contrasts with the case of one-dimensional equilibrium models which do not admit phase transitions if the interactions are short-ranged.

    Then in section IV. We do not endeavour to describe all possible nonequilibrium dynamics in the following - for example, self-organised criticality has also been studied using a Lee-Yang approach [25]. In their original papers on partition function zeros, Lee and Yang [2, 3] made use of the mapping between the Ising model of a magnet and the lattice gas.

    Essentially one associates up-spins with particles and down-spins with vacancies so that a positive interaction strength J in the Ising model results in a particle-particle attraction in the lattice gas, and a negative interaction strength gives rise to repulsive gas particles. Note that there is implicitly at most one particle per lattice site, so there is a hard-core exclusion in the lattice gas. As discussed in section III, one can realise the dynamics of the lattice gas through a set of transition rates that satisfy the detailed balance condition 20 with respect to the Boltzmann distribution.

    Thirty years after Lee and Yang's work, Katz, Lebowitz and Spohn KLS [20, 27] introduced a driven lattice gas model in which the rate at which particles hop in the direction of an external field is enhanced and the hop-rate against the field is suppressed. This model is well-studied and many results are discussed in [28]. As yet, the KLS model remains unsolved for general interaction strength, although in one dimension the steady state is known for some parameters [29].

    The mean-field approach predicts phase transitions in the steady state as parameters controlling the rate of insertion and extraction of particles at the boundaries are varied [31]. The existence of these phase transitions is confirmed through an exact solution of the ASEP [], achieved using a powerful matrix product approach [32, 34] which has subsequently been used to solve many generalisations of the ASEP.

    The details of the matrix product method are not necessary for the following-suffice to say that one ends up calculating a normalisation proportional to 23 through a product of matrices, often of infinite dimension. In this way one obtains some explicit formulas for the normalisation 22 which we shall use below 1. The asymmetric exclusion process with open boundaries is perhaps the simplest exactly solved nonequilibrium model that exhibits both a first-order and continuous phase transition in its steady state. Therefore it is an ideal candidate for testing the hypothesis outlined in section 3 that zeros of the normalisation should accumulate towards the positive real axis in the complex plane of transition rates.

    Before outlining the results of this analysis the details of which are presented in [23] we recall the definition of the ASEP with open boundaries. In this system, a particle on an N -site lattice can hop one site to the right at unit rate, as long as the receiving site is empty. Meanwhile, particles are inserted onto the leftmost lattice site if empty at a rate a and removed from the rightmost site if occupied at a rate b - see Fig.

    This is indicative of a first-order transition. The phase diagram for the model is shown in Fig. It is a simple matter to use a computer algebra package to solve this equation for its zeros in the complex- a plane at fixed N and b. In Fig. We immediately notice that the curve of zeros seems to intersect the real positive a axis at the correct transition point. Both of these observations are in accord with the results known for equilibrium partition function zeros discussed in section II.

    In this expression, A , J and g depend on a and b and the quantity J is the current of particles across the lattice. Although an electostatic analogy was used in [23] to find the zero distribution, the mathematical content is the same as that used to derive the two rules 10 and 15 in section II. Then, the second rule 15 gives the density of zeros m s on the circle as. That is, in the complex- x plane the zeros should become evenly distributed on a circle in the thermodynamic limit. Transforming the zeros of 24 obtained at different system sizes to the complex- x plane reveals this to be the case [23].

    These are precisely the properties of the equilibrium partition function zeros at a first-order transition point see section II. Recall from section II that a transition of n th order has the curve of zeros meeting the positive real axis at an angle of. Here the zeros clearly approach at an angle suggesting a second-order transition. In fact, one finds the density of zeros is s at a distance s along this line, confirming that the transition is second-order.

    In summary, we have found that the Lee-Yang theory of first-order and continuous phase transitions applies to the normalisation of the nonequilibrium asymmetric exclusion process just as it does to the partition function of equilibrium systems. Of course, this does not prove that the theory is generally applicable, and so there is some value in investigating other nonequilibrium steady states that exhibit phase transitions. One such state is that initially studied by Arndt, Heinzel and Rittenberg [35, 36]. When next to vacant sites, the positive particles hop to the right and negative particles to the left at unit rate, with hops in the opposite directions disallowed.

    Meanwhile, should two oppositely charged particles be next to one another, the following transitions can occur where the label above the arrow indicates the rate at which the transitions occur. These dynamics are illustrated in Fig. Although a matrix product solution for this model is known to exist [36], it is technically difficult to work out the solution in its full generality [37].

    However in the case of either a single vacancy [38] or a single negative particle, i. A slightly different Lee-Yang analysis, that precedes the work described above, has been used to study the nonequilbrium phase transition that occurs at a finite density of vacancies [22]. The motivation behind this study arose from computer simulations of the model [35, 36] for the case where the numbers of positive and negative particles were equal. Physically this approach is equivalent to placing the ring with nonequilibrium interactions in chemical equilibrium with a particle reservoir at fixed fugacity z.

    In fact, this is a standard 'trick' for dealing with closed systems in which the particle number is conserved, see e. The key point here, however, is that z is an equilibrium fugacity, and not a microscopic transition rate, and so the Lee-Yang theory of phase transitions described in section II ought to apply directly here, without reference to the discussion of section III. Unfortunately, an exact asymptotic i.

    This phenomenon has been explained as an abrupt increase in a correlation length to an anomalously large, but finite value [41]. It would be interesting, therefore, to extend the numerical computation of zeros performed in [22] to much larger systems, to see how the ellipses noted above develop.

    For example it might well happen that instead of approaching the positive-real fugacity axis, the zeros of 27 would terminate a short distance away from it. A large and important class of models with stochastic dynamics is provided by reaction-diffusion systems 2. In contrast to driven diffusive systems, where the particle-particle interactions imply conservation of particles, reaction-diffusion systems are characterised by dynamics that result in a change in particle number. Moreover there are a number of such systems that have absorbing states i.

    Phase transitions associated with whether the system has a finite probability of not being absorbed into such a state fall within the directed percolation and related universality classes [13]. We shall shortly discuss the second order phase transition associated with the directed percolation university class in a little detail. Meanwhile as a simple example of a reaction-diffusion system, we review a model for which the steady state that can be solved using the matrix product approach. The approach again provides us with an explicit expression for the normalisation 22 by virtue of which we can analyse its zeros in the plane of complex reaction rates.

    The system in question [42] has for the dynamics at neighbouring bulk sites on a one-dimensional lattice the processes. It was demonstrated [42] that the matrix product scheme used for the ASEP could be generalised to cater for the steady state of the present reaction-diffusion system on a lattice of N sites with reflecting boundary conditions i.

    In the matrix product approach, the normalisation Z N is given by a scalar derived from C N where C is a square matrix. A normalisation of this form leads to a complex free energy. Then, there is the possibility of a phase boundary when the magnitude of the two largest eigenvalues of C are equal. Note that this scenario contrasts with the transfer-matrix approach to one-dimensional equilibrium systems where the partition function is also written as a product of matrices.

    Since all elements of the transfer matrix are positive the largest eigenvalue cannot become degenerate and therefore there can be no phase transition [12]. However there is no such restriction on the elements of C , and so eigenvalue crossing is permitted and nonequilibrium one-dimensional phase transitions can occur. Also, since one of the eigenvalues does not depend on q , the free energy h q , k is a constant in one of the phases which, as the analysis of the normalisation zeros for the ASEP demonstrated, implies that the density of zeros on this circle is constant in the thermodynamic limit.

    In turn, this implies that the phase transition is first order, as confirmed by explicitly calculating the density profile in the two phases [43]. Introduced as a crude model of an epidemic [44], this contact process is known to exhibit a transition from a phase in which the absorbing state empty lattice is reached with certainty to a phase in which there is some probability that the epidemic remains active forever in the thermodynamic infinite system size limit [45].

    This transition occurs as the ratio between the decoagulation and decay rates is increased beyond a critical value. Although not proven, it is widely accepted on the basis of simulation and approximate methods such as series expansions that the transition just described is continuous and characterised by directed percolation DP exponents see [13] for an in-depth discussion of these issues.

    Directed percolation itself [46] is a geometric construction designed to model fluid flow through a random porous medium under the influence of gravity. Consider the rhombic lattice shown in Fig. The fluid is allowed to flow along a bond pointing diagonally downwards and then only if it is open. Each bond has a probability p of being open and hence a probability 1- p of being blocked , and the state of each bond is independent of any other and may not change with time. The main quantity of interest in this system is the percolation probability P n p that the fluid can penetrate to a depth n in the medium see Fig.

    If p is below some critical percolation threshold p c , the fluid only ever penetrates a finite distance-i. The importance of the DP transition is that a wide range of models that have a transition into an absorbing state are expected to have that transition characterised by the DP exponents, one of which is b [13]. Despite a huge amount of interest in directed percolation, none of the models expected to belong to its universality class has been solved exactly. Recently an attempt has been made to shed further light on the DP transition by studying the zeros of the percolation probability P n p [47, 48].

    At first glance, a connection between this probability and a partition function is not obvious. However, associating with each site a 'spin' state s i , that is up if site i is connected to a point on the n th layer and down otherwise, yields a form for P n p that has the structure of a transfer-matrix representation of a partition function for an equilibrium system with three-spin interactions [49]. It is not clear that being able to give P n p the appearance of a partition function necessarily implies that the Lee-Yang theory should hold.

    As is evident from Fig. By considering the sequences of zeros approaching the positive real axis, one can estimate the critical point p c and the density of zeros as it is reached [48]. The latter yields a prediction for one of the DP critical exponents and one finds good agreement with the most precise estimates of both the transition point and the associated exponent obtained through other means. Hence we have further evidence for the applicability of Lee and Yang's ideas concerning partition function zeros to a much wider range of statistical distributions than equilibrium steady states.

    In this work we have revisited the Lee-Yang description of equilbrium phase transitions with a view to seeing whether the ideas apply to more general nonequilibrium transitions. Recently there have been a number of studies of zeros of partition-function-like quantities that arise in systems with nonequilibrium dynamics, and we have seen in our review of these works that the Lee-Yang theory, as described in sectionII, seems to hold quite generally. We have argued that for dynamic models with a unique steady state, the normalisation defined as a sum over the steady-state configurational weights 22 serves as a suitable 'partition function' in the sense that its zeros, in the complex plane of any model parameter, should accumulate towards physical transition points in the thermodynamic limit.

    Furthermore, the density of zeros and angle of approach to the real axis indicate whether the transition is first-order manifested physically through phase-coexistence or continuous i. Thus studying the zeros of the normalisation 22 provides an unambiguous classification of nonequilibrium phase transitions as do the Lee-Yang zeros in the equilibrium case.

    The observation that backs up this scenario is embodied by equation 23 which reveals that the reciprocal of the steady-state normalisation is equal to the product of the characteristic relaxation times in the dynamics. Since near a phase transition one expects timescales to diverge, one also expects the normalisation to approach zero. However, a rigorous argument for this to be the case is still lacking. Moreover 23 implies a possible link between systems for which the steady state normalisation can be calculated and those for which eigenvalues of the transition matrix can in principle be calculated.

    A different class of systems encompasses those whose steady state is not unique. The contact process is, in fact, an example of such a model, in which the absorbing state is reached with certainty below the critical decoagulation rate, whereas above it, and on an infinite system, a second steady state can also be reached with some nonzero probability. Since this additional steady state exists only when the lattice size becomes infinite, one must take that limit first, before taking time to infinity. Otherwise, on a finite system the steady state is simply the absorbing state and the steady-state normalisation is trivially equal to a constant.

    Nevertheless the work of [47, 48], which we reviewed in section IV. Although we have not discussed this in great detail here, it should be noted that the Lee-Yang approach gives a method for extrapolating to the thermodynamic limit from solutions for small system sizes.

    In the work of [48], numerical solutions for small systems were used successfully to estimate the transition point and density of zeros as it is approached. From this information one learns about the nature of the phase transition and, for example, can estimate the values of critical exponents. This is a common technique in equilibrium statistical physics see e.

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