Discontinuous Phase Transition in an Exactly Solvable One-Dimensional Creation-Annihilation System
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un Discontinuous Phase Transition in an ExactlySolvable One-Dimensional Creation-AnnihilationSystem
F H Jafarpour ∗ and B Ghavami Bu-Ali Sina University, Physics Department, Hamadan, Iran
November 9, 2018
Abstract
An exactly solvable reaction-diffusion model consisting of first-classparticles in the presence of a single second-class particle is introduced ona one-dimensional lattice with periodic boundary condition. The numberof first-class particles can be changed due to creation and annihilationreactions. It is shown that the system undergoes a discontinuous phasetransition in contrast to the case where the density of the second-classparticles is finite and the phase transition is continuous.
One of the most important characteristics of non-equilibrium driven systems isthat their steady state consist of current of particles or energy. One-dimensionalreaction-diffusion models are examples of such systems which have attractedmuch attention during last decade [1, 2]. Phase transition and shock forma-tion in these systems are some of their interesting collecting behaviors. Thesesystems have also many applications in different fields of physics and biology.During recent years different models of this type have been studied widely andinteresting results have been obtained. The Asymmetric Simple Exclusion Pro-cess (ASEP) is a well known example. In this exactly solvable model, which isdefined on an open discrete lattice, particle are injected from the left bound-ary and extracted from the right boundary while hopping on the lattice to theleft and to the right randomly. This model has been shown to exhibit non-trivial steady-state phenomena such as phase transitions and shock formation[3]. In order to study the steady state properties of these shocks different mod-els have been proposed. It should be noted that the ASEP is not the onlyone-dimensional out-of-equilibrium system which exhibits shocks. It has beenshown that there are three families of two-states models in which a factorizedshock measure is invariant under the time evolution if some constraint on the ∗ Corresponding author’s e-mail:[email protected] A ∅ −→ ∅ A with rate α ∅ B −→ B ∅ with rate βAB −→ BA with rate 1 A ∅ −→ ∅∅ with rate λ ∅∅ −→ A ∅ with rate λ ′ . (1)As can be seen the number of A particles (first-class particles) is not conserved.In contrast, the number of B particles (second-class particles) is conserved sincethey only diffuse. It is assumed that in a system with at least one empty site onehas finite number of second-class particles with the density ρ B in the presenceof the first-class particles with fluctuating density. It has been shown that inthis case a continuous phase transition takes place if the order parameter of thesystem is taken to be the density of empty sites in the system ρ E . By taking α = β = 1 and defining ω := λλ ′ it turns out that ρ E is zero for ω < ω c whileit changes linearly as ρ E = ω ω − ρ B for ω > ω c in which ω c = ρ B − ρ B . Thecurrent of the second-class is always constant while the particle current of thefirst-class particles is given by different expressions in each phase. For the case α = 1 and β = 1 the transition point is obtained to be ω c = ρ B + α − − ρ B . Thedensity of the empty sites is zero below the transition point while it is given by ρ E = ω ω − ωρ B ω − α above this point. For ρ B = 0 the transition is still continu-ous.In present paper we assume that there exists only a single second-class particle inthe system which means their density goes to zero in the thermodynamic limit.Second-class or tagged particles are usually introduced to study the dynamicalproperties of the shocks in one-dimensional driven-diffusive systems; however,one of our major motivations for studying such limiting case is to investigateits effects on the critical behavior of the system and compare it with the pre-vious case in which the density of the second-class particles is non-zero in thethermodynamic limit. As we will see considering this limiting case changes thenature of phase transition from a continuous into a discontinuous one. As far aswe know such observation had not been reported before. Apart from the vastapplicability of such models in different fields of science (as mentioned above),classification of one-dimensional driven-diffusive models which are exactly solv-able using the Matrix Product Formalism (MPF) has been of great interests for2eople in this field (for a recent review see [11]). As we will see the model isstill exactly solvable using the MPF even in the limiting case ρ B → ω := λλ ′ and apply the MPF [3] to find the partitionfunction of the system. According to the MPF the stationary probability dis-tribution function of any configuration C of the system of length L + 1 with asingle second-class particle at the site L + 1 is given by P ( C ) = 1 Z T r [( L Y i =1 X i ) B ] (2)in which X i = E if the site i is empty otherwise X i = A . The normalizationfactor Z in the denominator of (2) will be called the partition function of thesystem. By applying the standard MPF the quadratic algebra of the model isobtained to be [10] AB = A + BAE = α EEB = β EE = ωα E . (3)By defining E = ωα | V ih W | in which h W | V i = 1 one finds from (3) AB = A + BA | V i = α | V ih W | B = β h W | . (4)This quadratic algebra has an infinite-dimensional representation given by thefollowing matrices and vectors A = α a · · · , B = β · · · a , | V i = , h W | = (cid:0) · · · (cid:1) (5)in which a = α + β − αβ . Since the stationary state of the system without vacanciesis trivial, we consider the partition function of the system with at least one emptysite which is defined by Z = T r [( A + E ) L B ] − T r [ A L B ] . (6)3sing (4-6) and after some straightforward calculations we find the followingexact expression for the partition function of the system Z = β + ωβ (1 + ω − α ) ( 1 + ωα ) L − − α ( 1 α ) L + ω ( α + β − β (1 + ω − α )(1 − α ) . (7)For α < α > L limit Z ∼ = β + ωβ ( β + ω − α ) ( ωα ) L for ω > α − ω ( α + β − β (1+ ω − α )(1 − α ) for ω < α − . (8)At the transition point the partition function of the system grows like O ( L ).Taking the density of the empty sites on the lattice given by ρ E = lim L →∞ ωL ∂∂ω ln Z (9)as the order parameter of the system, we find using (8) that ρ E = ω ω for ω > α −
10 for ω < α − . (10)At the transition point the density of the empty sites is obtained to be ρ E = ω ω ) . We should note that density of the empty sites for ω < α − ρ E ∝ O ( L ). Discontinuous changes of the density of the empty sites ρ E in the thermodynamic limit indicates a first-order phase transition in thesystem. As we mentioned earlier, in the case where the number of the second-class particles on the lattice is finite the density of the empty sites ρ E changedcontinuously over the transition point [10].In order to study the nature of the first-order phase transition one can applythe Yang-Lee theory. Recently it has been shown that the classical Yang-Leetheory can be applied to the out-of-equilibrium systems to study their phasetransitions (for a review see [12]). We have calculated the line of the Yang-Leezeros for our model in the complex- ω plan and found that they lie on a circle ofradius α . The center of this circle is at ( − ,
0) and intersects the real- ω axis at Re ( ω ) = α − π which again implies a first-order phase transitionat the transition point. The density of the zeros has also been found to be aconstant all over the circle.It is also interesting to calculate the density profile of the first class particleson the ring, as seen by the second-class particle, using the MPF. For a systemwith at least one empty site it is given by ρ A ( i ) = 1 Z ( T r [( A + E ) i A ( A + E ) L − i − B ] − T r [ A L B ]) 0 ≤ i ≤ L − . (11)4t turns out that (11) can be calculated exactly using (5) and here are the resultsin the large L limit ρ A ( i ) ∼ = ω + ω ( α + β − α ( β + ω ) e i − Lξ for ω > α − − α − α e − iξ for ω < α − ω + ω ω ( iL ) for ω = α − ξ = | ln( ωα ) | − . For ω > α − ω except just in front ofthe second-class particle where it increases exponentially to 1. In this phase thedensity of empty sites is ω ω . For ω < α − α to 1 in the bulk of the lattice. The density ofempty sites in this phase is nearly zero in the thermodynamic limit. As can beseen at the transition point the density profile of the particles is linear. Thisis a sign for a shock however since the number of first-class particles is nota conserved quantity the shock position fluctuates and therefore the resultingprofile is linear. This phenomenon has also been observed in the ASEP withopen boundaries on the first-order phase transition line where the injection andextraction rates become equal and smaller than one-half. The sock picturewill be more clear by calculating the connected two-point function of first-classparticles. Straightforward calculations result in the following exact expressionwhich is valid for i ≤ j and large system length h ρ A ( i ) ρ A ( j ) i c := h ρ A ( i ) ρ A ( j ) i − h ρ A ( i ) ih ρ A ( j ) i∼ = − ( ω − h ρ A ( i ) i )(1 − h ρ A ( j ) i ) . (13)We have also calculated the current of the first-class particles J A in the steadystate. In the large L limit we have found that the current of the first-classparticles does not depend on β and is given by J A = αω (1+ ω ) for ω > α −
10 for ω < α − ω ω ) for ω = α − . (14)On the other hand, the mean speed of the second-class particle defined as V = 1 Z ( βT r [( A + E ) L − EB ] + T r [( A + E ) L − AB ] − T r [ A L B ]) (15)can also be calculated exactly. It turns out that V is given by the followingexact expression in the thermodynamic limit V = αω + β ((1+ ω ) − αω )(1+ ω )( β + ω ) for ω > α −
11 for ω ≤ α − . (16)5 Ω Β= Β= Β= Figure 1: The mean speed of the second-class particle V as a function of ω forthree values of β . The length of the system is L = 100 and we have chosen α = 6.In Figure 1 we have plotted V as a function of ω for three different values of β on a lattice of length L = 100. For ω < α − ω while for ω < α −
1, abovethe transition point, for β < β >
1) the speed of the second-class particle isa decreasing (increasing) function of ω . For β = 1 the speed of the second-classparticle is always equal to unity.The model studied in this paper consists of a single second-class particle in thepresence of first-class particles with fluctuating density because of creation andannihilation of them. Comparing out results with those obtained in [10] oneshould note that the order of the phase transition is changed from two to onewhen the density of the second-class particles goes to zero. The physical ex-planation for such a macroscopic change can be as follows: for the two cases ρ B = 0 and ρ B = 0 and below the critical point there are empty sites on thering; however, their density goes to zero in the thermodynamic limit. Never-theless, it is less probable to find configurations of type A ∅ in the case ρ B = 0(in comparison to the case ρ B = 0) because in this case part of the system isoccupied by the second-class particles. Therefore as we increase ω above thecritical point it is more probable to create many empty sites in the case ρ B = 0than the case ρ B = 0 because as explained above we have more configurationsof type A ∅ in this case which result in the configuration ∅∅ . This means that aswe increase ω above the critical point we expect many empty sites to be createdat once in the case ρ B = 0 in contrast to the case ρ B = 0 where they are beingcreated slowly.It is also interesting to compare our results with those obtained in [13] where the6ame model as (1) has been considered except it does not contain the creationand annihilation of the first-class particles. It has been shown that the modelin this case has two phases: a condensate phase and a fluid phase. Obviouslyin our model by fixing the number of the first-class particles the phase in which ρ E = 0 will be destroyed; however, a condensate phase emerges in which thedensity profile of the first-class particles is no longer linear but an step-functionand this is in quite agreement with the results in [13].The MPF enables us to solve some of the one-dimensional driven-diffusive sys-tems exactly; however, by looking at these models we realize that they mighthave similar quadratic algebras. This means that a quadratic algebra can de-scribe different models with different physical properties and critical behaviors.Now the question is that whether or not other three-states models defined ona ring geometry can be described by (3). In fact we have found that there is afamily of such models which are exactly solvable and share the same quadraticalgebra [14]. So far two members of this family are introduced and studied in[10] and [15]. References [1] B. Schmittmann and R. K. P. Zia,
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