Discontinuous condensation transition and nonequivalence of ensembles in a zero-range process
DDiscontinuous condensation transitionand nonequivalence of ensemblesin a zero-range process
Stefan Grosskinsky ∗ and Gunter M. Sch ¨utz † November 24, 2018
Abstract
We study a zero-range process where the jump rates do not only de-pend on the local particle configuration, but also on the size of the system.Rigorous results on the equivalence of ensembles are presented, character-izing the occurrence of a condensation transition. In contrast to previousresults, the phase transition is discontinuous and the system exhibits ergod-icity breaking and metastable phases. This leads to a richer phase diagram,including nonequivalence of ensembles in certain phase regions. The paperis motivated by results from granular clustering, where these features havebeen observed experimentally. keywords. zero range process; discontinuous phase transition; equivalence ofensembles; metastability; ergodicity breaking; granular clustering
The zero-range processes is an interacting particle system introduced in [29],which has recently attracted attention due to the possibility of a condensationtransition. A prototype model with space homogeneous jump rates that exhibitscondensation has been introduced in [9]. When the particle density ρ in the sys-tem exceeds a critical value ρ c , the system phase separates in the thermodynamiclimit into a homogeneous background with density ρ c and a condensate, that con-tains all the excess particles. This phase transition is by now well understood on ∗ Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK,[email protected] † Forschungszentrum J¨ulich GmbH, Institut f¨ur Festk¨orperforschung, D-52425 J¨ulich, Ger-many, [email protected] a r X i v : . [ m a t h - ph ] J u l mathematically rigorous level for general zero-range processes [17], and hasbeen applied to model clustering phenomena in various fields (see [10] and ref-erences therein). In one dimension, a mapping to exclusion models gives riseto a criterion for non-equilibrium phase separation [19]. Further rigorous resultson the zero-range process include a proof of condensation even on finite lattices[12], and a refinement of the results in [17], which implies a limit theorem fortypical density profiles in case of condensation [24]. Regarding the backgrounddensity as order parameter, it has been shown in a general context (including dif-ferent particle species) that in spatially homogeneous zero-range processes witha stationary product measure condensation is always a continuous phase transi-tion [16]. Recently, investigations have been further extended to open boundarieswhere particles are injected and extracted [21] and heuristically to various gener-alized models. Those include a non-conserving zero-range process that exhibitsgeneric critical phases [2], zero-range processes with non-monotonic jump ratesleading to multiple condensate sites [28], or mass transport models with pair-factorised stationary measures that give rise to a spatially extended condensate[11].In this paper we study the condensation transition in a generalized zero-rangeprocess where the jump rates depend on the system size. The motivation forthis study comes from experiments on granular media reported in [26, 34, 32].Granular particles are distributed uniformly in a container which is divided inseveral compartments. When shaking the container, the particles start clusteringin some of the compartments and after equilibration, almost all particles forma ”condensate” in one of the compartments. The phenomenon is robust for avariety of shaking strengths and a gas-kinetic approach lead to a simplified modelequivalent to a zero-range process where the hopping rates depend on the numberof compartments [7, 34, 32, 33]. In an alternative activated-process approach itcan be modeled by a zero-range type process, where the jump rates depend onthe total number of particles in the system [23, 4], and both approaches have beensummarized in [30]. A heuristic analysis of the behaviour of the order parameteragrees with experimental observations and shows that generically the transition isdiscontinuous and the system exhibits hysteresis and metastability. This analysissuggests that the discontinuity is due to the dependence of the jump rates on thetotal number of particles or the number of compartments, respectively.To treat this phase transition on a rigorous level, we present a detailed analysisof a simple prototype model with system-size dependent jump rates, for which wederive results in the context of the equivalence of ensembles analogous to [17, 16].From a mathematical viewpoint our system provides an interesting example, sincethe origin of the phase transition is due to a non-standard behaviour of the grand-canonical measures, in particular the lack of a law of large numbers. This leads toa richer behaviour than in previous models, which can be fully understood onlyby studying the canonical measures as well, which is not the case for zero-range2rocesses with fixed jump rates [16]. The mathematical structure is also differentfrom standard results on systems with bounded Hamiltonians [8, 31]. We alsoshow how our findings can be directly generalized to a process where the jumprates depend on the total number of particles, rather than the size of the lattice.To establish the link between the stationary distribution and dynamics we includea discussion of metastability and the life times of metastable phases, which arecompared to Monte Carlo simulation data. Our results can be generalized heuris-tically to a large class of systems, including models of granular clustering, as isexplained in a forthcoming publication [18].The paper is organized as follows. In the next section we introduce the modeland show its phase diagram, which summarizes our results. In Section 3 we studycanonical and grand-canonical stationary measures and the equivalence of ensem-bles is discussed in Section 4. In Section 5 we present results on metastability andin Section 6 on the extension to a dependence on the number of particles in thesystem. In the discussion in Section 7 we give a detailed comparison with previ-ous results. We consider a zero-range process on a translation invariant lattice Λ L of size | Λ L | = L . The state space is given by the set of all particle configurations, X L = (cid:8) η = ( η x ) x ∈ Λ L : η x ∈ N (cid:9) , (1)where the number of particles per site can be any non-negative integer number.With rate g R ( η x ) one particle leaves site x ∈ Λ L , and jumps to another site y withprobability p ( y − x ) . To avoid degeneracies, we require the jump probabilities (cid:8) p ( x ) (cid:12)(cid:12) x ∈ Λ L (cid:9) to be irreducible and of finite range, i.e. p ( x ) = 0 if | x | > C for some C > . Under these conditions our main results are independent of theactual choice of p . Since they cover the basic novelties of the paper, we restrictourselves to the jump rates of the form g R ( k ) = (cid:26) c , k ≤ Rc , k > R for k ≥ , g (0) = 0 , (2)where c > c > . The rates are piecewise constant and the location of the jumpis given by the parameter R ≥ , which depends on the system size L , such that R → ∞ and R/L → a as L → ∞ , (3)where a ≥ is a system parameter. The most interesting case we will consider is a > , but we will also discuss a = 0 which depends on the asymptotic behaviourof R as L tends to ∞ . The same model has already been mentioned in [9] forfixed R . There is no phase transition in this case, but for large R one observes alarge crossover, i.e. convergence in the thermodynamic limit is very slow.3 (cid:144) CC (cid:144) F FF H E L Ρ c + a Ρ trans H a L Ρ c a Ρ F C (cid:144) FF (cid:144) CF H E L Ρ c + a Ρ c Ρ trans Ρ c Ρ bu l kd e n s it y Figure 1:
Stationary phase diagram for generic values of c > c . The four phases F ( E ) ( ρ ≤ ρ c ), F ( ρ c < ρ ≤ ρ c + a ), F/C ( ρ c + a < ρ < ρ trans ) and C/F ( ρ ≥ ρ trans ) areexplained in the text.Left: Phase diagram in terms of a (3) and the particle density ρ . Right: Backgrounddensity as a function of ρ for a = 0 . . Full lines are stable, broken lines metastable. The generator of the process is given by L f ( η ) = (cid:88) x,y ∈ Λ L g R ( η x ) p ( y − x ) (cid:0) f ( η x,y − f ( η ) (cid:1) . (4)It is defined for all continuous cylinder functions f ∈ C ( X L ) . Since we define theprocess only on finite lattices, there are no further restrictions on initial conditionsor the domain of the generator as opposed to zero-range processes on infinitelattices (cf. [1]). We do not specify the geometry or the dimension of the lattice,since our main results on the stationary distribution do not depend on these details.The only requirement is that the lattice is translation invariant, or more generally, φ x = const. is the only positive solution to the difference equation φ x = (cid:88) y ∈ Λ L φ y p ( x − y ) . (5)Note that no particles are created or annihilated and the number of particles is aconserved quantity. Under our assumptions on p and g there are no other conser-vation laws that would lead to degeneracies in the time evolution.For fixed L , also R is a fixed parameter and known results on stationary mea-sures for zero-range processes apply (see e.g. [10] and references therein). Thestationary weight w LR ( η ) for this process is of product form, w LR ( η ) = (cid:89) x ∈ Λ L w R ( η x ) , (6)where the single-site marginal is given by w R ( k ) = k (cid:89) i =0 g − R ( i ) = (cid:26) c − k , k ≤ Rc − R c R − k , k > R . (7)4ere the empty product (for k = 0 ) is understood to be unity.The results we derive in the following sections are summarized in the station-ary phase diagram in Figure 1 in terms of the conserved particle density ρ andthe parameter a (3). In the fluid phases F ( E ) , F and F/C the stationary mea-sure concentrates on homogeneous configurations with bulk density ρ . In phase F ( E ) for ρ ≤ ρ c the canonical and grand-canonical ensembles are equivalent (seeSection 4), and in phase F/C for ρ c + a < ρ ≤ ρ trans there exists an additionalmetastable condensed phase, which has a lifetime exponential in the system size(see Section 5). Typical condensed configurations have a ρ -independent homoge-neous bulk distribution with density ρ c < ρ , where the excess particles condenseon a single lattice site. In phase C/F , i.e. for ρ > ρ trans , the condensed phasebecomes stable and the corresponding fluid phase metastable. On top of metasta-bility the order parameters are discontinuous as a function of the density ρ , andtherefore the condensation transition is discontinuous. Since the state space X L is discrete we will identify measures µ (cid:0) { η } (cid:1) with theirmass functions µ ( η ) in the following to simplify notation. For each R and L thereexists a family of stationary product measures ν Lφ,R with single site marginal ν φ,R ( k ) = 1 z R ( φ ) w R ( k ) φ k . (8)The marginal is well defined for fugacities φ ∈ [0 , c ) , since the tail behaviourof the stationary weight (7) is w R ( k ) ∼ c − k for all fixed R . The single sitenormalization is given by the partition function z R ( φ ) = ∞ (cid:88) k =0 w R ( k ) φ k = c c − φ (cid:18) (cid:16) φc (cid:17) R +1 c − c c − φ (cid:19) (9)and the expected particle density under the measure ν Lφ,R is given by ρ R ( φ ) = (cid:10) η x (cid:11) ν φ,R = φ ∂ φ (cid:0) log z R ( φ ) (cid:1) == φc − φ + (cid:16) φc (cid:17) R +1 R + 1 + φ/ ( c − φ ) c − φc − c + ( φ/c ) R +1 . (10)Note that ρ R ( φ ) is strictly increasing in φ and that for every fixed R , ρ R ( φ ) → ∞ as φ → c . So for all densities ρ ≥ there exists φ R ( ρ ) such that the measure ν Lφ R ( ρ ) ,R has density ρ , i.e. product measures exist for all densities. But the singlesite marginals of these measures still depend on R and therefore on the system5ize L . Since R → ∞ as L → ∞ , the marginal (8) converges pointwise to asimple geometric distribution, i.e. for each k ∈ N , ν φ,R ( k ) → ν φ, ∞ ( k ) = 1 z ∞ ( φ ) ( φ/c ) k with z ∞ ( φ ) = c c − φ . (11)This convergence holds for each fixed φ < c , but it is not uniform in φ . Thelimiting product measure ν φ, ∞ is defined for all φ < c . We denote the particledensity with respect to this measure by ρ ∞ ( φ ) := (cid:10) η x (cid:11) ν φ, ∞ = φ ∂ φ (cid:0) log z ∞ ( φ ) (cid:1) = φc − φ , (12)and its inverse is given by φ ∞ ( ρ ) = c ρ ρ . (13)Since convergence (11) only holds for φ < c we define the critical density ρ c := ρ ∞ ( c ) = c c − c < ∞ . (14)Note that with this definition φ ∞ ( ρ c ) = c . In the following we summarize somestraightforward consequences of these definitions. Proposition 1
For all φ < c , ν Lφ,R → ν φ, ∞ weakly or, equivalently, (cid:104) f (cid:105) ν Lφ,R → (cid:104) f (cid:105) ν φ, ∞ as L → ∞ for all f ∈ C ,b ( X ) , (15) and ρ R ( φ ) → ρ ∞ ( φ ) . For all ρ ≥ we have φ R ( ρ ) → (cid:26) φ ∞ ( ρ ) , ρ<ρ c c , ρ ≥ ρ c and ν Lφ R ( ρ ) ,R → (cid:26) ν φ ∞ ( ρ ) , ∞ , ρ<ρ c ν c , ∞ , ρ ≥ ρ c , (16) where φ R is the inverse of (10) and the second convergence holds in the weaksense as in (15). Proof. (11) implies pointwise convergence of arbitrary n -point marginals ν nφ,R and in general this is equivalent to convergence of expected values of cylinder testfunctions, as long as they are bounded. This does not directly imply convergenceof the unbounded test function η x which yields the density, but ρ R ( φ ) → ρ ∞ ( φ ) follows by direct computation from (10).Since ρ R ( φ ) and its inverse are continuous for φ < c or equivalently ρ < ρ c ,we have φ R ( ρ ) → φ ∞ ( ρ ) . Since φ ∞ ( ρ ) < c and z ∞ is a continuous function,inserting φ R ( ρ ) in (9) yields as L → ∞ z R (cid:0) φ R ( ρ ) (cid:1) = z ∞ (cid:0) φ R ( ρ ) (cid:1)(cid:18) (cid:16) φ R ( ρ ) c (cid:17) R +1 c − c c − φ R ( ρ ) (cid:19) → z ∞ (cid:0) φ ∞ ( ρ ) (cid:1) . (17) C ,b ( X ) denotes the set of all bounded, continuous cylinder functions f : X → R . A cylinderfunction depends only on the particle configuration on a fixed finite number of lattice sites. ν Lφ R ( ρ ) ,R → ν φ ∞ ( ρ ) , ∞ weakly for ρ < ρ c analogous to above. For ρ > ρ c to leading order φ R ( ρ ) (cid:39) c − (cid:16) c c (cid:17) R/ c (cid:112) z ∞ ( c )( ρ − ρ c ) → c as L → ∞ . (18)For ρ = ρ c the correction has a different power (cid:0) c c (cid:1) R/ which leads to the samebehaviour as for ρ > ρ c . Inserting in (9) this yields analogous to (17) z R (cid:0) φ R ( ρ ) (cid:1) = z ∞ (cid:0) φ R ( ρ ) (cid:1)(cid:18) (cid:16) c c (cid:17) R/ (cid:114) ρ − ρ c z ∞ ( c ) (cid:19) → z ∞ ( c ) as L → ∞ (19)so that ν Lφ R ( ρ ) ,R → ν c , ∞ weakly. (cid:50) Note that by Proposition 1 the density does not converge if ρ > ρ c since ρ R (cid:0) φ R ( ρ ) (cid:1) = ρ (cid:54)→ ρ c = ρ ∞ ( c ) , (20)and the variance of η x even diverges as V ar ( η x ) = φ ∂ φ ρ R ( φ ) (cid:12)(cid:12) φ = φ R ( ρ ) (cid:39) c (cid:16) c c (cid:17) R/ − (21)Therefore there is no standard law of large numbers for the measures ν Lφ R ( ρ ) ,R when ρ > ρ c . In particular one can show the following. Proposition 2
For each L let η L , . . . , η LL be iid random variables with distribu-tion ν φ R ( ρ ) ,R and assume that R (cid:29) log L . Then L (cid:88) x ∈ Λ L η Lx → (cid:26) ρ , ρ < ρ c ρ c , ρ ≥ ρ c almost surely . (22) Proof. see appendixNote that for ρ > ρ c (20) holds due to very large values η x ∼ (cid:0) c c (cid:1) R/ having verysmall probabilities (cid:0) c c (cid:1) R/ , which also leads to divergence of the variance (21). Inturn, the small probabilities lead to almost sure convergence of the sample meanto ρ c < ρ , which is a non-standard strong law of large numbers. The breakdown ofthe standard strong law coincides with the region of nonequivalence of ensembles,as has been observed also in the context of spin systems [8, 31].As noted before, the limiting product measures ν φ, ∞ (11) exist for all φ < c and for reasons explained below, we call the family of measures (cid:8) ν φ ∞ ( ρ ) , ∞ : ρ ≥ (cid:9) the fluid phase . The pressure of the fluid phase is given by p fluid ( φ ) := lim L →∞ L log z L ∞ ( φ ) = log z ∞ ( φ ) = log c c − φ (23)7 fluid p gcan p gcan log z R L = c c log 2 ¥ Φ p s fluid s gcan Ρ c - Ρ s Figure 2:
Properties of fluid and grand-canonical measures for c = 2 , c = 1 as givenin (23) to (26). Left: Pressure p gcan (full red line), p fluid (broken red line) and log z R for L = 2 , , (dashed blue lines), demonstrating the fast convergence to p gcan . Right:Entropy densities s gcan (full red line) and s fluid (broken red line). and we define the entropy density by the negative Legendre transform s fluid ( ρ ) := − sup φ ≥ (cid:0) ρ log φ − p fluid ( φ ) (cid:1) = p (cid:0) φ ∞ ( ρ ) (cid:1) − ρ log φ ∞ ( ρ ) == (1 + ρ ) log(1 + ρ ) − ρ (log c + log ρ ) , (24)where the supremum is attained for φ = φ ∞ ( ρ ) (13). Note that the fluid pressureand entropy density are different from the grand-canonical quantities, because z R ( φ ) = ∞ for φ ≥ c (9). This yields p gcan ( φ ) := lim L →∞ L log z LR ( φ ) = (cid:26) p fluid ( φ ) , φ < c ∞ , φ ≥ c (25)and the negative Legendre transform of the pressure is given by s gcan ( ρ ) = (cid:26) s fluid ( ρ ) , ρ ≤ ρ c s fluid ( ρ c ) − ( ρ − ρ c ) log c , ρ > ρ c . (26)Note that the Legendre transform of the pressure is usually called the free energydensity. In thermodynamics, the free energy F is related to the entropy S via F = U − T S , where U is the internal energy and T the temperature. Since thereis no energy and temperature in our case, we define the entropy density as thenegative free energy density. The functions (23) to (26) are illustrated in Figure2. In analogy to previous results [17, 16] we expect a condensation transitionfor ρ > ρ c . But the non-standard behaviour of the grand-canonical measures, inparticular the lack of a law of large numbers (22), will lead to a richer behaviourthan in previous studies, which can be fully understood only in the context of theequivalence of ensembles. In particular, the grand-canonical approach alone doesnot provide a complete picture of the phase transition.8 .2 Canonical measures The canonical measures are given by π L,N := ν Lφ,R ( . | Σ L = N ) where Σ L ( η ) := (cid:88) x ∈ Λ L η x , (27)i.e. they are given by a grand-canonical measure conditioned on a fixed number N of particles. Their mass functions are independent of φ and given in terms ofthe stationary weights (6) by π L,N ( η ) = 1 Z L,N w LR ( η ) δ (Σ L ( η ) , N ) , (28)concentrating on configurations X L,N = (cid:8) η ∈ X L (cid:12)(cid:12) Σ L ( η ) = N (cid:9) . (29)The partition function is now given by the finite sum Z L,N = w LR ( X L,N ) = (cid:88) η ∈ X L,N w LR ( η ) . (30)In the following we analyze the limiting behaviour of this quantity. In the dis-cussion configurations with many particles on a small number of sites turn out toplay an important role. Therefore we define the disjoint sets of configurations X mL,N = (cid:8) η ∈ X L,N (cid:12)(cid:12) η x > R for exactly m sites x ∈ Λ L (cid:9) (31)with more than R particles on exactly m sites. Theorem 1
Suppose R (cid:29) log L , i.e. log LR → as L → ∞ . Then the limit s can ( ρ ) := lim L →∞ L log Z L,N , where N/L → ρ , (32) exists and is called the canonical entropy density. It is given by s can ( ρ ) = (cid:26) s fluid ( ρ ) , ρ ≤ ρ trans s fluid ( ρ c ) + s cond ( ρ, ρ c ) , ρ > ρ trans , (33) where s cond ( ρ, ρ c ) = lim L →∞ L log w R (cid:0) ( ρ − ρ c ) L (cid:1) . (34) The transition density ρ trans ( a ) is given by the unique solution of a = (cid:16) s fluid ( ρ c ) − ( ρ − ρ c ) log c − s fluid ( ρ ) (cid:17)(cid:46) log c c , (35) where ρ trans ( a ) ≥ ρ c + a with equality if and only if a = 0 . s cond ( ρ, ρ c ) = − ( ρ − ρ c ) log c − a log c c . (36)As a special case, taking a = 0 we have ρ trans = ρ c as the unique solution of (35),and comparing (33) with (26) yields s can ( ρ ) = s gcan ( ρ ) for all ρ ≥ . (37)On the other hand, both entropies are different whenever a > . Proof of Theorem 1.
Using (31) we decompose the state space X L,N = (cid:83) Mm =0 X mL,N .The maximal number M of sites containing more than R particles is certainlybounded by M =: (cid:100) N/R (cid:101) . Notice that M → ρ/a ∈ (0 , ∞ ] as L → ∞ , and inparticular M/L → . We can estimate the number of “uncondensed” configura-tions where no site has more than R particles by the following Lemma, which isproved in the appendix. Lemma 1
For all
L, N ≥ and M as above we have | X L,N | (cid:0) L + MM (cid:1)(cid:14) ( L − M ) R ≤ | X L,N | ≤ | X L,N | . (38) This includes for all ρ ≥ and N/L → ρ lim L →∞ L log | X L,N | = lim L →∞ L log | X L,N | = χ ( ρ ) , (39) where χ ( ρ ) := (1 + ρ ) log(1 + ρ ) − ρ log ρ .Furthermore, if R (cid:29) √ L , then lim L →∞ | X L,N | / | X L,N | = 1 for all ρ ≥ . Now we split the partition function accordingly Z L,N = M (cid:88) m =0 Z mL,N , where Z mL,N = w LR ( X mL,N ) . (40)For the term m = 0 we get with Lemma 1 L log Z L,N = 1 L log (cid:0) c − N | X L,N | (cid:1) → χ ( ρ ) − ρ log c = s fluid ( ρ ) . (41)The contributions of the other terms are given by Z mL,N = (cid:18) Lm (cid:19) c − mR N (cid:88) k = m ( R +1) c − ( N − k )0 c − ( k − mR )1 (cid:12)(cid:12) X L − m,N − k (cid:12)(cid:12)(cid:18) k − mR − m − (cid:19) . (42)10ere we have chosen m sites on which we distribute k particles such that eachsite contains at least R + 1 particles, giving rise to the first and last combinatorialfactor. The N − k remaining particles are distributed on L − m sites such thatnone contains more than R particles. The sum can be approximated by an integraland evaluated by the saddle point method. The saddle point equation reads log c c − LL − m χ (cid:48) (cid:16) N − kL − m (cid:17) + log k − mR − k − m ( R + 1) = 0 . (43)This has a solution if and only if N − ( L − m ) ρ c ≥ m ( R + 1) or, equivalently ρ ≥ ρ c + m a . (44)In this case, to leading order the solution to (43) is given by k (cid:39) N − ( L − m ) ρ c (45)where we have used that m/L → for all m ≤ M . On the other hand, for ρ < ρ c + m a the sum in (42) is maximized for the boundary value k = m ( R + 1) .We get in leading exponential order Z mL,N (cid:39) , ρ ≤ m a (cid:0) Lm (cid:1)(cid:0) c c (cid:1) m c − N e ( L − m ) χ ( ρ − ma ) , m a<ρ<ρ c + m a (cid:0) Lm (cid:1)(cid:0) c c (cid:1) mR +( L − m ) ρ c c − N e ( L − m ) χ ( ρ c ) , ρ ≥ ρ c + m a . (46)For ρ > ρ c + a we get a rough estimate by adding both cases, M (cid:88) m =2 Z mL,N ≤ Z L,N (cid:18) LM (cid:19)(cid:18) ML (cid:16) c c (cid:17) N − M − R − ( L − ρ c + (cid:16) c c (cid:17) R CL M − (cid:19) , (47)where C = exp (cid:16) ( c /c ) ρ c e − χ ( ρ c ) (cid:17) . Now, to leading order L log (cid:32)(cid:18) LM (cid:19)(cid:16) c c (cid:17) R C L M − (cid:33) (cid:39) − ML (cid:16) ML + ML (cid:17) − log LL − RL log c c + ML log L → − a log c c ≤ as L → ∞ , (48)since M/L → , R/L → a ≥ . This holds only if M (cid:28) L/ log L or, equiva-lently, R (cid:29) log L . Since ρ > ρ c + a the first summand on the right-hand side of(47) vanishes with an analogous argument. Therefore L log (cid:18) M (cid:88) m =2 Z mL,N (cid:46) Z L,N (cid:19) → , (49)11nd the only exponential contribution to (47) is given by Z L,N . Thus we have,using (46), lim L →∞ L log M (cid:88) m =1 Z mL,N = lim L →∞ L log Z L,N == lim L →∞ L log (cid:18) L (cid:16) c c (cid:17) R +( L − ρ c c − N (cid:12)(cid:12) X L − , ( L − ρ c (cid:12)(cid:12)(cid:19) == ( a + ρ c ) log c c − ρ log c + χ ( ρ c ) = s fluid ( ρ c ) + s cond ( ρ, ρ c ) . (50)This is a linear function in ρ with the same slope − log c as s gcan ( ρ ) (26). Notethat for ρ → ∞ the first term (41) behaves as s fluid ( ρ ) (cid:39) − ρ log c + log(1 + ρ ) + 1 . (51)Therefore, whereas for small ρ (41) dominates the partition function, (50) dom-inates for large ρ , since it has larger asymptotic slope − log c > − log c . Thetransition density ρ trans as a function of a is found by equating both contributionswhich leads directly to (35). Differentiating the right-hand side of this equationyields a (cid:48) ( ρ ) = 1 − log 1 + ρρ (cid:46) log c c . (52)Thus a (cid:48) ( ρ c ) = 0 and a (cid:48) ( ρ ) ∈ (0 , for all ρ > ρ c . Since also a ( ρ c ) = 0 , (35) hasa unique solution ρ trans ( a ) ≥ ρ c for all a ≥ . Further we have ρ (cid:48) trans ( a ) = 1 a (cid:48) ( ρ trans ( a )) > for all ρ ≥ ρ c , (53)and thus ρ trans ( a ) ≥ ρ c + a with equality if and only if a = 0 . (cid:50) Now, if a > then M as defined after (31) is bounded and converges to ρ/a , andthus (47) implies that M (cid:88) m =2 Z mL,N (cid:46) Z L,N → as L → ∞ . (54)This is significantly stronger than (48) and it is easy to see that it still holds for a = 0 , as long as R (cid:29) √ L log L . Thus for L → ∞ the canonical measureconcentrates on certain parts of the state space, and from the proof of Theorem1 (47) the rate of convergence is faster than polynomial in L . Therefore we canimmediately deduce the following. 12 orollary 1 For R (cid:29) √ L log L we have ρ < ρ trans ⇒ π L,N (cid:0) X L,N (cid:1) → , L n π L,N (cid:0) X L,N \ X L,N (cid:1) → ,ρ > ρ trans ⇒ π L,N (cid:0) X L,N (cid:1) → , L n π L,N (cid:0) X L,N \ X L,N (cid:1) → , (55) for all n ∈ N as L → ∞ and N/L → ρ . This implies in analogy to (33), that for ρ > ρ trans a typical configuration consistsof a homogeneous background with density ρ c and the ( ρ − ρ c ) L excess particlesconcentrate in a single lattice site. We expect this kind of behaviour actuallyalready for R (cid:29) log L , since w R has an exponential tail and maximal fluctuationsunder w LR in the occupation number are of order log L . Our estimates are notstrong enough to deduce this, but we are primarily interested in a > , which iscovered by the above result. The same is true for the last statement of Lemma 1.For ρ = ρ trans the contributions of condensed and fluid configurations to thecanonical entropy are equal (33). This is true on the exponential scale and todeduce the behaviour on the transition line we need a finer estimate, given in thefollowing Theorem. Theorem 2
For
N/L → ρ trans and a > we have w LR ( X L,N ) /w LR ( X L,N ) = O ( L / ) → ∞ as L → ∞ , (56) which implies π L,N ( X L,N ) → . So in case of a discontinuous transition (i.e. a > ) the transition line belongs tothe condensed phase C/F . For a = 0 the transition is continuous and therefore ρ = ρ c belongs to the fluid phase F ( E ) . Proof.
According to (42) in the proof of Theorem 1, w LR ( X L,N ) =
L c − R N (cid:88) k = R +1 c − ( N − k )0 c − ( k − R )1 (cid:12)(cid:12) X L − ,N − k (cid:12)(cid:12) == L (cid:16) c c (cid:17) R + ρ c ( L − c − N (cid:12)(cid:12) X L − ,ρ c ( L − (cid:12)(cid:12) (cid:0) o (1) (cid:1)(cid:90) NR +1 exp (cid:18) χ (cid:48)(cid:48) ( ρ c ) L ( L − ( k − ¯ k ) (cid:19) dk (57)where ¯ k = N − ( L − ρ c + o ( L ) is the solution to the saddle point equation (43). Inaddition to the proof of Theorem 1 we consider the next order of the expansion toget the correct asymptotic behaviour. Since with Lemma 1, χ (cid:48)(cid:48) ( ρ ) = − ρ (1+ ρ ) < for all ρ > and ¯ k ∈ ( R + 1 , N ) , the asymptotic behaviour of the Gaussian13ntegral with variance σ = − L/χ (cid:48)(cid:48) ( ρ c )(1 + o (1)) is given by its normalizationand we get w LR ( X L,N ) == L / (cid:16) c c (cid:17) R + ρ c ( L − c − N (cid:12)(cid:12) X L − ,ρ c ( L − (cid:12)(cid:12) (cid:112) πρ c (1 + ρ c ) (cid:0) o (1) (cid:1) . (58)With C = (cid:112) πρ c (1 + ρ c ) this leads to w LR ( X L,N ) w LR ( X L,N ) = CL / (cid:0) c c (cid:1) R + ρ c L c − N (cid:12)(cid:12) X L − ,ρ c ( L − (cid:12)(cid:12) c − N (cid:12)(cid:12) X L,N (cid:12)(cid:12) (cid:0) o (1) (cid:1) == CL / (cid:0) c c (cid:1) R + ρ c L c − N (cid:0) ( L − ρ c ) − L − (cid:1) c − N (cid:0) L + N − L − (cid:1) (cid:0) o (1) (cid:1) , (59)where we have used the third statement of Lemma 1 that holds for a > . Weuse Stirling’s formula for the binomial coefficients and note that due to Theorem1 the exponential terms in the ratio vanish, which leaves us with w LR ( X L,N ) w LR ( X L,N ) = CL / (cid:0) o (1) (cid:1) → ∞ as L → ∞ , N/L → ρ trans . (60)Together with Theorem 1 this implies that w LR ( X L,N \ X L,N ) → as L → ∞ , N/L → ρ trans , (61)which implies the last statement of the Theorem. (cid:50) In Table 1 we summarize the results of the previous section in connection withthe phase diagram shown in Figure 1. In particular, for a = 0 the phases F and F/C are empty since ρ c = ρ trans , and we have s can ( ρ ) = s gcan ( ρ ) for all ρ ≥ asnoted already in (37). This implies that the canonical entropy density is concaveand the condensation transition is continuous. On the other hand, for a > we have equivalence of ensembles only in phase F ( E ) , the canonical entropydensity is non-concave, and the transition is discontinuous. These results concernequivalence of ensembles in terms of convergence of entropies of the canonicaland the grand-canonical measure. In Figure 3 they are illustrated by numericalcalculations of the canonical entropy density using the recursion relation Z L,N = N (cid:88) k =0 w R ( k ) Z L − ,N − k . (62)14anonical entropy grand-canonical entropyphase s can ( ρ ) s gcan ( ρ ) F(E) s fluid ( ρ ) s fluid ( ρ ) F, F/C s fluid ( ρ ) C/F s fluid ( ρ c ) − ( ρ − ρ c ) log c − a log c c s fluid ( ρ c ) − ( ρ − ρ c ) log c Table 1:
Summary of the results of Section 3: Comparison between canonical and grand-canonical entropy density. Equivalence of ensembles holds only in phase F ( E ) . ‰ ‰ ‰ ‰ ‰ ‰‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ œœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœ c = c =
1, a = s fluid H Ρ L s gcan H Ρ L s fluid H Ρ c L + s cond H Ρ, Ρ c L Ρ c + a Ρ c Ρ trans Ρ s can H Ρ L ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ c = c = = s fluid H Ρ L s gcan H Ρ L s fluid H Ρ c L + s cond H Ρ, Ρ c L Ρ c + a Ρ c Ρ trans Ρ s can H Ρ L ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ œœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœ c = c = = s fluid H Ρ L s gcan H Ρ L Ρ c Ρ trans - - - Ρ s can H Ρ L Figure 3:
Canonical entropy density s can ( ρ ) for various values of c , c and a . Datapoints are calculated numerically according to (62) with L = 100 ( × ) ,
200 (+) ,
400 ( (cid:51) ) ,and show good agreement with the theoretical predictions for the thermodynamic limit(see Table 1).
15s can be seen, the grand-canonical entropy density is equal to the concave hullof s can ( ρ ) which itself is not concave for a > . The canonical entropy densityfurther coincides with the one of the fluid phase, up to the point when it becomesmetastable and the condensed phase becomes stable. This point has been derivedexactly by studying the dominating terms in the canonical partition function.We can make a connection to other formulations of the equivalence of ensem-bles, using the specific relative entropy h ( π L,N , ν
Lφ,R ) := 1
L H ( π L,N , ν
Lφ,R ) = 1 L (cid:68) log π L,N ( η ) ν Lφ,R ( η ) (cid:69) π L,N . (63)With the identity π LL,N = ν Lφ,R ( . | Σ L = N ) , this can be expressed in two usefulforms, h ( π L,N , ν
Lφ,R ) = − L log ν Lφ,R (cid:0) Σ L = N (cid:1) == log z R ( φ ) − NL log φ − L log Z L,N . (64)The derivation of these expressions is straightforward, see e.g. [17]. The follow-ing is a direct consequence of our results on the canonical measure in Theorem1. Corollary 2
Choosing φ = φ R ( ρ ) according to (16) we get for all ρ ≥ h ( π L,N , ν Lφ R ( ρ ) ,R ) → s gcan ( ρ ) − s can ( ρ ) . (65) Proof.
We use the second expression in (64) for the specific relative entropy.Choosing φ = φ R ( ρ ) , the first two terms converge log z R ( φ R ( ρ )) − log φ R ( ρ ) NL → s gcan ( ρ ) (66)to the grand-canonical entropy density (26), since with Proposition 1, analogousto (17) and (19) z R (cid:0) φ R ( ρ ) (cid:1) → (cid:26) z ∞ (cid:0) φ ∞ ( ρ ) (cid:1) , for ρ < ρ c z ∞ ( c ) , for ρ ≥ ρ c . (67)Convergence of the third term in (64) has been shown in Theorem 1, which fin-ishes the proof. (cid:50) We can read from Table 1 that s gcan ( ρ ) − s can ( ρ ) = , ρ ≤ ρ c s fluid ( ρ c ) − s fluid ( ρ ) − ( ρ − ρ c ) log c , ρ c <ρ<ρ trans a log( c /c ) , ρ ≥ ρ trans . (68)16n particular, for a = 0 we have ρ c = ρ trans and h ( π L,N , ν Lφ R ( ρ ) ,R ) → for all ρ ≥ , (69)whereas for a > this holds only for ρ ≤ ρ c . By a standard result [6], conver-gence in specific relative entropy implies weak convergence, i.e. convergence ofexpectations of bounded cylinder test functions f ∈ C ,b ( X ) , (cid:12)(cid:12)(cid:12) (cid:104) f (cid:105) π L,N − (cid:104) f (cid:105) ν LφR ( ρ ) ,R (cid:12)(cid:12)(cid:12) → as L → ∞ , N/L → ρ . (70)This is another formulation of the equivalence of ensembles.Furthermore, we can compare the canonical measures with the expected fluidmeasures for the background. Theorem 3
Let a > . Choosing φ = φ ∞ ( ρ ) according to (13) we get h ( π L,N , ν Lφ ∞ ( ρ ) , ∞ ) → s fluid ( ρ ) − s can ( ρ ) = 0 for ≤ ρ < ρ trans , (71) whereas for φ = c h ( π L,N , ν Lc , ∞ ) → ( ρ − ρ c ) log c c > for ρ ≥ ρ trans . (72) Now let a = 0 and R (cid:29) √ L log L . We have ρ trans = ρ c and (71) holds for ≤ ρ ≤ ρ c , (72) for ρ > ρ c . Proof.
According to the definition (63) we have h ( π L,N , ν Lφ ∞ ( ρ ) , ∞ ) = 1 L (cid:88) η ∈ X L,N π L,N ( η ) log π L,N ( η ) ν Lφ ∞ ( ρ ) , ∞ ( η ) = p fluid (cid:0) φ ∞ ( ρ ) (cid:1) − NL log φ ∞ ( ρ ) c − L log Z L,N + 1 L (cid:88) η ∈ X L,N π L,N ( η ) log w LR ( η ) , (73)where we have used the definitions (11) and (28), ν φ, ∞ ( k ) = 1 z ∞ ( φ ) ( φ/c ) k , π L,N ( η ) = 1 Z L,N w LR ( η ) δ (Σ L ( η ) , N ) . (74)Splitting the last term of (73) and using Corollary 1 we see that L (cid:88) η ∈ X L,N π L,N ( η ) log c − N + 1 L (cid:88) η ∈ X L,N \ X L,N π L,N ( η ) log w LR ( η ) → − ρ log c , (75)as L → ∞ , N/L → ρ , as long as ρ < ρ trans . Therefore, with definition (24), h ( π L,N , ν Lφ ∞ ( ρ ) , ∞ ) → s fluid ( ρ ) − s can ( ρ ) = 0 for ρ < ρ trans . (76)17his holds for all a ≥ as long as R (cid:29) √ L log L . For ρ > ρ trans we also use(73) where φ ∞ ( ρ ) is replaced by c . Again with Corollary 1 the main contributionto the last term comes now from η ∈ X L,N . The sum can be computed by thesaddle point method analogous to the proof of Theorem 1 and we get L (cid:88) η ∈ X L,N π L,N ( η ) log w LR ( η ) → a log c c − ρ c log c − ( ρ − ρ c ) log c . (77)The same also holds for ρ = ρ trans , since with Theorem 2 π L,N concentrates on X L,N also in this case. The first terms in (73) are now p fluid (cid:0) c (cid:1) − NL log c c → s fluid ( ρ c ) + ρ log c − ( ρ − ρ c ) log c , (78)and together with the behaviour of s can from Theorem 1 we get for ρ ≥ ρ trans h ( π L,N , ν Lc , ∞ ) → ( ρ − ρ c ) log c c > , (79)finishing the proof of Theorem 3. Note that a = 0 is included as a special case inthe above derivation as long as R (cid:29) √ L log L . (cid:50) (71) allows us to identify the limit measure and we have (cid:104) f (cid:105) π L,N → (cid:104) f (cid:105) ν φ ∞ ( ρ ) , ∞ as L → ∞ , N/L → ρ . (80)As a direct consequence of the relative entropy inequality ([5], Lemma 3.1), thisholds not only for bounded cylinder test functions f , but for the larger class with (cid:104) e (cid:15)f (cid:105) ν φ ∞ ( ρ ) , ∞ < ∞ for some (cid:15) > . Since the fluid measures have finite expo-nential moments, this includes local occupation numbers f ( η ) = η x , which areunbounded. This ensures convergence of densities for ρ < ρ trans ( ρ ≤ ρ c for a = 0 ), i.e. in the fluid phases F ( E ) , F and F/C . (72) may suggest that the limiting distribution of the background in the condensedphase C/F is more complicated than the expected fluid measure ν c , ∞ . Togetherwith Corollary 1 we can show that the non-zero specific relative entropy is onlydue to the contribution of the single condensate site and indeed the backgrounddistribution is as expected. In the following we attach some (arbitrary) orderingto the lattice sites and identify Λ L = { , . . . , L } . On X L − we define the measure ˆ π L,N as a marginal on the first L − coordinates ˆ π L,N := π L,N ( . | L ∈ argmax) ,..,L − (81)18here π L,N ( . | L ∈ argmax) denotes the measure π L,N conditioned on the eventthat η L ≥ η x for all x = 1 , . . . , L − . Since π L,N is invariant under site permuta-tions, we have ˆ π L,N := π L,N ( . | y ∈ argmax) Λ L \{ y } for all y ∈ Λ L . (82)Note that ˆ π L,N concentrates on a subset of X L − , ˆ X L − := (cid:8) ˆ η ∈ X L − (cid:12)(cid:12) Σ L − (ˆ η ) < N, ˆ η , .., ˆ η L − ≤ N − Σ L − (ˆ η ) (cid:9) , (83)and in case of condensation it can be interpreted as the distribution of the back-ground. Theorem 4
For ρ ≥ ρ trans we have as L → ∞ , N/L → ρ , H (ˆ π L,N , ν L − c , ∞ ) → , (84) and thus for bounded cylinder test functions (cid:104) f (cid:105) π L,N → (cid:104) f (cid:105) ν c , ∞ . (85)Note that the first statement (84) involves the total rather than the specific relativeentropy and is therefore much stronger than Corollary 2 and Theorem 3. This im-plies convergence in total variation norm [5]. Such a result is not possible belowcriticality, since the conditioning on the particle number in the canonical mea-sures leads to divergence of the relative entropy. Above criticality, this conditionis accounted for purely by the condensate site and does not affect the background,which shows the same fluctuations as i.i.d. random variables. Following recent re-sults in [24], this enables to show that the stationary density profiles converge to aBrownian motion with a jump at the location of the condensate. Below criticalitythe corresponding expected behaviour would be a Brownian bridge, but there isno proof so far.The second statement (85) is a direct consequence of the first but not a very strongone, since it would also follow from convergence in specific relative entropy. Thesite with maximum occupation number will be in the support of the cylinder testfunction only with probability of order /L . But due to this possibility, the testfunction has to be bounded, not necessarily by a constant but by a number of or-der o ( L ) . This excludes f ( η ) = η x as expected, since the expected density doesnot converge for ρ ≥ ρ trans . Note also that with (72) and (85) this system is anexample where weak convergence is strictly weaker than convergence in specificrelative entropy. Proof. (81) and (83) imply that ˆ π L,N (ˆ η ) = π L,N (cid:0) ˆ η , N − Σ L − (ˆ η ) (cid:1) π L,N ( L ∈ argmax) ˆ X L − (ˆ η ) , (86)19here (ˆ η , N − Σ L − (ˆ η )) ∈ X L,N denotes the concatenated configuration. Bypermutation invariance we get π L,N ( L ∈ argmax) = 1 L π
L,N ( X L,N ) + ˜ R L,N = 1 L (cid:0) o (1) (cid:1) (87)where ≤ ˜ R L,N ≤ π L,N ( X L,N \ X L,N ) . So the error is exponentially small in thesystem size for ρ > ρ trans (see Corollary 1) and of order L − / for ρ = ρ trans .Now we can compute the relative entropy H (ˆ π L,N , ν L − c , ∞ ) = (cid:88) ˆ η ∈ ˆ X L − ˆ π L,N (ˆ η ) log π L,N (cid:0) ˆ η , N − Σ L − (ˆ η ) (cid:1) π L,N ( L ∈ argmax) ν L − c , ∞ (ˆ η ) == (cid:88) ˆ η ∈ ˆ X L − ˆ π L,N (ˆ η ) log w L − R (ˆ η ) c − R c − ( N − Σ L − (ˆ η ) − R )1 Lz L − ∞ ( c )( c /c ) Σ L − (ˆ η ) Z L,N (1 + o (1)) + R L,N , (88)where analogous to (31) ˆ X L − = (cid:8) ˆ η ∈ ˆ X L − (cid:12)(cid:12) ˆ η , . . . , ˆ η L − ≤ R (cid:9) . (89)Therefore we have | R L,N | = (cid:12)(cid:12)(cid:12)(cid:12) (cid:88) ˆ η ∈ ˆ X L − \ ˆ X L − ˆ π L,N (ˆ η ) log π L,N (cid:0) ˆ η , N − Σ L − (ˆ η ) (cid:1) Lν L − c , ∞ (ˆ η ) (1 + o (1)) (cid:12)(cid:12)(cid:12)(cid:12) ≤≤ Cπ L,N (cid:0) X L,N \ ( X L,N ∪ X L,N ) (cid:1) L → as L → ∞ , (90)using Corollary 1 and Theorem 2, since the argument of the logarithm is at mostexponential in L . On ˆ X L − we have w L − R (ˆ η ) = c − Σ L − (ˆ η )0 and thus H (ˆ π L,N , ν L − c , ∞ ) = π L,N ( X L,N ) log ( c /c ) R c − N L c L − Z L,N ( c − c ) L − + o (1) , (91)where we have used ˆ π L,N ( ˆ X L − ) = π L,N ( X L,N ) . With Theorems 1 and 2 wehave for ρ ≥ ρ trans Z L,N = w LR ( X L,N ) (cid:0) o (1) (cid:1) == L / (cid:16) c c (cid:17) R + ρ c L c − N (cid:12)(cid:12) X L − ,ρ c ( L − (cid:12)(cid:12) (cid:112) πρ c (1 + ρ c ) (cid:0) o (1) (cid:1) , (92)and according to Lemma 1 (cid:12)(cid:12) X L − ,ρ c ( L − (cid:12)(cid:12) = (cid:18) (1 + ρ c )( L − − L − (cid:19)(cid:0) o (1) (cid:1) . (93)A careful application of Stirling’s formula, which was not necessary in the proofof Theorem 2, yields (cid:18) (1+ ρ c )( L − − L − (cid:19) = (cid:16) c c − c c ρ c (cid:17) L − (cid:0) πρ c (1+ ρ c ) L (cid:1) − / (cid:0) o (1) (cid:1) , (94)20here we have used in the exponential term that ρ c = c / ( c − c ) . Pluggingeverything into (91) this leads to a perfect cancellation and we get as L → ∞ H (ˆ π L,N , ν L − c , ∞ ) = π L,N ( X L,N ) log (cid:0) o (1) (cid:1) + o (1) → , (95)which finishes the proof of the first statement.Let f ∈ C ,b ( X ) be a cylinder test function bounded by C and supported onthe lattice sites supp ( f ) ⊂ N with (cid:12)(cid:12) supp ( f ) (cid:12)(cid:12) = n . In the following let L > max supp ( f ) such that supp ( f ) (cid:40) Λ L . We have (cid:104) f (cid:105) π L,N = (cid:88) η ∈ X L,N π L,N ( η ) f ( η ) = (cid:88) η ∈ X L,N π L,N ( η ) f ( η ) + R L,N , (96)where due to Corollary 1 | R L,N | = (cid:12)(cid:12)(cid:12) (cid:88) η ∈ X L,N \ X L,N π L,N ( η ) f ( η ) (cid:12)(cid:12)(cid:12) ≤ Cπ L,N ( X L,N \ X L,N ) → , (97)Since | argmax( η ) | = 1 for all η ∈ X L,N , we have (cid:88) η ∈ X L,N π L,N ( η ) f ( η ) = 1 L (cid:88) y ∈ Λ L (cid:88) η ∈ X L,N π L,N (cid:0) η (cid:12)(cid:12) argmax = { y } (cid:1) f ( η ) == 1 L (cid:88) y ∈ Λ L \ supp ( f ) (cid:88) ˆ η ∈ ˆ X L − π L,N (cid:0) ˆ η (cid:12)(cid:12) argmax = { y } (cid:1) Λ L \{ y } f (ˆ η ) + R L,N = L − nL (cid:88) ˆ η ∈ ˆ X L − ˆ π L,N (ˆ η ) f (ˆ η ) + R L,N (98)due to (82), where boundedness of f implies | R L,N | = (cid:12)(cid:12)(cid:12)(cid:12) L (cid:88) y ∈ supp ( f ) (cid:88) η ∈ X L,N π L,N (cid:0) η (cid:12)(cid:12) argmax = { y } (cid:1) f (ˆ η ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ nL C π L,N ( X L,N ) → as L → ∞ . (99)Note that this is the only place where we crucially require that f is bounded by aconstant of order o ( L ) . With Corollary 1 we get (cid:88) ˆ η ∈ ˆ X L − ˆ π L,N (ˆ η ) f (ˆ η ) = (cid:104) f (cid:105) ˆ π L,N + R L,N (100)where | R L,N | = (cid:12)(cid:12)(cid:12)(cid:12) (cid:88) ˆ η ∈ ˆ X L − \ ˆ X L − ˆ π L,N (ˆ η ) f (ˆ η ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ π L,N ( ˆ X L − \ ˆ X L − ) == C π
L,N (cid:0) X L,N \ ( X L,N ∪ X L,N ) (cid:1) → as L → ∞ . (101)21ogether with (84) shown above, this implies (cid:104) f (cid:105) π L,N = (cid:104) f (cid:105) ˆ π L,N + R L,N + R L,N + R L,N → (cid:104) f (cid:105) ν c , ∞ , (102)since convergence in total relative entropy implies weak convergence. (cid:50) In the previous section the role of the density ρ c + a remains open. A first hintappears in the proof of Theorem 1, where the saddle point equation for condensatecontributions (43) only has solutions for ρ ≥ ρ c + a . But in the context of theequivalence of ensembles we are not able to distinguish the phases F and F/C in the phase diagram (Figure 1), as can be seen in Table 1. A further analysisof the canonical measures in terms of the order parameter of the model, i.e. thebackground density ρ bg of uncondensed particles, will clarify this point. Since for a = 0 the condensation transition is continuous, we only consider the case a > throughout this section. We define the observable Σ bgL ( η ) := Σ L ( η ) − max x ∈ Λ L η x , (103)which can be interpreted as the number of particles in the background, since atmost one site contributes to the condensate. Theorem 5
Let S , S , . . . ∈ N be any sequence with S L /L → ρ bg > . Then thelimit I ρ ( ρ bg ) := − lim L →∞ L log π L,N (cid:0) Σ bgL = S L (cid:1) ∈ [0 , ∞ ] (104) exists for all ρ > ( N/L → ρ ), and defines the rate function for the events (cid:8) Σ bgL = S L (cid:9) . For ρ bg > ρ , I ρ ( ρ bg ) = ∞ and for ρ bg ≤ ρ it can be written as I ρ ( ρ bg ) = s can ( ρ ) − s fluid ( ρ bg ) ++ (cid:26) ( ρ − ρ bg ) log c , ρ bg ≥ ρ − a ( ρ − ρ bg ) log c + a log( c /c ) , ρ bg ≤ ρ − a . (105) Proof.
For ρ bg > ρ , S L > N eventually and thus π L,N (cid:0) Σ bgL = S L (cid:1) = 0 eventually.For ρ bg ≤ ρ we use the identity π L,N = ν Lφ R ( ρ ) ,R ( . | Σ L = N ) = ν Lφ R ( ρ ) ,R (cid:0) . ∪ { Σ L = N } (cid:1) ν Lφ R ( ρ ) ,R (Σ L = N ) (106)and the fact that (64) and (65) imply − L log ν Lφ R ( ρ ) ,R (Σ L = N ) = h (cid:0) π L,N , ν Lφ R ( ρ ) ,R (cid:1) → s gcan ( ρ ) − s can ( ρ ) . (107)22urthermore, Corollary 1 implies that lim L →∞ L log π L,N ( . ) = lim L →∞ L log π L,N (cid:0) . ∩ ( X L,N ∪ X L,N ) (cid:1) , (108)and therefore we get lim L →∞ L log π L,N (cid:0) Σ bgL = S L (cid:1) = s gcan ( ρ ) − s can ( ρ )++ lim L →∞ L log ν φ R ( ρ ) ,R ( η L = N − S L ) ++ lim L →∞ L log ν L − φ R ( ρ ) ,R (cid:0) Σ L − = S L , η , .., η L − ≤ ( N − S L ) ∧ R (cid:1) . (109)For the last two terms we have fixed the maximum to be on site L , since the cor-responding polynomial correction vanishes on the logarithmic scale in the limit.With the definition of the single site measure (8) the second last term is given by lim L →∞ L log ν φ R ( ρ ) ,R ( η L = N − S L ) == (cid:26) ( ρ − ρ bg ) log( φ gcan ( ρ ) /c ) , ρ bg ≥ ρ − a ( ρ − ρ bg ) log( φ gcan ( ρ ) /c ) − a log( c /c ) , ρ bg ≤ ρ − a , (110)where (cf. (16)) φ gcan ( ρ ) := lim L →∞ φ R ( ρ ) = (cid:26) φ ∞ ( ρ ) = c ρ/ (1 + ρ ) , ρ ≤ ρ c c , ρ ≥ ρ c . (111)Due to the condition η , . . . , η L − ≤ ( N − S L ) ∧ R in the last term, which followsfrom (108) and (103), all configurations in that event have the same probabilityand we get lim L →∞ L log ν L − φ R ( ρ ) ,R (cid:0) Σ L − = S L , η , .., η L − ≤ ( N − S L ) ∧ R (cid:1) == lim L →∞ L log (cid:18) ( φ R ( ρ ) /c ) S L z R ( φ R ( ρ )) L − | ˜ X L − ,S L | (cid:19) == ρ bg log φ gcan ( ρ ) c − p (cid:0) φ gcan ( ρ ) (cid:1) + χ ( ρ bg ) == ( ρ bg − ρ ) log φ gcan ( ρ ) − s gcan ( ρ ) + s fluid ( ρ bg ) . (112)where ˜ X L − ,S L = (cid:8) η ∈ X L − ,S L (cid:12)(cid:12) η , . . . , η L − ≤ ( N − S L ) ∧ R (cid:9) . (113)Due to the more restrictive condition this is only a subset of X L − ,S L , but com-pletely analogously to Lemma 1 one can show that as L → ∞ L log | ˜ X L − ,S L | → (1+ ρ bg ) log(1+ ρ bg ) − ρ bg log ρ bg = χ ( ρ bg ) . (114)23 = Ρ= Ρ=Ρ c + a0 0.5 Ρ c Ρ c + a 2 Ρ trans Ρ bg I Ρ H Ρ bg L Ρ= Ρ= Ρ=Ρ trans Ρ c Ρ c + a 2 Ρ trans Ρ bg I Ρ H Ρ bg L Figure 4:
The rate function I ρ ( ρ bg ) for c = 2 , c = 1 , a = 0 . and various values of ρ .For ρ > ρ c + a the function has a local minimum at ρ bg = ρ c , which becomes the globalminimum for ρ > ρ trans . Inserting (110) and (112) into (109) finishes the proof. (cid:50)
Figure 4 shows that the distribution of Σ bgL concentrates on values of the order ρL for ρ < ρ trans and on values of the order ρ c L for ρ > ρ trans . These two casescorrespond to the phases F/C and
C/F , respectively, and have been identifiedalready in the previous section. But in Figure 4 also the role of ρ c + a can beidentified. For ρ < ρ c + a , the rate function I ρ ( ρ bg ) (104) has only one minimum I ρ ( ρ ) = 0 , whereas for ρ > ρ c + a it has an additional local minimum at ρ bg = ρ c ,i.e. the condensed phase becomes metastable. For ρ > ρ trans this local minimumbecomes the global one, and the fluid phase becomes metastable. For ρ = ρ trans the rate function vanishes for both phases, but the finer analysis of Theorem 2reveals that the fluid phase is already metastable in this case.By definition, the observable Σ bgL ( η ) changes at most by ± during each jumpof a particle. So the process (cid:0) Σ bgL ( η ( t )) (cid:1) t ≥ is a one-dimensional simple randomwalk (or a birth-death process) on { , , . . . , N } , whose stationary large deviationrate function is I ρ . The minima of this rate function correspond to the fluid phasefor ρ bg = ρ and the condensed phase for ρ bg = ρ c . For finite L the system has twoquasi-stationary distributions π L,N ( . | X L,N ) and π L,N ( . | X L,N ) , (115)corresponding to the fluid and the condensed phase, respectively. Analogous to(81) we define ˜ π L,N = π L,N ( . | X L,N , L ∈ argmax) ,..,L − (116) Proposition 3
In the limit L → ∞ , N/L → ρ , we have for all ρ ≥ h (cid:0) π L,N ( . | X L,N ) , ν Lφ ∞ ( ρ ) , ∞ (cid:1) → and π L,N ( . | X L,N ) → ν φ ∞ ( ρ ) , ∞ , (117)24 nd for all ρ ≥ ρ c + aH (cid:0) ˜ π L,N , ν L − c , ∞ (cid:1) → and π L,N ( . | X L,N ) → ν c , ∞ . (118) In both cases the second convergence is weakly with respect to bounded cylindertest functions.
As in Theorem 4, we can show convergence in total relative entropy (118) for thecondensed phase, which is much stronger than convergence in specific relativeentropy (see comments in the previous section).
Proof.
The first statements in (117) and (118) can be proved analogous to Theo-rem 3 and Theorem 4, respectively. Since π L,N ( η | X L,N ) = π L,N ( η ) π L,N ( X L,N ) X L,N ( η ) = 1 | X L,N | X L,N ( η ) (119)is the uniform measure on X L,N , we get for (117) analogous to (73) h (cid:0) π L,N ( . | X L,N ) , ν Lφ ∞ ( ρ ) , ∞ (cid:1) == 1 L (cid:88) η ∈ X L,N π L,N ( η | X L,N ) log z ∞ (cid:0) φ ∞ ( ρ ) (cid:1) L (cid:0) φ ∞ ( ρ ) /c (cid:1) N | X L,N | == p fluid (cid:0) φ ∞ ( ρ ) (cid:1) − NL log φ ∞ ( ρ ) c − L log | X L,N |→ s fluid ( ρ ) + ρ log c − χ ( ρ ) = 0 as L → ∞ (120)for all ρ ≥ , using Lemma 1. Note that here we are a priori restricted to X L,N sothat there is no error term as in (73).The same holds for a modification of (88) to derive (118). Using ˜ π L,N (ˆ η ) = π L,N (ˆ η , N − Σ L − (ˆ η )) π L,N (argmax =
L, X L,N ) argmax= L,X L,N (cid:0) ˆ η , N − Σ L − (ˆ η ) (cid:1) == L ( c /c ) − Σ L − (ˆ η ) − R c − N π L,N ( X L,N ) Z L,N argmax= L,X L,N (cid:0) ˆ η , N − Σ L − (ˆ η ) (cid:1) (121)and π L,N ( X L,N ) → , we get in direct analogy to the proof of Theorem 4 H (˜ π L,N , ν L − c , ∞ ) = (cid:88) ˆ η ∈ ˆ X L − ˜ π L,N (ˆ η ) log L ( c /c ) − R c − N z ∞ ( c ) L − π L,N ( X L,N ) Z L,N == π L,N ( X L,N ) log (cid:0) o (1) (cid:1) → as L → ∞ . (122)The second statements in (117) and (118) follow completely analogously tothe proofs of Theorem 3 and Theorem 4. (cid:50) ρ < ρ trans and ρ ≥ ρ trans , respectively,where the quasi-stationary distributions converge to the stationary distribution.For finite L , both phases have life-times of the order ∼ e ξ ( ρ ) L exponential inthe system size for all ρ > ρ c + a , where the exponential rate ξ ( ρ ) depends on thedensity. It can be calculated using the hitting times τ fluidL ( ρ ) := inf (cid:8) t ≥ (cid:12)(cid:12) max x ∈ Λ L η x ( t ) > R (cid:9) ,τ condL ( ρ ) := inf (cid:8) t ≥ (cid:12)(cid:12) max x ∈ Λ L η x ( t ) ≤ R (cid:9) , (123)which depend on the initial configuration as well as the time evolution. Due tothe effective one-dimensional random walk picture mentioned above, the quasi-stationary expectations of these random variables are determined by the rate func-tions at the locations of local minima and maxima. These are I ρ ( ρ ) = s can ( ρ ) − s fluid ( ρ ) (min.) I ρ ( ρ − a ) = s can ( ρ ) − s fluid ( ρ − a ) + a log c (max.) I ρ ( ρ c ) = s can ( ρ ) − s fluid ( ρ c ) + a log c c + ( ρ − ρ c ) log c (min.) , (124)where the last two are only defined for ρ > ρ c + a (cf. Figure 4). Note that I ρ ( ρ ) = 0 for ρ ≤ ρ c , whereas I ρ ( ρ c ) = 0 for ρ ≥ ρ c . For ρ > ρ c + a we thenhave ξ fluid ( ρ ) := lim L →∞ L log (cid:28) τ fluidL ( ρ ) (cid:29) π L,N ( . | X L,N ) ,e L t = I ρ ( ρ − a ) − I ρ ( ρ ) == s fluid ( ρ ) − s fluid ( ρ − a ) + a log c ,ξ cond ( ρ ) := lim L →∞ L log (cid:28) τ condL ( ρ ) (cid:29) π L,N ( . | X L,N ) ,e L t = I ρ ( ρ − a ) − I ρ ( ρ c ) == s fluid ( ρ c ) − s fluid ( ρ − a ) + ( ρ c + a − ρ ) log c (125)where (cid:28) .. (cid:29) π L,N ( . | X L,N ) ,e L t denotes the average with respect to a quasi-stationaryinitial distribution and the time evolution given by the generator L (4). Note thatfor ρ < ρ c + a , ξ fluid ( ρ ) = ∞ and ξ cond ( ρ ) is not defined, since the condensedphase is not stable. The asymptotic behaviour as ρ → ∞ is given by ξ fluid ( ρ ) (cid:39) log 1 + ρ ρ − a → ξ cond ( ρ ) (cid:39) ρ log c c − log(1 + ρ − a ) + const. → ∞ . (126)For all a > we have ξ fluid ( ρ c + a ) > ξ cond ( ρ c + a ) and ξ fluid ( ρ trans ) = ξ cond ( ρ trans ) , as expected. This behaviour is illustrated in Figure 5 for some spe-cific values of the parameters. The predictions are in very good agreement with26 œ ´´œ Ξ cond H Ρ L Ξ fluid H Ρ L Ρ c + a 2 Ρ trans ¥ ΡΞ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰œ œ œ œ œ œ œ œ fluidcondensed40 60 80 100 12010 L ` Τ L H Ρ t r a n s L p ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰‰ + + + + + ++ + + + + + + ++ ++ ++ ++ œ œ œœ œœ œ œœ œ œ œ œ œ œœ œ œœ œ œ D D D DDD D D DDDD D DD D DDDD D L = H ´ L H + L H œ L H D L Exp H - t L P H Τ L (cid:144) ` Τ L p ³ t L Figure 5:
Life-times of fluid and condensed phase for c = 2 , c = 1 , a = 0 . .Top: The exponential rate ξ ( ρ ) of the life-time as a function of the density. × and (cid:5) denote Monte Carlo data, errors are of the size of the symbols. Bottom left: Expectedlife-times as used in (125) in a logarithmic plot as a function of L for ρ = ρ trans . Bottomright: Tail distribution of the normalized lifetimes τ condL ( ρ trans ) / (cid:28) τ condL ( ρ trans ) (cid:29) ,compared with the tail of an Exp (1) random variable. ρ = ρ trans we see that the expected lifetimes for the condensedphase are larger than for the fluid phase, which is in accordance with Theorem 2.There appears to be a polynomial correction in the condensed phase, but the dataare not good enough to measure the power in L .Note that the last part of the derivation in this section is not rigorous, sincestrictly speaking (cid:0) Σ bgL ( η ( t )) (cid:1) t ≥ is not a Markov process. Still one could use apotential theoretic approach analogous to [3], to show rigorously that the aver-age life times of both phases are exponential in L . However, getting the righttimescale with this approach would require quite some technical effort. Besidesthe exponential growth rate of the life times with the system size L , simulationsalso indicate that the distribution of the lifetimes is actually exponential, as canbe seen in Figure 5 on the bottom right. This is to be expected, since the systemeffectively jumps between the two metastable phases in a Markovian way. In this section we consider the case where the jump rates depend on the number ofparticles in the system rather than the lattice size, which is also the case in somemodels for granular clustering [23, 4], one of our main motivations for this study.We modify our original model (2), g R ( k ) = (cid:26) c , k ≤ Rc , k > R for k ≥ , g (0) = 0 , (127)where R is now a function of the number of particles Σ L ( η ) . For simplicity weconcentrate on the specific choice R = a Σ L ( η ) with a ∈ [0 , , since a ≥ isnot interesting for this model. So in principle, the jump rates do not only dependon the local occupation number but on the global configuration. But restricted toa subset X L,N with fixed particle number Σ L ( η ) = N , R is just a parameter, theprocess is well defined and standard results on stationary measures apply. There-fore the canonical measures are well defined as in (28). In particular, Theorems1 to 4 still hold and the proofs apply directly, where a should be replaced by aρ , since now R/L → ρ a . So analogous to (35) the transition density ρ trans isdetermined by the relation a = (cid:16) s fluid ( ρ c ) − ( ρ − ρ c ) log c − s fluid ( ρ ) (cid:17)(cid:46)(cid:16) ρ log c c (cid:17) , (128)and s fluid is given as in (24). The canonical entropy density s can is still given by(33), but the contribution of the condensate, which determines the behaviour forlarge ρ , is now given by s cond ( ρ, ρ c ) = − ρ (cid:16) a log c c + log c (cid:17) + ρ c log c . (129)28his leads to s can ( ρ ) = (cid:40) s fluid ( ρ ) , ρ ≤ ρ trans s fluid ( ρ c ) − ρ (cid:16) a log c c + log c (cid:17) + ρ c log c , ρ > ρ trans . (130)To study the equivalence of ensembles, one has to define the grand-canonicalmeasures. This is not as straightforward as in (8), since the number of particles Σ L ( η ) and thus R is now a random variable. However, we know that the set of allstationary measures is convex, and the extremal points are the canonical measures(see e.g. [22] or [16]). So the grand-canonical measures can be defined as convexcombinations of canonical measures, ν Lφ,R ( η ) = (cid:89) x ∈ Λ L w LR ( η x ) φ η x (cid:46) ∞ (cid:88) N =0 φ N Z L,N . (131)If the weights w R ( k ) depended only on the system size L , this would be equiv-alent to (8), but here the measures are obviously not of product form since theweights depend on the total number of particles through R = a Σ L ( η ) . Also thenormalizing partition function Z R ( φ ) := ∞ (cid:88) N =0 φ N Z L,N (132)does not factorize, since now Z L,N = w LaN ( X L,N ) . Nevertheless we can definethe pressure p gcan ( φ ) := lim L →∞ L log ∞ (cid:88) N =0 φ N Z L,N , (133)and by a saddle point argument analogous to the proof of Theorem 1 this is welldefined and given by p gcan ( φ ) = sup ρ ≥ (cid:0) ρ log φ + s can ( ρ ) (cid:1) , (134)the Legendre transform of the negative canonical entropy density (130). With(129) we have for ρ > ρ trans ρ log φ + s can ( ρ ) = ρ (cid:16) log φ − a log c c − log c (cid:17) + ρ c log c , (135)which, analogously to (25), implies p gcan ( φ ) = (cid:26) p fluid ( φ ) , φ < φ c ( a ) ∞ , φ ≥ φ c ( a ) . (136)29 fluid p gcan a = c = c = c Φ c H a L c ¥ Φ p ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ œœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœ c = c =
1, a = s fluid H Ρ L s gcan H Ρ L s flluid H Ρ c L + s cond H Ρ , Ρ c L Ρ meta Ρ c H a L Ρ trans Ρ s can Figure 6:
Pressure and entropies for a = 0 . , c = 2 and c = 1 as given in (136), (130)and (139). Data points are calculated numerically according to (62) with L = 100 ( × ) ,
200 (+) ,
400 ( (cid:51) ) , and show good agreement with the theoretical predictions for thethermodynamic limit. The difference is that now the pressure is finite up to φ c ( a ) := c (cid:16) c c (cid:17) a ≥ c , (137)which is strictly bigger than the value c in (25) for all a ∈ (0 , . Note that for φ = φ c ( a ) the saddle point argument (134) does not apply and (133) diverges,so p gcan (cid:0) φ c ( a ) (cid:1) = ∞ , see Figure 6 left. As a consequence of (136), the criticaldensity defined as in (14) is now a -dependent and given by ρ c ( a ) := ρ ∞ (cid:0) φ c ( a ) (cid:1) = c − a c − a − c − a , (138)where ρ ∞ is still given by (12). So analogous to (26), the grand-canonical entropydensity is given by the negative Legendre transform of (136), s gcan ( ρ ) = (cid:26) s fluid ( ρ ) , ρ ≤ ρ c ( a ) s fluid ( ρ c ( a )) − ( ρ − ρ c ( a )) log φ c ( a ) , ρ > ρ c ( a ) . (139)By definition, this is again the concave hull of s can ( ρ ) , as can be seen in Figure6, right. The canonical entropy density is calculated numerically using (62) fordifferent values of L and N , and as before the results agree very well with thepredictions.As in the original model, the case a = 0 leads to a continuous phase transitionand this line of the phase diagram is identical to Figure 1. But for all a ∈ (0 , , ρ c ( a ) > ρ c (0) , which is the value in (14) for the original model. So the phase re-gion F ( E ) in the phase diagram is larger than in the original model (see Figure 7).To complete the phase diagram, we have to derive the analogue of the transitionline ρ c + a , which we call ρ meta in the following. This is defined by the emergenceof a metastable condensed phase, characterized by a second local maximum of the30 (cid:144) CC (cid:144) F F H E L Ρ meta H a L Ρ trans H a L Ρ c H a L Ρ c H L Ρ Figure 7:
Stationary phase diagram of the process (127) for c = 2 , c = 1 . The phases F ( E ) , F/C and
C/F are defined in Section 2, F ( E ) and F/C overlap (shaded region)and phase F is empty. rate function I ρ ( ρ bg ) in Theorem 5. The proof of this theorem makes use of thegrand-canonical measures, and since these are now of different form, it does notapply directly. However, with some effort the proof can be written purely in termsof canonical measures (not shown here), and so the result (105) still applies, ofcourse with a replaced by ρ a . An analysis similar to Section 5 reveals that therate function has an additional local minimum I ρ (cid:0) ρ c (0) (cid:1) for ρ > ρ meta = ρ c (0)1 − a . (140)So there exists a metastable condensed phase with background density ρ c (0) ,which is still the same as in the previous model, independent of a . This is tobe expected, since the outflow of the condensate site has to match the backgroundcurrent. A simple heuristic argument along these lines provides a general frame-work to understand the transition, and is presented in detail in [18]. Note that incomparison with (138), ρ meta ( a ) ≤ ρ c ( a ) for all a ∈ [0 , , (141)with equality if and only if a = 0 . This follows immediately from the elementaryinequality x − a − ≤ (1 − a )( x − . So the phase regions F ( E ) and F/C asdefined in Section 2 overlap (see shaded region in Figure 7), and the region F isempty. In contrast to our previous model, the equivalence of ensembles still holdsin the presence of a metastable condensed phase. In the following we discuss differences in the condensation transition betweenzero-range processes with and without size-dependence in the jump rates. To31implify matters we concentrate on the rates (2) for L -dependent jump rates, butthe features we discuss should hold in general. • Without L -dependence the condensation transition in zero-range processesis continuous, i.e. the background density ρ bg is a continuous function of thetotal particle density ρ . For model (2) this is only true if a = 0 , for a > thebackground density ρ bg = ρ c < ρ trans is smaller than the transition densityand the transition is discontinuous. • If the jump rates do not depend on L , the equivalence of ensembles holdsfor all densities, and for ρ ≥ ρ c the entropy density is linear in ρ which isoften characterized as partial equivalence of ensembles [8, 31]. The reasonis that the contribution of the condensate to the entropy density vanishes as L → ∞ . In model (2) this contribution does not vanish, cf. Theorem 1, andtherefore we have only equivalence of ensembles for ρ ≤ ρ c and nonequiv-alence for larger densities. As a consequence of this, the canonical entropydensity is non-concave, whereas it is concave in case of no L -dependence. • Another striking feature of model (2) is that it exhibits ergodicity breaking,i.e. for ρ > ρ c + a there are two phases, fluid and condensed, with life-times exponential in L , one of which is metastable depending on the density.Without L -dependence in the jump rates this does not occur, and for alldensities there is only one stable phase, either fluid for ρ ≤ ρ c or condensedfor ρ > ρ c .So far a discontinuous transition in a zero-range process has only been observedheuristically in a two-species system where the stationary state is not known [15].The above features only concern the stationary measure, and for systems without L -dependence they have been shown rigorously in a general context [16]. In thefollowing we comment on further differences regarding equilibration and station-ary dynamics, which have been studied only heuristically so far. • If we prepare a system without L -dependence in a homogeneous distribu-tion with density ρ > ρ c it exhibits coarsening [17, 13]. Initially, clustersform all over the lattice, and as time progresses the larger cluster sites gainparticles on the expense of the smaller cluster, leading to a self-similar timeevolution. The driving force for this behaviour is the fact that there is nostable fluid phase with density ρ > ρ c . This is not the case in model (2),which does not exhibit coarsening for that reason. Instead, it takes a timeof order e ξ fluid L before the condensate appears. • In a similar setting metastability has been reported as a precursor of thecoarsening regime, i.e. before coarsening to a single condensate sets in[20]. Unlike in the present case, heuristic theoretical analysis supported by32onte-Carlo simulation shows that the life time of these metastable config-urations does not grow exponentially with system size. • For systems without L -dependence, in the condensed phase the distributionof the homogeneous background has a sub-exponential tail [16]. In con-nection to this, the stationary time scale for movement of the condensatelocation (once a single condensate has build up) is also sub-exponential in L , as was found heuristically in [14] in case of a power law. For model(2) the background distribution is just ν c , ∞ ( k ) ∼ ( c /c ) − k (see Theorem4), which has an exponential tail. Therefore condensates can move only bydissolving completely and, after a time of order e ξ fluid L in the coexistingfluid phase, forming on a different site. So the time scale for the stationarymotion of a condensate is exponential in the system size.The long time it takes to form a condensate in the present model is observed inMonte Carlo simulations and is explained heuristically by a random walk picturein Section 5. The time it takes for the transition between the phases depends onthe specific model as well as the definition of the phases. In any case its order issubexponential in the system size, and for the model (2) it is actually of order L .Moreover, if ρ > ρ c + na for n ≥ also more than one condensate is possible.But as can be seen in the proof of Theorem 1, the contribution to the partitionfunction of such a configuration is negligible. Therefore one typically observesonly one condensate, which is a common feature with the stationary behaviourof a system without L -dependence, although both cases have very different lifetimes. In [27] a hydrodynamic theory is developed for the time evolution underEulerian scaling above the condensation threshold. This leads to a generic picturefor the evolution of a space-dependent initial density profile with total supercriti-cal density in systems with rates that do not depend on L . It would be interestingto study this problem in the present model.Finally, we would also like to stress an intriguing difference to the usual theoryof first order phase transitions in statistical mechanics. In systems with finite localstate space or with bounded Hamiltonians, such as spin systems (Ising model)or exclusion models, the pressure p is defined for all fugacities φ ≥ , and afirst order phase transition is a result of the pressure being non-analytic (see e.g.[25, 35]). In the model we studied here, the pressure (25) is defined only for φ Acknowledgments Both authors would like to thank Pablo Ferrari for inspiring discussions and use-ful comments about essential parts of the manuscript, and for an invitation toNUMEC at the University of S˜ao Paulo, which was supported by FAPESP. Theauthors are also grateful for the hospitality of the Isaac Newton Institute in Cam-bridge, where this work was initiated during the programme “Principles of theDynamics of Non-Equilibrium Systems”. Appendix A.1 Proof of Proposition 2 For ρ < ρ c (22) follows by standard results and for ρ > ρ c we have as L → ∞ ν φ R ( ρ ) ,R ( η Lx > R ) (cid:39) (cid:114) ρ − ρ c z ∞ ( c ) (cid:16) c c (cid:17) R/ . (142)34herefore if we define the truncated occupation numbers ˆ η Lx = η Lx ∧ R , we have ν Lφ R ( ρ ) ,R (cid:16) L (cid:88) x ∈ Λ L ( η Lx − ˆ η Lx ) (cid:54) = 0 (cid:17) = ν Lφ R ( ρ ) ,R (cid:0) at least one η Lx > R (cid:1) == 1 − (cid:16) − ν φ R ( ρ ) ,R ( η x > R ) (cid:17) L ≤ CL (cid:16) c c (cid:17) R/ . (143)With R (cid:29) log L this bound is summable and the Borel-Cantelli Lemma impliesthat L (cid:88) x ∈ Λ L η Lx − L (cid:88) x ∈ Λ L ˆ η Lx → a.s. as L → ∞ . (144)Moreover (cid:104) ˆ η x (cid:105) = ρ c + O (cid:0) ( c c ) R/ (cid:1) and V ar (ˆ η x ) ≤ c c ( c − c ) + O (cid:0) R ( c c ) R/ (cid:1) andtherefore by the usual strong law we have (cid:80) x ∈ Λ L ˆ η Lx → ρ c a.s. .Taken together, this implies (22) for ρ > ρ c , and ρ = ρ c works analogously withthe power R/ replaced by R/ . A.2 Proof of Lemma 1 Each configuration in X L,N \ X L,N has at least one site with more than R particlesand we denote the number of such sites by E ( η ) := (cid:88) x ∈ Λ L η x >R ( η ) . (145)Note that for η ∈ X L,N \ X L,N we have ≤ E ( η ) ≤ M = (cid:100) N/R (cid:101) , (146)where M is as defined in (31). For each configuration we define S ( η ) := (cid:0) η x ∧ R (cid:12)(cid:12) x ∈ Λ L (cid:1) ∪ (cid:0) η x − R (cid:12)(cid:12) x ∈ Λ L , η x >R (cid:1) ∈ X L + E ( η ) ,N . (147)If E (cid:0) S ( η ) (cid:1) > , we have to repeat this mapping at most M times such that ¯ η := S M ( η ) ∈ X L + l ( η ) ,N , (148)where l ( η ) ≤ M denotes the total number of extra coordinates. For l ( η ) < M wecan identify ¯ η by a configuration in X L + M,N , by setting all remaining coordinatesequal to zero. By this construction it is clear that for each η ∈ X L,N \ X L,N thereexists a unique ¯ η ∈ X L + M,N , i.e. (cid:12)(cid:12) X L,N \ X L,N (cid:12)(cid:12) = (cid:12)(cid:12) X L,N (cid:12)(cid:12) − (cid:12)(cid:12) X L,N (cid:12)(cid:12) ≤ (cid:12)(cid:12) X L + M,N (cid:12)(cid:12) . (149)35urther, each ¯ η has the special property that at least l ( η ) sites contain exactly R particles, and there are only L sites whose occupation number can be less or equalthan that. Therefore we can improve the above estimate as (cid:12)(cid:12) X L,N (cid:12)(cid:12) − (cid:12)(cid:12) X L,N (cid:12)(cid:12) ≤ (cid:18) L + MM (cid:19) (cid:12)(cid:12) X L,N − R (cid:12)(cid:12) ≤ (cid:18) L + MM (cid:19) (cid:12)(cid:12) X L,N (cid:12)(cid:12) ( L − M ) R , (150)where the combinatorial factor counts the number of positions of sites with R particles. We also used the fact that for all k = 1 , . . . , R (cid:12)(cid:12) X L,N − k +1 (cid:12)(cid:12) ≥ ( L − M ) (cid:12)(cid:12) X L,N − k (cid:12)(cid:12) , (151)since there are at least L − M positions to put an additional particle withoutviolating the constraint η x ≤ R for all x . Together with the obvious fact that (cid:12)(cid:12) X L,N (cid:12)(cid:12) ≤ (cid:12)(cid:12) X L,N (cid:12)(cid:12) , this proves the first statement of the lemma, i.e. 11 + (cid:0) L + MM (cid:1)(cid:14) ( L − M ) R (cid:12)(cid:12) X L,N (cid:12)(cid:12) ≤ (cid:12)(cid:12) X L,N (cid:12)(cid:12) ≤ (cid:12)(cid:12) X L,N (cid:12)(cid:12) . (152)With Stirling’s formula we get lim L →∞ L log (cid:18) L + ML (cid:19) = lim L →∞ (cid:16)(cid:0) ML (cid:1) log (cid:0) ML (cid:1) − ML log ML (cid:17) = 0 . (153)since M/L → as L → ∞ . Therefore L log 11 + (cid:0) L + MM (cid:1)(cid:14) ( L − M ) R → (154)and (152) certainly includes the second statement of the lemma. 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