1.6. Hipótesis
2.2.8. Nuevas bases teóricas que interactúan con la Recepción Comunicacional
Results on stationary product measures discussed above can be applied in more general models, but in this thesis we are only interested in closed finite systems and their scaling limits. The total number of particles in such systems is conserved and there is no restriction on the number of particles per site. We further assume the weightsw(n)>0 are sub-exponential in the sense that
w(n+ 1) w(n) =
v(n)
u(n 1) !1, as n! 1. (2.18) If the limit is di↵erent from 1, it is equal to c and by rescaling the rates we can always fix c = 1. The only exception is that w has super-exponential decay, then
c =1 and there is no condensation, so we do not consider this case.
With product measures, the canonical measures⇡⇤,N on irreducible sub- sets X⇤,N have explicit formulae. Since the number of particles is conserved, the conditioned measures ⌫⇤(d⌘|X⇤,N) are also stationary, and since the process is er- godic on X⇤,N, these conditional measures are equal to ⇡⇤,N and independent of the fugacity . Taking = 1 for simplicity, we then have
⇡⇤,N[d⌘] =⌫1⇤[d⌘|X⇤,N] = 1 Z⇤,N Y x2⇤ w(⌘x)d⌘, (2.19)
whereZ⇤,N =⌫1⇤[X⇤,N] is the normalisation.
The set of all stationary measures of processes with dynamics (2.11) is a convex subset of measures on X⇤ (see, e.g., [43, Proposition I.1.8]). On a finite lattice⇤, the canonical measures ⇡⇤,N are the extreme points for this set, and the grand-canonical product measures⌫⇤ can be written as a convex combination
⌫⇤= X
N2N
⌫⇤[X⇤,N]⇡⇤,N,
which are not extremal. On finite lattices, it can be shown that there are no other extremal measures than the canonical ones, and therefore the full set of stationary distributions is given by their convex hull. However, on infinite lattice the problem is more complicated. In spatially homogeneous systems the grand-canonical mea- sure are extremal, but there may be more non-homogeneous extremal measures, which are analogous to blocking measures for exclusion process (see, e.g., [43, Chap- ter.VIII]).
2.2. Condensation and equivalence of ensembles 19 In the thermodynamic limit
L=|⇤|, N ! 1, such that N/L!⇢ 0,
the grand-canonical measures (with simple product structures) are usually expected to provide a good approximation to the sequence of canonical measures. In statistical mechanics this is called the equivalence of ensembles and one convenient way of quantifying the distance between two distributions is relative entropy. We do not use this technique directly in this thesis, and for its application in zero-range processes see [12] and more general discussion in [77]. Notice, for inclusion processes, we need to consider an adapted parameter-dependent thermodynamic limit to see the condensation where the equivalence of ensembles technique is not valid, which is discussed in details in Section 2.3.3.
Recall that the average particle density R( ) (2.15) is strictly increasing in withR(0) = 0, we can define its critical limit as the following.
Definition 2.9. The critical density⇢c 2[0,1] is defined as
⇢c := lim
% c
R( ), withR( ) defined in (2.15), (2.20) and the system exhibits condensation if⇢c <1.
It is clear that c < 1 is a necessary condition for condensation, see, e.g., [73, Lemma II.3.3] for a proof in a special case. If the stationary weights had super-exponential decay, for example independent random walkers where⌘xare i.i.d Poisson random variables, we have c =1and⇢c=1and there is no condensation. This general connection between condensation and critical density works well in the thermodynamic limit for both homogeneous and inhomogeneous systems. It also works for systems with size-dependent parameters, e.g. inclusion processes. For other scaling limits such as N ! 1 on a fixed lattice ⇤, the above definition has to be adapted, see, e.g., [54]. Here we only review results on the connections between condensation, stationary currents as well as equivalence of ensembles for homogeneous systems. For results on inhomogeneous or more generalised systems, see [42, Section 4] and references therein.
Recall marginals (2.13) ⌫ [⌘x =n] = 1 z( )w(n) n, with z( ) =X1 n=0 w(n) n,
2.2. Condensation and equivalence of ensembles 20 then the critical density (2.20) is
⇢c =R(1) = 1 z(1) 1 X n=0 nw(n)2(0,1].
It is easy to showz(1) =1implies⇢c =1(see, e.g., [73, Lemma II.3.3]). Therefore, the system exhibits condensation with ⇢c < 1 if and only if nw(n) is summable, i.e. w(n) must decay fast enough like a sub-exponential distribution, the measures are then defined for all 2 [0,1] = D and the range of densities is given by R(D ) = [0,⇢c]. Therefore, for ⇢c <1 the range of densities attainable by grand- canonical measures is a strict subset of [0,1). For typical stationary configurations under canonical distributions ⇡⇤,N withN/L=⇢>⇢c, the system phase separates into a condensed and a fluid phase. It can be shown (see, e.g., a review in [42, Section 3]) that the bulk phase is distributed as the product measure ⌫1 at the
critical density⇢c, and that the condensed phase containing a macroscopic amount of order (⇢ ⇢c)L particles concentrates on a vanishing fraction of the lattice. This result is analogous to classical results on phase separation in the Ising model with spin-exchange (Kawasaki) dynamics (see, e.g., [43, Chapter 4]), where the main di↵erence is that the models we discussed above have unbounded local state spaces and the condensed phase contributes only sub-extensively to the total entropy (or free energy) of the system. For the special cases where w(n) have power law or stretched exponential tails, a series of papers [12, 26, 54, 55, 56] have shown that the condensed phase occupies a single site on the lattice.
The stationary current in a general lattice gas model is defined as the expected net number of particles crossing a bond in a (specified) positive direction per unit of time. The full current depends on the lattice geometry and vanishes for reversible systems, in which case one has to consider the di↵usivity. The main interest for us will be the average jump rate of a particle per connecting bond. In the rest of this chapter we will simply call this thecurrentfor ease of presentation, even though in symmetric systems it is rather the activity. Since ⌫⇤ is a homogeneous product measure, the grand canonical current can be defined for an arbitrary pair of sitesx6=y2⇤ as
jgc :=⌫⇤(u(⌘x)v(⌘y)) =⌫1(u)⌫1(v) = (⌫1(v))2, (2.21) where for the last representation we have used the following recursive property of
2.2. Condensation and equivalence of ensembles 21 the stationary measures
¯
⌫ (n+ 1)¯⌫ (k 1)u(n+ 1)v(k 1) = ¯⌫ (n)¯⌫ (k)u(k)v(n), 8n 0, k 1, which is implied by the form (2.13) and (2.14) of the marginals. Similarly, the canonical currentcan be defined as
j⇤,N :=⇡⇤,N(u(⌘x)v(⌘y)), (2.22) which is still independent ofx 6=y 2⇤ since the canonical measures are permuta- tion invariant in homogeneous systems. The thermodynamic limit of the canonical current
j(⇢) = lim
L,N!1j⇤,N for all ⇢ 0,
is usually named thecurrent-density relationor thefundamental diagramof the process.
To compare both currents, it is often convenient to also view the grand- canonical one as a function of the density using the one-to-one relation⇢=R( ) in (2.21), and in this case we writejgc(⇢) which exists only for densities in [0,⇢c].
The following theorem shows the weak convergence of the canonical measure to the grand-canonical measures on finite lattices. It was first published in [12] in terms of relative entropy and provided first rigorous results on the equivalence of ensembles.
Theorem 2.4. 1. We have weak convergence of the canonical measures to the grand-canonical measures in the sense that, forf 2Cb(X),
⇡⇤,N(f)! 8 < : ⌫ (f) with R( ) =⇢ for ⇢⇢c ⌫ c(f) for ⇢>⇢c asL, N ! 1 and N/L!⇢>0.
2. For all ⇢ > ⇢c we also have weak law of large numbers: denote ML := maxx2⇤⌘x, then ML (⇢ ⇢c)L ! 1 in distribution, i.e. 8✏>0, ⇡⇤,N h ML (⇢ ⇢c)L 1 >✏ i !0 as L, N ! 1, N/L!⇢>⇢c . Above theorem has been generalised in [42] for subcritical and supercritical cases. This result implies that the canonical measures converge locally to the grand- canonical measures for⇢<⇢c, and for⇢>⇢cthe canonical measure start to converge
2.3. Models 22