Chapter 6 Condensation in the inclusion process
6.3 Fixed diffusion rate
Firstly we consider the case of fixed diffusion ratedL=dindependent of the system size.
In this case there is no condensation transition and we find that there is equivalence of ensemble at all densities.
6.3.1 Determining the relevant scale (aL)
First we find the relevant scaleaLthat describes the large deviations of the density under
the reference measure. This is the scale on which the canonical and grand canonical entropy densities are defined. The thermodynamic pressure on scaleaL is given by,
p(µ) = lim L→∞ 1 aL logνL eaLµSL = lim L→∞ dL aL log 1−edµ0 1−edµ0+aLL−1µ (6.23)
following (6.15). By applying L’Hˆopital’s rule we find the relevant scale aL at which
large deviations of the density under the reference measure decay; Case 1: aLL, p(µ) = 0 ifµ≤0 ∞ ifµ >0 (6.24) Case 2: aL=L, p(µ) = dlog 1−edµ0 1−edµ0+µ ifµ <−dµ0 ∞ ifµ≥ −dµ0 (6.25) Case 3: aLL, p(µ) = ¯ρµ (6.26)
It follows from the G¨artner-Ellis Theorem [34, 35], that large deviations of the density under the reference measure decay on the scale L, this is therefore the correct scale to derive non-trivial results on the equivalence of ensembles from the relative entropy method (c.f. Theorem 3.7). We therefore fix aL=Lfor the non size-dependent case.
Now the grand canonical measures as defined in Section 6.2.3 for chemical potentials inµ∈ Dp = (−∞,−dµ0) are given by,
νµL[η] = e LµSL(η)ν[η] νL(eLµSL) = Y x∈ΛL eµηxν L[ηx] z(µ) (6.27)
by, z(µ) =νL(eµη) = 1−edµ0 1−edµ0+µ d . (6.28)
The average density under the grand canonical measures is also independent of L for fixed µand is given by,
R(µ) =νµL(SL) =∂µp(µ) =
dedµ0+µ
1−edµ0+µ (6.29)
and we denote its inverse by µ(ρ) (the Lindex is not required in this case),
µ(ρ) = log ρ
d+ρ −dµ0 . (6.30)
So the Legendre-Fenchel transform of the grand canonical pressures is, p∗(ρ) =−sgcan(ρ) =ρµ(ρ)−p(µ(ρ))
=ρ(logρ−dµ0)−(d+ρ) log (d+ρ) +dlog
d 1−edµ0
, (6.31)
which is strictly convex on (0,∞).
The restricted pressures, see Definition 3.6, are equal to the thermodynamic pressure p on Dp by Corollary 3.9, so we have Dpm =Dp and
pm(µ) =p(µ) for all µ∈(−∞,−dµ0). (6.32)
Therefore the restricted grand canonical measures, see Definition 3.7, ¯
νµ,mL [η] =νµL[η|ML< mL], (6.33)
exist for all m ∈ (0,∞) and µ ∈ Dp. Also, by Corollary 3.9, for each restriction m
the restricted measures at chemical potential µ converge weakly to the grand canoni- cal measure at µ. This result is intuitively very clear, since the grand canonical single site marginals have exponential tails the maximum ofL independent samples from this distribution is of order logL, the grand canonical measures therefore concentrate (ex- ponentially) on configurations with maximum less than mLL.
6.3.2 Equivalence of ensembles
Since the thermodynamic pressure (6.25) is steep at its boundary, −dµ0, the Legendre-
Fenchel transform is strictly convex on the entire domain (0,∞). Equivalence of ensem- bles follows immediately from Theorem 3.7.
Proposition 6.1. The canonical measures conditioned on densityρLconverge in specific
relative entropy, on the scaleaL, to the grand canonical measure with chemical potential
chosen to fix the average density,
lim
L→∞
1
LH(π
L
ρ|νµ(ρ)L ) =sgcan(ρ)−scan(ρ) = 0 for allρ >0 . (6.34)
So πρL ←→L νµ(ρ)L , and we have convergence of expected values of bounded cylinder test
functions (see Definition 3.11).
Also the canonical current converges to the grand canonical current, given by (6.17), jLcan(ρL)→jgc(ρ) =ρ(ρ+d) asL→ ∞and ρL→ρ . (6.35)
Note that this does not simply follow from the equivalence, since the current in the inclusion process is an unbounded test function (this is not the case in the zero-range process). The on site occupation numbers are asymptotically independent and dis- tributed according to νµ(ρ)L . Also, by following the comment made after Theorem 3.7, the equivalence result can be strengthened to,
lim
L→∞
1 ˆ aL
H(πLρ|νµ(ρ)L ) = 0 for all ˆaLlogL . (6.36)
Although it is not required for finding the large deviations of the density, that give rise to equivalence of ensembles, we may still calculate the large deviations of the maximum. Following Section 3.4.4 the large deviations of the maximum can be expressed in terms of the restricted canonical entropy density and the condensate contribution. The condensed entropy contribution is given by,
scond(m) = lim L→∞
1
LlogνL[mLL] formL→mas in (3.41) (6.37)
= (dµ0)m . (6.38)
Since the reference measures have exponential tails we observe that the exponential ‘cost’ of a macroscopic maximum of sizemis linear inm. The entropy density of the restricted canonical measures is given by the Legendre-Fenchel transform of the grand canonical pressure, since the restricted pressures are equal to the thermodynamic pressure (6.25), and the Legendre-Fenchel transform of the pressure is strictly convex on its essential domain. That is, by Theorem 3.10, we have
scan,m(ρ) = lim L→∞ν
L[S
L=ρL|ML≤mL]
=−p∗(ρ) , (6.39)
wherep∗(ρ) is given explicitly in (6.31). By Applying Theorems 3.7 and 3.10 the large deviations of the joint maximum and density under the reference measure are described
by the rate function,
I(ρ, m) =−(scond(m) +sgcan(ρ−m))
=p∗(ρ−m)−(dµ0)m .
Proposition 6.2. As ρL →ρ and mL→ m, following (3.18)and (3.41), the canonical large deviations of the maximum are give by,
Iρ(m) : = lim L→∞
1
Lπ
L
ρ[ML=mL] =−sgcan(ρ−m)−scond(m) +sgcan(ρ)
=p∗(ρ−m)−µ0md−p∗(ρ) (6.40) = (ρ−m) log ρ−m d+ρ−m −ρlog ρ d+ρ +dlog d+ρ d+ρ−m