The strong ergodicity for the stochastic dynamics of unbounded lattice spin systems^{1}

Nobuo Yoshida^{2}

The purpose of this talk is to present some results on the ergodic property for unbounded spin systems on the lattice. The talk will be organized as follows:

1. Basic definitions Gibbs state (equilibrium state) Associated stochastic dynamics (relaxation to the equilibrium described by an SDE)

2. Results Thm. 2.1: ”High temparature” ⇒ Log-Sobolev inequality (LSI) Thm. 2.2: LSI ⇒ ”The relaxation occurs very fast.”

3. Outline of the proofs

1 Basic definitions

The configuration spaces: The configuration spaces are defined as follows;

R^{Λ} = {σ = (σ_{x})_{x∈Λ} ; σ_{x} ∈ R}, Λ ⊂ Z^{d},
Ω = R^{Z^{d}},
S′ = ⋃_{p≥1}{σ ∈ Ω; lim_{|x|↗∞} |x|^{−p}|σ_{x}| = 0}.

A configuration in S′ is called a tempared configuration. Gibbs states. We consider a spin system on Z^{d} with the following Hamiltonian; for Λ ⊂⊂ Z^{d} and its boundary condition ω ∈ R^{Λ^{c}}, we define a function H^{Λ,ω} : R^{Λ} ⟶ R by

H^{Λ,ω}(σ) = −\frac{1}{2} ∑_{{x,y∈Λ \atop ∥x−y∥_{1}=1}} Jσ_{x}σ_{y} − ∑_{{x∈Λ,y∉Λ \atop ∥x−y∥_{1}=1}} Jσ_{x}ω_{y} + ∑_{x∈Λ} U(σ_{x}),  (1.1)

where J ∈ (0, ∞) and U : R → R is a function on which some reasonable assumptions are imposed. To make the exposition simpler, let us just consider the following example (ϕ^{4}- potential):

U(s) = s^{4} − bs^{2}, b ∈ R  (1.2)

The important aspect of U in (1.2) is that arbitrarily deep double wells are allowed so that the phase transition may occur for d ≥ 2 and large enough J.

The finite volume Gibbs state is then defined by;

E^{Λ,ω}(dσ) = \frac{exp −H^{Λ,ω}(σ)}{normalization} ∏_{x∈Λ} dσ_{x}  (1.3)

1Notes for a talk on September 11, 1998 at Kyushu University

2Division of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan, email:nobuo@kusm.kyoto-u.ac.jp, URL:http://www.kusm.kyoto-u.ac.jp/ nobuo

A Borel probability measure μ on R^{Z^{d}} is called a tempared Gibbs state if the following two conditions are satisfied;

DLR-equation; ∫ μ(R^{Λ} × dω_{Λ^{c}})E^{Λ,ω}(dσ_{Λ}) = μ(dσ_{Λ} × dω_{Λ^{c}}), ∀Λ ⊂⊂ Z^{d}.

tempared; ∃p > 0, lim_{|x|↗∞} |x|^{−p}μ(|σ_{x}|) = 0.

The following are known;

• G_{t} := {tempared Gibbs state} ≠ ϕ.

• J << 1 ⇒ ♯G_{t} = 1.

• d ≥ 2, J >> 1 ⇒ ♯G_{t} ≠ 1 for some b > 0 (phase transition).

The stochastic dynamics. For a set Λ ⊂ Z^{d} (Λ ⊂⊂ Z^{d} or Λ = Z^{d}) and ω ∈ Ω, consider the following stochastic differential equation (SDE) for the unknown process σ_{t}^{Λ,ω} = (σ_{t,x}^{Λ,ω})_{x∈Z^{d}};

σ_{t,x}^{Λ,ω} = \{ \begin{matrix}ω_{x} + θ_{t,x} − \frac{1}{2}∫ _{0}^{t}ds\frac{∂H^{Λ,ω}}{∂σ_{x}} (σ_{s}^{Λ,ω}) & if x ∈ Λ, \\ ω_{x} & if x ∉ Λ, \end{matrix}  (1.4)

wher (θ_{x})_{x∈Z^{d}} are independent standard Brownian motions on a probability space (Θ, P).

• For ω ∈ S′ the above SDE has a unique solution.

• For Λ ⊂⊂ Z^{d}, σ_{t}^{Λ,ω} is reversible with respect to E^{Λ,ω}.

• For Λ = Z^{d}, σ_{t}^{ω} := σ_{t}^{Z^{d},ω} is reversible with respect to any tempared Gibbs state.

2 Results

We introduce a function space C as follows;

C = \{f : Ω ⟶ R | \begin{matrix}There is Λ ⊂⊂ Z^{d} such that \\ f depends only on (σ_{x})_{x∈Λ}, \\ f ∈ C^{1}(R^{Λ} → R), \\ \frac{∂}{∂σ_{x}} f are bounded for ∀x ∈ Λ. \end{matrix} \} .  (2.1)

Theorem 2.1 ([Y97]) Let d ≥ 1 and b ∈ R be arbitrary. Then, for sufficiently small J > 0, there is γ ∈ (0, ∞) such that

E^{Λ,ω} (f^{2} log \frac{f^{2}}{E^{Λ,ω}(f^{2})}) ≤ γE^{Λ,ω} (∑_{x∈Λ} |\frac{∂f}{∂σ_{x}} |^{2}) ,  (2.2)

for all Λ ⊂⊂ Z^{d}, ω ∈ R^{Λ^{c}} and f ∈ C. In particular, the unique tempared Gibbs state μ satisfies

μ (f^{2} log \frac{f^{2}}{μ(f^{2})}) ≤ γμ ( ∑_{x∈Z^{d}} |\frac{∂f}{∂σ_{x}} |^{2})  (2.3)

for all f ∈ C.

Remark 2.1 LSI,s for unbounded spin models are studied in some other recent papers;

• A direct application of the Bakry-Emery criterion ([BE85]) for LSI is discussed in [AKR95]. This is possible when b < 0.

• d = 1, ([Z96, Theorem 4.1]); In this case, the B-E criterion is no longer available in general. Instead, one takes advantage of a mixing condition which comes from the dimentionionality.

Remark 2.2 The assumptions for U imposed in [Y97] are mild enough to include examples like (1.2) which are often discussed in physical litratures. However, from the mathematical point of view, these assupmtions are not minimal to prove Theorem 2.1 with. In a very recent paper [BH98], T. Bodineau and B. Helffer succeeded in reducing the assumptions for U to the presumably the minimal ones. Their proof uses the result of B. Helffer [He97] as well as the argument in [Y97].

Theorem 2.2 ([Y98]) Suppose that (2.2) holds and that C_{1} > γ. Then, there exist coefficients {C_{2}(f, ω) ∈ (0, ∞) | f ∈ C, ω ∈ S′} such that the following holds for any t ≥ 1, Λ ⊂⊂ Z^{d}, f ∈ C and ω ∈ S′;

|Pf(σ_{t}^{Λ,ω}) − E^{Λ,ω}f| ≤ C_{2}(f, ω) exp (−t/C_{1}) ,  (2.4)
|Pf(σ_{t}^{ω}) − μf| ≤ C_{2}(f, ω) exp (−t/C_{1}) .  (2.5)

where μ is the unique tempared Gibbs state.

Remark 2.3 Results of type (2.5) can also be found in [Z96].

3 Outline of the proofs

Thm. 2.2; We use the relative entropy as follows;

|Pf(σ_{t}^{Λ,ω}) − E^{Λ,ω}f| ≈ |Pf(σ_{t}^{Λ(t),ω}) − E^{Λ(t),ω}f|
∼^{<} C(f)H(P(σ_{t}^{Λ(t),ω} ∈ ·)|E^{Λ(t),ω})  (3.1)

where Λ(t) is a certain (time-dependent) subset of Λ and H(·|·) denotes the relative entropy. We estimate (3.1) by using Girsanov formula and show that it decays exponentially.

Thm. 2.1;

J << 1 \overrightarrow{}_{1} a mixing condition (M) \overrightarrow{}_{2} spectral gap (SG)
(M)+(SG) \overrightarrow{}_{3} (LSI)

• \overrightarrow{}_{1}; The main part of this implication is proved by generalizing the argument (”surgery

construction”) in [DSh85]. Some techniques from [AKRT97, COPP78] are also used to handle the Vasserstein distance.

• \overrightarrow{}_{2} and \overrightarrow{}_{3}; Both of these implications are proved by generalizing the argument in

[LY93].

References

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[AKRT97] Albeverio, S., Kondratiev, Yu, G. and Röckner, M. and Tsikalenko, T. V.: Uniqueness of Gibbs states for quantum lattice systems, Prob.Th.Rel.Fields. 108,193–218, (1997) .

[BE85] Bakry, D. and Emery, M.: Diffusions hypercontractives, Séminaire de Probabilités XIX, Springer Lecture Notes in Math., 1123, 177–206, (1985).

[BH82] Bellissard, J. and Hϕegh-Krohn, R.: Compactness and maximal Gibbs state for random Gibbs fields on the lattice, Commun. Math. Phys. 84,297–327, (1982).

[BH98] Bodineau, T. and Helffer, B.: Log-Sobolev inequality for unbounded spin systems, preprint, (1998).

[COPP78] Cassandro, M., Olivieri, E., Pellegriotti, A. and Presutti, E.: Existence and uniqueness of DLR measures for unbounded spin systems, Z. Warsch. verw. Gebiete 41, 313–334, (1978).

[DSh85] Dobrushin, R. L. and Shlosman, S. : Constructive criterion for the uniqueness of Gibbs field, in “Statistical Physics and Dynamical Systems” ed. by J. Fritz, A. Jaffe and D. Szasz, Birkhäuser (1985).

[He97] Helffer, B. Remarks on the decay of correlations and Witten Laplacians III–Application to logarithmic Sobolev inequalities.

[LY93] Lu, S. L. and Yau, H. T.: Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics, Commun. Math. Phys. 156, 399–433, (1993).

[SZ92] Stroock, D. W. and Zegarlinski, B. : The logarithmic Sobolev inequality for discrete spin systems on the lattice, Commun. Math. Phys. 149, 175-193 (1992).

[Y97] Yoshida, N.: The log-Sobolev inequality for weakly coupled lattice fields, preprint (1997).

[Y98] Yoshida, N. :Application of log-Sobolev inequality to the stochastic dynamics of unbounded spin systems on the lattice, preprint, 1998.

[Z96] Zegarlinski, B. : The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice. Commun. Math. Phys. 175, 401–432, (1996).