Diffusive long-time behavior of Kawasaki dynamics.

*(English)*Zbl 1083.60077The authors consider a semigroup associated with a Kawasaki dynamics in \(\mathbb Z^d\). The model is a lattice gas in \(\mathbb Z^d\) whose phase space is given by \(\Omega=S^{\mathbb Z^d}\), where \(S=\{0,1\}\). On the phase space a Hamiltonian \(H(\eta)\) is defined generated by a short range, translation invariant bounded interaction \(J(\eta)\). Under a suitable assumption on \(J\), called by the authors condition (USM), which is implied, except for a uniformity condition, by R. L. Dobrushin and S. B. Shlosman’s complete analyticity [J. Stat. Phys. 46, No. 5/6, 983–1014 (1987; Zbl 0683.60080)], it is possible to prove the existence of only one Gibbs state \(\mu\) associated with the interaction.

The dynamics on \(\Omega\) is given by a formal generator \(\mathcal L\) defined in the folloving way: \[ (\mathcal L F)(\sigma)=\sum_{<x,y>}c_{x,y}(\sigma)\nabla_{x,y}f, \] where the sum goes over the nearest neighbors and \(\nabla_{\cdot,\cdot}\) is the discreted gradient. The assumptions on the transition rates \(c_{\cdot,\cdot}(\cdot)\) of the process are: a) finite range, b) detailed balance, c) positivity and boundedness. Under all these assumptions the authors prove the \(L_2\)-convergence of the evolution. More precisely, let \(P_t\) be the semigroup generated by \(\mathcal L\), then for any \(\varepsilon\) and for all local functions \(f\) on \(\Omega\) there is a positive constant \(A(\varepsilon,f)\) such that \[ \mu \left [ (P_tf-\mu(f))^2\right]\leq A(\varepsilon,f)t^{-(d/2 -\varepsilon)}. \] This result is proved for \(d\geq 3\), because the cases \(d=1,2\) were studied by N. Cancrini and F. Martinelli [J. Math. Phys. 41, No. 3, 1391–1423 (2000; Zbl 0977.82031)].

The dynamics on \(\Omega\) is given by a formal generator \(\mathcal L\) defined in the folloving way: \[ (\mathcal L F)(\sigma)=\sum_{<x,y>}c_{x,y}(\sigma)\nabla_{x,y}f, \] where the sum goes over the nearest neighbors and \(\nabla_{\cdot,\cdot}\) is the discreted gradient. The assumptions on the transition rates \(c_{\cdot,\cdot}(\cdot)\) of the process are: a) finite range, b) detailed balance, c) positivity and boundedness. Under all these assumptions the authors prove the \(L_2\)-convergence of the evolution. More precisely, let \(P_t\) be the semigroup generated by \(\mathcal L\), then for any \(\varepsilon\) and for all local functions \(f\) on \(\Omega\) there is a positive constant \(A(\varepsilon,f)\) such that \[ \mu \left [ (P_tf-\mu(f))^2\right]\leq A(\varepsilon,f)t^{-(d/2 -\varepsilon)}. \] This result is proved for \(d\geq 3\), because the cases \(d=1,2\) were studied by N. Cancrini and F. Martinelli [J. Math. Phys. 41, No. 3, 1391–1423 (2000; Zbl 0977.82031)].

Reviewer: Alessandro Pellegrinotti (Roma)

##### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82C22 | Interacting particle systems in time-dependent statistical mechanics |