[Paper Review] Logarithmic Sobolev inequality for the inhomogeneous zero range process

Hanna Jankowski|ArXiv.org|Jun 30, 2006

Point processes and geometric inequalities16 references3 citations

TL;DR

This paper establishes that the logarithmic Sobolev constant for the inhomogeneous symmetric nearest-neighbor zero range process on a $N^d$-sized cube scales as $N^2$, under uniform Lipschitz and monotonicity conditions on the jump rates. The proof leverages grand canonical measures, spectral gap estimates, and moment bounds via Taylor expansion and martingale techniques, extending known results from the homogeneous case to inhomogeneous dynamics with spatially varying rates.

ABSTRACT

We prove that the logarithmic Sobolev constant for the inhomogeneous symmetric nearest neighbour zero range process on a cube of size N^d grows as N^2. We apply this result to the inhomogeneous process which arises in the study of the homogeneous version of the zero range interacting particle system with colours.

Motivation & Objective

• To establish a logarithmic Sobolev inequality for the inhomogeneous symmetric nearest-neighbor zero range process on a $d$-dimensional cube of size $N^d$.
• To extend known results on spectral gap and logarithmic Sobolev constants from the homogeneous to the inhomogeneous case, where jump rates depend on site location.
• To analyze the decay to equilibrium of the process under inhomogeneous dynamics, particularly in the context of nonequilibrium fluctuations and hydrodynamic scaling.
• To provide a quantitative bound on the logarithmic Sobolev constant $C_{ ext{LS}}$ in terms of system size $N$, showing it grows as $N^2$ under uniform regularity conditions on the rate functions.
• To develop technical tools—particularly moment estimates and measure comparisons—enabling the analysis of entropy dissipation in non-homogeneous particle systems.

Proposed method

• Uses grand canonical measures $ u_{ ilde{R}}$ indexed by overall density $ ilde{ ho}$ to analyze the invariant measure of the inhomogeneous process.
• Applies a decomposition of the Dirichlet form involving local averages $m_j = AV_{y otin B_j} ho_y$ and local rate deviations $F_j(m_j)$ to control variance terms.
• Employs the Schwarz inequality to bound cross-term contributions in the entropy dissipation estimate, reducing the problem to moment bounds on $F_j(m_j)$.
• Applies Proposition 4.11 and Corollary 4.11 to switch from microcanonical to grand canonical measures, enabling the use of Taylor expansion and moment estimates.
• Uses the uniform Lipschitz condition $ ext{sup}_{k,x} |c_x(k+1) - c_x(k)| o a_1 < igcirc$ and monotonicity $ ext{inf}_{k,x} ig[c_x(k+k_0) - c_x(k)ig] o a_2 > 0$ to control higher-order moments.
• Selects $l = ho^{-d}$ to balance error terms and achieve the desired $N^2$ scaling in the final bound on the logarithmic Sobolev constant.

Experimental results

Research questions

• RQ1 How does the logarithmic Sobolev constant behave for the inhomogeneous symmetric nearest-neighbor zero range process on a $d$-dimensional cube of size $N^d$?
• RQ2 Can the logarithmic Sobolev inequality be established under uniform Lipschitz and monotonicity conditions on the site-dependent jump rates $c_x(k)$?
• RQ3 What is the dependence of the logarithmic Sobolev constant on the system size $N$ in the inhomogeneous setting, compared to the homogeneous case?
• RQ4 How do local fluctuations in particle density and rate functions affect the entropy dissipation and spectral gap in the inhomogeneous model?
• RQ5 Can the martingale method of Lu and Yau be adapted to inhomogeneous zero range processes with spatially varying rates?

Key findings

• The logarithmic Sobolev constant $C_{ ext{LS}}$ for the inhomogeneous symmetric nearest-neighbor zero range process on a $N^d$-sized cube grows as $N^2$ under uniform Lipschitz and monotonicity conditions on the rate functions.
• The entropy dissipation inequality (1.3) holds uniformly for positive functions $f$, with the constant $C_{ ext{ED}}$ scaling as $N^2$, consistent with the $N^2$ scaling of $C_{ ext{LS}}$.
• The spectral gap $C_{ ext{SG}}^{-1}$ is bounded below by a constant times $N^{-2}$, implying exponential decay to equilibrium in $L^2( u_{ ilde{R}})$ at rate $N^{-2}$.
• The proof relies on a decomposition of the Dirichlet form into local contributions and uses moment bounds on deviations of local rates from their mean, controlled via Taylor expansion and grand canonical measure estimates.
• The error terms in the moment estimates are shown to be $Oig(rac{1}{| ext{Box}|} rac{1}{l^d} ilde{ ho}ig)$, and by choosing $l = ho^{-d}$, the dominant $N^2$ scaling is preserved.
• The result holds under the assumption $ ilde{ ho} > ho_0$, ensuring the validity of the measure switching argument used in the proof.