Macroscopic evolution of particle systems with short- and long-range interactions
Macroscopic evolution of particle systems with short- and long-range interactions
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具有短程和长程相互作用的粒子系统的宏观演化
DOI：
10.1088/0951-7715/13/6/314
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发表时间：
2000
期刊：
影响因子：
1.7
通讯作者：
Rossana Marra
中科院分区：
文献类型：
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作者：
G. Giacomin;J. Lebowitz;Rossana Marra
我们考虑具有一般短程相互作用的晶格气体和范围为 γ-1、γ>0 的 Kac 势 Jγ(r)，通过粒子跳跃到最近邻空位点演化，其速率满足相对于平衡测量的详细平衡。缩放类空间 γ-1 和类时间 γ-2，我们证明在极限 γ→0 下，宏观密度分布 ρ(r,t) 满足方程 这里 σs(ρ) 是参考系统的迁移率，其中 J≠0，并且 (ρ) = ∫[fs(ρ(r))-½ρ(r)∫J(r-r')ρ(r') dr dr']，其中fs(ρ) 是参考系统的（严格凸）自由能密度。除了 J 上的正则性条件之外，此结果的唯一要求是参考系满足 Varadhan-Yau 定理的假设，导致 J ≠ 0 (*)。因此，如果在非恒定密度分布上达到最小值，则 (*) 也成立，这包括发生相偏析的情况。使用相同的技术，我们还推导出具有 A 和 B 颗粒之间的长程排斥相互作用的双组分 A-B 混合物的密度的流体动力学方程。密度 ρA 和 ρB 的方程的形式为 (*)。他们描述了在低温下的分层转变，其中通过空位发生分离，即跳跃到空位点。在空位非常少的限制下，该问题变得类似于连续介质系统中所谓的不可压缩极限中的相分离。
We consider a lattice gas with general short-range interactions and a Kac potential Jγ(r) of range γ-1, γ>0, evolving via particles hopping to nearest-neighbour empty sites with rates which satisfy detailed balance with respect to the equilibrium measure. Scaling spacelike γ-1 and timelike γ-2, we prove that in the limit γ→0 the macroscopic density profile ρ(r,t) satisfies the equation Here σs(ρ) is the mobility of the reference system, that with J≡0, and (ρ) = ∫[fs(ρ(r))-½ρ(r)∫J(r-r')ρ(r') dr dr'], where fs(ρ) is the (strictly convex) free energy density of the reference system. Beside a regularity condition on J, the only requirement for this result is that the reference system satisfy the hypotheses of the Varadhan-Yau theorem leading to (*) for J≡0. Therefore, (*) also holds if achieves its minimum on non-constant density profiles and this includes the cases in which phase segregation occurs. Using the same techniques we also derive hydrodynamic equations for the densities of a two-component A-B mixture with long-range repulsive interactions between A and B particles. The equations for the densities ρA and ρB are of the form (*). They describe, at low temperatures, the demixing transition in which segregation takes place via vacancies, i.e. jumps to empty sites. In the limit of very few vacancies the problem becomes similar to phase segregation in a continuum system in the so-called incompressible limit.