## Hydrodynamic limits for one-dimensional particle systems with moving boundaries.(English)Zbl 0869.60085

A new class of one-dimensional interacting particle systems with exclusion dynamics is considered. The authors study the additional feature of random boundaries of the systems with a random motion that is coupled to the local particle density, and show that the hydrodynamic limiting behavior in these systems is characterized by the solution of an appropriate Stefan (free-boundary) equation. The case study of the two basic examples is given. Actually both of examples can be regarded as crude microscopic models of the dynamics of a liquid-solid system with an interface. One case (Model $$\#1)$$ corresponds to the melting of a solid and the other (Model $$\#2)$$ to the freezing of a supercooled liquid.
More precisely, the particle system (Model $$\#1)$$ with a melting boundary is described by the particle configuration $$\zeta$$ and the generators $$\Omega_{\mathbb{L}_N}$$, $$\Omega_{\partial \mathbb{L}_N}$$ where $$\mathbb{L}_N: =\{-N,-N+1, \dots, N-1,N\}$$ and $$\Omega_{\partial \mathbb{L}_N}$$ is a generator of describing particle transfers to the endpoints $$\partial \mathbb{L}_N$$ of $$\mathbb{L}_N$$. For any positive time $$S$$, let $$\rho_F(t)$$, $$0\leq t\leq S$$ (resp. $$\rho_0(x)$$, $$0\leq x\leq 1)$$ be piecewise smooth functions satisfying $$0\leq\rho_F(t)$$ (resp. $$\rho_0(x))\leq 1)$$. Set $$N_r: =2^rN_0$$ for a positive integer $$r$$. Let $$\rho (x,t)$$ denote the density for the classical one-sided Stefan problem and $$B(t)$$ the free boundary. Formally the Model $$\#1$$ (melting) may be expressed as a diffusion equation for an enthalpy function $$a(x,t)$$ with a diffusion coefficient that depends discontinuously on the value of $$a$$. In this case we have $$a(x,t)=\rho(x,t)$$ for $$x>B(t)$$ where the boundary is simply $$B(t)= \sup\{x;\;a(x,t)= -1\}$$ with the initial position $$B(0)=0$$. The spatial domain is $$[-1,+1]$$ and $$\rho(1,t)= \rho_F(t)$$, $$\rho(x,0) =\rho_0(x)$$. The corresponding Stefan condition is given by $$dB/dt= -\nabla \rho(B(t),t)$$, which indicates the outward displacement of the boundary (melting). By using the instantaneous configuration $$\zeta_t$$ we define the enthalpy function $$a_r$$ in the $$r$$-th particle system by $$a_r(x,t): =\zeta_{N^2_rt} (N_rx)$$. Set $$\overline a_r(x,t)= \mathbb{E} [a_r(x,t)]$$. Suppose that each realization of the particle system comes equipped with an initial particle configuration. Then $$\overline a_r(x,t)$$ converges weakly to an enthalpy function $$a(x,t)$$ in $$L^2$$, where $$a(x,t)$$ is the unique solution to the Stefan problem, i.e., $\begin{split} \int^{+1}_{-1} \bigl\{a(x,s) G(x,s)- a(x,0) G(x,0)\bigr\} dx=\\ =\int^{+1}_{-1} \int^s_0 \left\{a {\partial G\over\partial t} +H(a) {\partial^2G \over\partial x^2} \right\} dxdt-\int^s_0 \rho_F(t) {\partial G\over \partial x} (1,t)dt, \end{split} \tag{1}$ with $$H(a)=0$$ (if $$a\leq 0)$$, and $$=a$$ (if $$a \geq 0)$$, where $$G(x,t)$$ denotes any smooth test function with $$G(1,t)=0$$. Moreover, it is also derived that, with probability 1, $$a_r(x,t)$$ itself converges weakly to $$a(x,t)$$ in $$L^2$$, where $$a(x,t)$$ is the solution to the same Stefan problem (1) with the prescribed initial and boundary conditions. For the second system (Model $$\#2)$$ with a freezing boundary, a similar statement is proved as well.
Reviewer: I.Dôku (Urawa)

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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### References:

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