##
**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.

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) |

### Keywords:

exclusion process; interacting particle systems; hydrodynamic limiting behavior; freezing of a supercooled liquid; Stefan problem
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\textit{L. Chayes} and \textit{G. Swindle}, Ann. Probab. 24, No. 2, 559--598 (1996; Zbl 0869.60085)

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[13] | LOS ANGELES, CALIFORNIA 90024 SANTA BARBARA, CALIFORNIA 93106 E-mail: lchay es@math.ucla.edu E-mail: swindle@bernoulli.ucsb.edu |

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