Hydrodynamic limit for two-species exclusion processes

1. Problem Statement

The paper investigates the hydrodynamic limit of a system of interacting particles known as the two-species exclusion process on a $d$-dimensional discrete torus ($\mathbb{T}^d_N$).

• System Dynamics: The system consists of two types of mechanically distinguishable particles, denoted as $+$ (charge $+1$) and $-$ (charge $-1$), and empty sites (charge $0$). The state space is $\{-1, 0, 1\}^{\mathbb{T}^d_N}$.
• Constraints: The exclusion principle applies: at most one particle can occupy any site.
• Interactions: The dynamics include:
  1. Hopping: Particles jump to neighboring empty sites with rates $C_+$ and $C_-$.
  2. Exchange: Neighboring particles of opposite types ($+,-$) exchange positions with rate $C_E$.
  3. Annihilation: Neighboring $+$ and $-$ particles annihilate (become empty) with rate $C_A$.
  4. Creation: Two neighboring empty sites are filled with $+$ and $-$ particles with rate $C_C$.
• Conservation Law: The paper focuses on the regime where $C_A > 0$ or $C_C > 0$. In this case, the total charge $Q = \sum_{x} \eta(x)$ is the unique conserved quantity.
• Goal: To prove that under diffusive rescaling (space scaled by $N$, time by $N^2$), the empirical density of the charge converges to the solution of a macroscopic nonlinear diffusion equation.

2. Methodology

The author employs the standard entropy method and martingale approach common in the theory of interacting particle systems, adapted for non-gradient systems. The proof strategy is divided based on the parameters of the model:

A. Classification of Cases

The paper analyzes three distinct cases based on the rates of annihilation ($C_A$) and creation ($C_C$):

1. Case 1: $C_A > 0$ and $C_C > 0$. (General non-gradient case).
2. Case 2: $C_A > 0$ and $C_C = 0$. (Assumes a gradient condition).
3. Case 3: $C_A = 0$ and $C_C > 0$. (Assumes a gradient condition).

B. Key Technical Steps

1. Martingale Decomposition:
   The empirical measure $\pi^N_t$ is analyzed via a martingale $M^H_N(t)$. The core challenge is handling the term involving the generator $L_N$. The author decomposes the microscopic current $W$ into a linear combination of gradients (which converge to the macroscopic flux) and a term in the range of the generator (which vanishes in the limit).
   $$W \approx \sum D_{ij}(\rho) \nabla \eta + L_N f$$

2. Spectral Gap Estimates (Crucial for Case 1):
   Since Case 1 is generally non-gradient, the standard replacement of local functions by their averages requires a spectral gap estimate for the generator confined to finite volumes.

   • The author proves a spectral gap of order $O(N^{-2})$ for the generator on a finite box with fixed total charge.
   • This is achieved by comparing the specific two-species process to a mean-field type process and utilizing a duality argument between $+$ and $-$ particles.
   • This estimate allows the application of the One-Block and Two-Block estimates to replace local empirical densities with the macroscopic density.

3. Characterization of Closed Forms:
   To handle the non-gradient nature, the paper characterizes the space of "closed forms" (functions orthogonal to the range of the generator). The author establishes a one-to-one correspondence between the closed forms of this two-species model and those of the generalized exclusion process, allowing the transfer of known algebraic characterizations.

4. Diffusion Coefficient:
   The diffusion coefficient matrix $D(\rho)$ is derived variationally. A significant result is the proof that $D(\rho)$ is a diagonal matrix with identical diagonal entries ($D(\rho) = d(\rho)I$), despite the complex interactions. This simplifies the macroscopic equation significantly.

5. Uniqueness of Weak Solutions:
   The paper addresses the uniqueness of the solution to the resulting nonlinear parabolic equations. Since the diffusion coefficient may not be smooth (especially in Case 2), the author relies on the fact that the diffusion matrix is diagonal to prove uniqueness of weak solutions for the specific class of equations derived.

3. Key Contributions and Results

Main Theorems

The paper proves that the empirical density field converges to the unique weak solution $\rho(t, u)$ of a nonlinear parabolic equation:

• Case 1 ($C_A, C_C > 0$):
  The limit satisfies:
  $$\partial_t \rho = \Delta (\tilde{d}(\rho))$$
  where $\tilde{d}(\rho) = \int_{-1}^\rho d(\gamma) d\gamma$. The diffusion coefficient $d(\rho)$ is given by a variational formula involving the spectral gap and current correlations. No gradient condition is assumed.

• Case 2 ($C_A > 0, C_C = 0$, Gradient Condition):
  Under the condition $C_+ + C_- - C_A - 2C_E = 0$, the limit satisfies a two-phase Stefan problem:
  $$\partial_t \rho = \Delta P(\rho)$$
  where $P(\rho) = C_+ \rho \mathbb{1}_{\{\rho>0\}} - C_- \rho \mathbb{1}_{\{\rho<0\}}$. This describes a system with distinct diffusion coefficients for positive and negative charge densities.

• Case 3 ($C_A = 0, C_C > 0$, Gradient Condition):
  Under the condition $C_+ + C_- - 2C_E = 0$, the limit satisfies the standard heat equation:
  $$\partial_t \rho = C_E \Delta \rho$$

Spectral Gap Result

A significant independent contribution is Theorem 3.11, which establishes a spectral gap of order $N^{-2}$ for the two-species exclusion process on a finite domain with fixed charge. This is a non-trivial result for non-gradient systems and is essential for proving the hydrodynamic limit in Case 1.

Explicit Diffusion Coefficients

The paper provides explicit expressions for the diffusion coefficient $d(\rho)$ in terms of the model parameters ($C_+, C_-, C_A, C_C, C_E$) and the static compressibility $\chi(\rho)$.

4. Significance

• Extension of Hydrodynamic Theory: This work extends the hydrodynamic limit theory to non-gradient multi-species systems with creation and annihilation mechanisms, which are more complex than simple exclusion processes.
• Spectral Gap for Non-Gradient Systems: The proof of the spectral gap for this specific non-gradient model is a major technical achievement, as such estimates are notoriously difficult to obtain without the gradient condition.
• Connection to Interface Dynamics: The results are directly applicable to the 1-dimensional SOS (Solid-On-Solid) interface model. The two-species exclusion process corresponds to the evolution of height differences in the SOS model. Thus, this paper provides a rigorous derivation of the macroscopic evolution equations for these interface models.
• Handling Discontinuous Diffusion: The analysis of Case 2 provides a rigorous framework for understanding Stefan problems (phase transitions) arising from microscopic particle systems with discontinuous macroscopic fluxes.

In summary, Sasada provides a rigorous mathematical foundation for the macroscopic behavior of complex two-species particle systems, bridging the gap between microscopic stochastic dynamics and macroscopic nonlinear diffusion equations, even in the absence of gradient conditions.