Variational calculation of transport coefficients in diffusive lattice gases

A diffusive lattice gas is characterized by the diffusion coefficient depending only on the density. The Green-Kubo formula for diffusivity can be represented as a variational formula, but even when the equilibrium properties of a lattice gas are analytically known, the diffusion coefficient can be computed only in the exceptional situation when the lattice gas is gradient. In the general case, minimization over an infinite-dimensional space is required. We propose an approximation scheme based on minimizing over finite-dimensional subspaces of functions. The procedure is demonstrated for one-dimensional generalized exclusion processes in which each site can accommodate at most two particles. Our analytical predictions provide upper bounds for the diffusivity that are very close to simulation results throughout the entire density range. We also analyze nonequilibrium density profiles for finite chains coupled to reservoirs. The predictions for the profiles are in excellent agreement with simulations.

Figure 1

Illustration of the GEP in one dimension, where each site is occupied by at most two particles. The arrows indicate possible transitions with rates (3). Two particles occupying the same site jump independently, each with a rate $\frac{1}{2}{p}_{2s}$.

Figure 2

Illustration of the procedure of computing $Q\left(f\right)$ for $n=1$ when the local function $f\left(\tau \right)=f\left({\tau }_1\right)$ depends only on a single site and hence only $f^0\equiv f$ and $f^1$ contribute to (30).

Figure 3

Illustration of the procedure for $n=2$ when the local function $f$ depends on two adjacent sites, and only $f, f^1$, and $f^2$ contribute to (30).

Figure 4

(a) Zeroth- and second-order approximations of the diffusivity over the entire density range. (b) Zeroth-, second-, and third-order approximations in the density range $0.9\le \rho \le 1.1$. Numerical results (dotted lines) were obtained by simulating an open system of length $L=1024$.

Figure 5

Illustration of the procedure for $n=3$ when the local function $f$ depends on three adjacent sites, and only $f, f^1, f^2$, and $f^3$ contribute to (30).

Figure 6

Difference between $n\text{th}$ and $(n-1)\text{st}$ order approximations vs $\rho$ (a) and $n$ (b). The plots are for the 2-GEP with particle-uniform rates (11). The quality of approximation is the worst around half-filling; in the examples of (b), the quality is worst when $\lambda =2.81$, corresponding to $\rho =1.38$.

Figure 7

Illustration of the 2-GEP with open boundaries. The “sites” 0 and $L$ correspond to reservoirs with (in general) different densities. These boundary conditions are realized by transitions of the states at sites 1 and $L-1$, which are indicated by arrows.

Figure 8

(a) $\rho \left(x\right)$ for the system of size $L=1024$ and ${\rho }^0\left(x\right)$ obtained from Eq. (51) with replacement $D\to D_0$. (b) The difference between ${\rho }^0\left(x\right)$ and $\rho \left(x\right)$ observed in simulations. The difference ${\rho }^0\left(x\right)-{\rho }^3\left(x\right)$ is shown for comparison. (c) The difference ${\rho }^0\left(x\right)-\rho \left(x\right)$ for $x=3/8$ and $7/8$ as a function of system size. The difference ${\rho }^0\left(x\right)-{\rho }^3\left(x\right)$ for the same two values of $x$ is also shown. For the three panels, we choose the boundary densities $({\rho }_0,{\rho }_L)=(2,0)$.

Figure 9

The scaled correlation function $L{\mathrm{\Gamma }}_i$ vs the scaled coordinate $i/L$ for three different system sizes. The data collapse supports the emergence of the scaling expressed by (54).

Figure 10

Correlation functions (59) multiplied by the system size $L$. The data collapse supports the emergence of the scaling behavior (60). The sum rules (61a) and (61b) are obeyed.

Figure 11

(a) $L(C_i^{21}-C_i^{12})$ vs $x=i/L$ for three different system sizes; ${\mu }_3$ [Eq. (70)] is shown for comparison. (b) Simulation results for $I={\int }_0^{2}D\left(\rho \right)d\rho$ and $\widehat{I}={\int }_0^2{D}_0\left(\rho \right)d\rho$. The estimates $I_n$ with $n=0,2,3$ are also shown.