The hydrodynamic limit of the symmetric simple exclusion process with speed change is given by a diffusive equation in the appropriate scale. Following the nongradient method introduced by Varadhan and the Navier-Stokes methods developed by Yau, we prove that in the same scale, the next order correction is given by a third order equation for dimension $d \geq 3$.
References
[1] Dobrushin, R. L. (1989). Caricatures of hydrodynamics. In IXth International Congress on Mathematical Physics (I. M. Davies, B. Simon and A. Truman, eds.) 117-132. Hilger, Bristol, UK. MR91a:82056 0725.76005[1] Dobrushin, R. L. (1989). Caricatures of hydrodynamics. In IXth International Congress on Mathematical Physics (I. M. Davies, B. Simon and A. Truman, eds.) 117-132. Hilger, Bristol, UK. MR91a:82056 0725.76005
[2] Esposito, R. and Marra, R. (1993). On the derivation of the incompressible Navier-Stokes equation for Hamiltonian particle systems. J. Statist. Phys. 74 981-1004. MR1268784 0831.76076 10.1007/BF02188213[2] Esposito, R. and Marra, R. (1993). On the derivation of the incompressible Navier-Stokes equation for Hamiltonian particle systems. J. Statist. Phys. 74 981-1004. MR1268784 0831.76076 10.1007/BF02188213
[3] Esposito, R., Marra, R. and Yau, H. T. (1994). Diffusive limit of asymmetric simple exclusion. In On Three Levels (M. Fannes, ed.) 324 43-53. NATO, Brussels. 0841.60082[3] Esposito, R., Marra, R. and Yau, H. T. (1994). Diffusive limit of asymmetric simple exclusion. In On Three Levels (M. Fannes, ed.) 324 43-53. NATO, Brussels. 0841.60082
[4] Guo, M., Papanicolaou, G. C. and Varadhan, S. R. S. (1988). Nonlinear diffusion limit for a system with nearest neighbor interactions. Comm. Math. Phys. 118 31-59. 0652.60107 MR954674 10.1007/BF01218476 euclid.cmp/1104161907
[4] Guo, M., Papanicolaou, G. C. and Varadhan, S. R. S. (1988). Nonlinear diffusion limit for a system with nearest neighbor interactions. Comm. Math. Phys. 118 31-59. 0652.60107 MR954674 10.1007/BF01218476 euclid.cmp/1104161907
[6] Landim, C., Olla, S. and Yau, H. T. (1996). Some properties of the diffusion coefficient for asymmetric simple exclusion processes. Ann. Probab. 24 1779-1808. MR98a:60150 0872.60078 10.1214/aop/1041903206 euclid.aop/1041903206
[6] Landim, C., Olla, S. and Yau, H. T. (1996). Some properties of the diffusion coefficient for asymmetric simple exclusion processes. Ann. Probab. 24 1779-1808. MR98a:60150 0872.60078 10.1214/aop/1041903206 euclid.aop/1041903206
[7] Landim, C., Olla, S. and Yau, H. T. (1997). First order correction for the hydrodynamic limit of asymmetric simple exclusion processes in dimension d 3. Comm. Pure Appl. Math. 50 149-203. MR98c:60143 0866.76003 10.1002/(SICI)1097-0312(199702)50:2<149::AID-CPA2>3.0.CO;2-C[7] Landim, C., Olla, S. and Yau, H. T. (1997). First order correction for the hydrodynamic limit of asymmetric simple exclusion processes in dimension d 3. Comm. Pure Appl. Math. 50 149-203. MR98c:60143 0866.76003 10.1002/(SICI)1097-0312(199702)50:2<149::AID-CPA2>3.0.CO;2-C
[8] Reed, M. and Simon, B. (1975). Methods of Modern Mathematical Physics 2. Academic Press, New York. MR493420 0308.47002[8] Reed, M. and Simon, B. (1975). Methods of Modern Mathematical Physics 2. Academic Press, New York. MR493420 0308.47002
[9] Rezakhanlou, F. (1991). Hydrodynamic limit for attractive particle systems on Zd. Comm. Math. Phys. 140 417-448. MR93f:82035 0738.60098 10.1007/BF02099130 euclid.cmp/1104248092
[9] Rezakhanlou, F. (1991). Hydrodynamic limit for attractive particle systems on Zd. Comm. Math. Phys. 140 417-448. MR93f:82035 0738.60098 10.1007/BF02099130 euclid.cmp/1104248092
[11] Yau, H. T. (1991). Relative entropy and hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys. 22 63-80. MR93e:82035 0725.60120 10.1007/BF00400379[11] Yau, H. T. (1991). Relative entropy and hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys. 22 63-80. MR93e:82035 0725.60120 10.1007/BF00400379