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A nonequilibrium molecular dynamics method is proposed to produce nonequilibrium states flexibly. In this method, virtual points are set in a simulation box, and coarse-grained physical quantities at these points are constrained using Gauss's principle of least constraint. The coarse-grained physical quantities are evaluated by averaging microscopic quantities with an appropriate weight. To obtain the weight to evaluate the coarse-grained physical quantities, a shape function matrix is initially constructed from the particle configuration. This matrix expresses an interpolation of the physical quantities at particle positions from the coarse-grained quantities at the virtual points. Then, a matrix form of the weight is calculated as the Moore-Penrose pseudoinverse matrix for the shape function matrix. This method is applied to constrain the coarse-grained kinetic energy and produce a nonuniform temperature field in the system. The temperature profile at a nonequilibrium steady state depends on the method for constructing the shape function matrix. In particular, a local temperature coincides with the coarse-grained temperature when the shape function matrix is constructed based on a higher-order interpolation.

The dynamical behavior of a system consisting of a (heavy) piston and two rectangular boxes, each containing two hard disks and in contact with the piston, is studied based on a projection operator method for a microcanonical ensemble. We derive a coupled set of nonlinear equations for slow variables of the system and solve it to confirm that our theory with no adjustable parameters reproduces experimental results fairly well. Some limitations of the theory are discussed from the viewpoint of the separation of slow and fast time scales and ergodicity.