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For the two problems to be equivalent, the initial density for the symmetric channel Φ (X,Z,0) must be invariant under reflection about the X axis. When we consider symmetric channels, we will restrict our attention to initial conditions that satisfy this requirement.
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An approximate 1D equation of motion has also been developed for diffusion in a narrow symmetric 2D channel in the presence of a constant external field. Numerical tests of the validity of this equation may be found in 10.1103/PhysRevLett.96.130603;
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