8.1 Results of Linear Response Theory

Let be the Hamiltonian of the given system when it is isolated. If we apply a weak disturbing field that couples to some property (with ) the Hamiltonian of the perturbed system is given by

(8.1) |

(8.2) |

(8.3) |

Combining this with the above equation we find that , or

Thus we may determine the transport coefficient either from an equilibrium simulation using equ. 8.4, or from a non-equilibrium simulation with applied field using equ. 8.5 Generally the second method yields better statistics but is more prone to nonlinearity problems (large fields); also, systems responding to an external field must be thermostated.

(8.6) |

(8.7) |

Now consider the conductivity . It may be determined in two ways:

- In an equilibrium simulation, using the Green-Kubo relation

(8.8) - In a non-equilibrium simulation, using the measured response to an applied
field
:

(8.9)

(8.10) | |||

(8.11) |

will be consistent with equ. 8.5 if only

(Note that we have switched to the Hamiltonian formalism.)

Example: In the Hamiltonian case the equations of motion are

(8.13) | |||

(8.14) |

Therefore we have and , and the requirement 8.12 is trivially fulfilled.

F. J. Vesely / University of Vienna