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Classical linear response theory

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  • Consider Hamilton's equations in the form

    \(\underline {\dot{q}_i}\) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ \({\partial H \over \partial p_i} \)
    \(\underline {\dot{p}_i}\) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ \( -{\partial H \over \partial q_i} \)

    We noted early in the course that an ensemble of systems evolving according to these equations of motion would generate an equilibrium ensemble (in this case, microcanonical). Recall that the phase space distribution function \(f (x, t) \) satisfied a Liouville equation:

    \[ {\partial f \over \partial t} + iLf = 0 \]

    where \( iL = \{...,H\} \). We noted that if \( \partial f/\partial t=0 \), then \(f = f (H) \) is a pure function of the Hamiltonian which defined the general class of distribution functions valid for equilibrium ensembles.

    What does it mean, however, if \( \partial f/\partial t \neq 0 \)? To answer this, consider the problem of a simple harmonic oscillator. In an equilibrium ensemble of simple harmonic oscillators at temperature \(T\), the members of the ensemble will undergo oscillatory motion about the potential minimum, with the amplitude of this motion determined by the temperature. Now, however, consider driving each oscillator with a time-dependent driving force \(F (t)\). Depending on how complicated the forcing function \(F (t) \) is, the motion of each member of the ensemble will, no longer, be simple oscillatory motion about the potential minimum, but could be a very complex kind of motion that explores large regions of the potential energy surface. In other words, the ensemble of harmonic oscillators has been driven away from equilibrium by the time-dependent force \(F (t) \). Because of this nonequilibrium behavior of the ensemble, averages over the ensemble could become time-dependent quantities rather than static quantities. Indeed, the distribution function \(f (x, t) \), itself, could be time-dependent. This can most easily be seen by considering the equation of motion for a forced oscillator

    \[ m\ddot{x} = -m\omega^2 x + F(t) \]

    The solution now depends on the entire history of the forcing function \(F (t) \), which can introduce explicit time-dependence into the ensemble distribution function.

    Contributors and Attributions

    Mark Tuckerman (New York University)