Irreversible and Nonlinear Dynamics of Open Systems

  • E. B. Davies
Part of the NATO Advanced Study Institutes Series book series (volume 57)


In the generally accepted mathematical description of the dynamics of a fixed number of non-relativistic particles, the state of the system at a particular time is described by a unit vector ψ(t) in a suitable complex Hilbert space H, say L2 (R3n) for n distinguishable spinless particles, and the time evolution is determined by the Schrödinger equation
$$ \frac{{d\psi }}{{dt}} = - iH\psi \left( t \right) $$
where the self-adjoint operator H is called the Hamiltonian of the system and is obtained by a well-known prescription. There are circumstances, arising particularly in statistical mechanics, where one has to consider more complicated states of the system, called mixed states. These are density matrices, or more precisely trace class linear operators ρ:HH such that p is self-adjoint, non-negative and of trace equal to one. Here trace is defined by
$$ tr\left[ \rho \right] = \sum\limits_n { < \rho {\psi _n}} ,{\psi _n} > $$
where {ψn} is any orthonormal basis of H. The evolution equation for mixed states corresponding to (1) is
$$ \frac{{d\rho }}{{dt}} = - i\left[ {H,\rho } \right] $$
with solution
$$ \rho \left( t \right) = {e^{ - iHt}}\rho \left( 0 \right){e^{iHt}} $$


Mixed State Gibbs State Bloch Equation Weak Coupling Limit Nonlinear Schrodinger Equation 
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Copyright information

© Plenum Press, New York 1980

Authors and Affiliations

  • E. B. Davies
    • 1
  1. 1.Mathematical InstituteUniversity of OxfordOxfordEngland

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