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Homogeneous and Nonlinear Generalized Master Equations: A New Type of Evolution Equations (pp.251-302) $100.00
Authors:  (Victor F. Los, Institute of Magnetism, National Academy of Sciences of Ukraine, Kiev, Ukraine)
Deriving the kinetic (irreversible) equations from the reversible microscopic dynamics
of the many-body systems remains one of the principal tasks of statistical
physics. One of the key assumptions used to deriving the Boltzmann equation is
the Bogoliubov principle of weakening of initial correlations which implies that on
a sufficientl large time scale all initial correlations (existing at the initial instant) are
damped. This approach leads to the evolution equations that are not valid on all time
scales and do not allow considering all the stages of the system evolution, in particular,
the short-term and transient regimes, which are of scientifi and practical interest for
considering the ultrafast relaxation and non-Markovian processes, decoherence phenomena
and onset of irreversibility. To take initial correlations into account, a method,
based on the conventional time-independent projection operator technique, that allows
converting the conventional linear inhomogeneous (containing a source caused by initial
correlations) time-convolution generalized master equation (TC-GME) and timeconvolutionless
GME (TCL-GME) into the homogeneous form exactly, is proposed.
This approach results in the exact linear time-convolution and time-convolutionless homogeneous
generalized master equations (TC-HGME and TCL-HGME) which take
the dynamics of initial correlations into account via modifie memory kernels governing
the evolution of the relevant part of a distribution function of a many-particle
system. These equations describe the evolution of the relevant part of a distribution
function on all time scales including the initial stage when the initial correlations matter.
The obtained TC-HGME is applied to the spatially homogeneous dilute gas of
classical and quantum particles. However, to derive the desired nonlinear equations
(the Boltzmann equation in particular) from this actually linear equation, we should
make an additional approximation neglecting the time-retardation of a one-particle
distribution function which restricts the time scale by times much smaller than the relaxation
time for a one-particle distribution function. To obtain the actually nonlinear
evolution equations and avoid any restrictions on the time scales, we develop a new
method, based on using a time-dependent operator (generally not a projection operator)
converting a distribution function of a total system into the relevant form, that
allows deriving the new exact nonlinear generalized master equations. The derived
inhomogeneous nonlinear GME is a generalization of the linear Nakajima-Zwanzig
TC-GME and can be viewed as an alternative to the BBGKY chain. To include the
initial correlations into consideration, we convert the obtained inhomogeneous nonlinear
GME into the homogeneous form by the method which we used for conventional
linear GMEs. The obtained exact homogeneous nonlinear GME describes all evolution
stages of the system of interest and treats initial correlations on an equal footing
with collisions by means of the modifie memory kernel. As an application, we obtain
a new homogeneous nonlinear equation retaining initial correlations for a one-particle
distribution function of the spatially inhomogeneous nonideal gas of classical particles.
We show that on the kinetic time scale, the time-reversible terms resulting from
initial correlations vanish (if the particle dynamics have the ergodic mixing property)
and this equation can be converted into the Vlasov-Landau and Boltzmann equations
without any additional commonly used approximations. 

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Homogeneous and Nonlinear Generalized Master Equations: A New Type of Evolution Equations (pp.251-302)