Derivation of Coarse-grained Transport Equation- Instruction for Application -

Fumihiko SakataProfessor Emeritus, Ibaraki University

Member of the board of directors, Gushinkai Foundation

Presentation given atBeijing Normal University, October. 13, 2011

“new property could not be predicted from a knowledge of the

lower-level properties”“the whole is greater than the

composition of its parts”

emergence is specific for biological (living) system,

complex system, self-organizing system,

system of network, …Finite, self-sustained,Strongly interacting,nonlinear complex

system

Introduction

Organ System Animal

Emergence

Nucleus

Searching for Dynamics of Emergence (Evolution of Matter)

DOE-Project

Theoretical Nuclear Physics Ab-initio Calculation of Nuclear Structure and Reaction Green’s function Monte Carlo (GFMC) No-core shell model (NCSM) Nuclear Energy Density Functional Theory (EDFT)

What do we learn from numerical calculation

CoveringPhysics, Climate, Groundwater,

Fusion Energy, Life SciencesMaterials & Chemistry

Microscopic Equation

High-PerformanceComputer

Experiment

Science in the Era of High Performance Computer

Dynamicsof

Emergence

Black Box ?

京 (K) (Kobe, RIKEN AICS)天河 1A 号 (Tianhe-1A)(Tianjin, NSCC)Jaguar(Tennessee, ORNL)

There are no general quantum many-body reaction theory !!

What actually happens in HI-DIC, nuclear fusion and fission ・ microscopic origin of dissipation and fluctuation described in macroscopic variables・ role of one- and two-body dissipation (mean-field and correlation/macro and micro)・ role of adiabatic- and diabatic-mean field・ role of macro (mean field) variables (window, wall, neck, asymmetry, friction,…)・ role of various reaction process (one, two and many nucleon transfer, successive transfer,…)

http://euroball.lnl.infn.it/ECTbigs/talks/Bob_Chapman.pdf

V.I.Zagrebaev, PRC64,034606(2001)

superposition of nn-scattering two body scattering

How to understand Emergence Dynamics in Nuclei

Microscopic Equation

Time-Dependent Variational Principle (principle of least action)There are no general quantum many-body reaction theory !! (Later discussion)Simulation is needed like gravitational n-body problem

TDVP associates to a Hamiltonian quantum system a set of trajectories running on a classical phase space

Canonical Eqs. of Motion

Classical image (e.g.; Wigner transformation)

Numerical Simulation describing - Change of Nuclear Structure as well as Reaction Process -

Time-Dependent Hartree Fock (mean field: ph-amplitudes) Fermion Molecular Dynamics (wave packet: central values of coordinate & momentum)

Gaussian wave packet

Trial state for N nucleon system

Basic equation of FMD

How to describe different reaction channel

spatial⊗ spin&isospin

Elastic, Inelastic A(a, a’)A’ Transfer ReactionA(a, b)B : B=A+n, a=n+b

+ + various sequential Reactions

j(A) Intrinsic coordinate of system A

Elastic and inelastic Collision

Single-Nucleon Transfer Reaction

General Hamiltonian in parameter space

Hamiltonian in elastic and inelastic collision

Center of mass + Intrinsic + Relative + Coupling

How to understand Numerical Simulation

Liouville Equation for Distribution Function (t)

Time Development of Dynamical Variable B(t)

There holds a relation

Multi-Channel Reaction

Relevant dofIrrelevant dof

is determined by numerical simulation

Suppose an exact calculation has been done, and the following two conditions are satisfied;

Consider a time evolution from to We already have

Let us introduce projection of dynamical variable onto relevant space at time

During time interval t (macroscopically short, but microscopically long), coupling effects from the relevant dof onto the irrelevant system are negligible

relavant motion is described by single (mean) trajectory (like Langevin), irrelevant motion by distribution function (like molecular motion for Brownian particle)

Coarse-grained dimension

Coarse-grained transport equation

Expand with respect todissipation force and random force are given perturbatively

Generally, tI dependent time-independent when chaos develops in system

Let us introduce

Approximation

with

is defined by the conditionGeneralized local FD theorem

TDHF basic dynamics and anti-symmetrization are included FMD quantum mechanical

V.I.Zagrebaev, PRC64,034606(2001)

Some Remarks

Simplifications are needed same width of wave packets anti-symmetrization: let us start with a few high momentum nucleons are transferred (N+M), (N+1,M-1)+/-(N-1,M+1),…are important wall (mean potential ) of initial nuclei are important

determinant expanded in small determinants

Extension of parameter space nucleon expressed by two Gaussian