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

Fumihiko SakataProfessor Emeritus, Ibaraki University

Member of the board of directors, Gushinkai Foundation

Presentation given atInstitute of Theoretical Physics, October. 11, 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 PrincipleThere 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

xj(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 r(t)

Time Development of Dynamical Variable B(t)

There holds a relation

Multi-Channel Reaction

Relevant dofIrrelevant dof

is determined by numerical simulation

Modified Time-Dependent Projection Operator Method

During Dt (macroscopically short, 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 Brownian motion)

Two basic assumptions

is well described by ?

Formal solution

Initial condition

IrrelevantRelevant

Project onto relevant space

mean-effect fluctuation effects

Taking account of order-to-chaos transition in irrelevant system

Coarse-Grained Macroscopic Hamiltonian

Integration over instead of

Expand in terms of origin of random force as well as dissipation force for relevant motion is evaluated term by term appearing in perturbative treatment of .

Define small time interval (microscopically long)

next step ; initial condition to get

c.f. PTP125(2011)359

=

Coarse-grained equation

Coarse-grained dimension e

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 of initial nuclei are important