1. Non-local Transport of Strongly Coupled Plasmas Satoshi Hamaguchi, Tomoyasu Saigo, and August Wierling Department of Fundamental Energy Science, Kyoto University

2. Outline 1.Introduction 2.Theory 3.MD simulation methods 4.Results 5.Summary

3. dusty plasmas, colloidal suspension, non-neutral plasmas, etc., e i Fine Particle -Q Debye screening clouds Introduction Yukawa potential ： inverse of Debye length ： Charge of fine particle

4. Screening parameter ： Coupling parameter ： ： inter particle spacing ： particle temperature Yukawa System = a system of particles interacting through Yukawa potential strongly coupled system  : one component plasma

5. G. E. Morfill: Max-Planck-Institute for Extraterrestrial Physics http://www.mpe.mpg.de/www_th/plasma-crystal/index_e.html Plasma Crystal Experiments

6. Molecular Dynamics Simulation

7. phase diagram of Yukawa systems

8. Motivation  want to have fluid equations valid for up to Strongly Coupled Systems Ordinary Hydrodynamics: valid only for : relaxation time : shear viscosity : bulk viscosity

9. Non-local transport coefficients  i.e., wavenumber and frequency dependent transport coefficients Generalized Hydrodynamics: non-local effects : generalized shear viscosity : generalized bulk viscosity

10. Simple Assumption If we assume Linearized equation Fourier-Laplace transform

11. Under the simple assumption For example, transverse wave ka

12. More generally… Linearized Generalized Hydrodynamics equation in Laplace-Fourier space Fourier-Laplace transform of

13. goals determine the generalized shear viscosity –in general, the simple assumption mentioned in the previous page does not hold: the wavenumber dependence of τ R cannot be ignored. determine the relaxation time τ R as a function of Γ (or system temperature T) and .

14. Theory microscopic analysis ↔ hydrodynamics analysis Microscopic flux (current): Assumption: hydrodynamic j should behave similarly at least if. Consider Transverse Current Autocorrelation Function

15. current autocorrelation functions longitudinal: transverse:

16. current correlation functions q = 0.619 q =2.48q =2.23q =1.96 q =1.75q = 1.24

17. Under the simple assumption For example, transverse wave ka

18. Transverse Current Autocorrelation Function Under Navier-Stokes Eqn: However microscopically…. Low wave number high wave number

19.  hydrodynamic approximation (no memory)  Relaxation Time Approximation(RTA) Memory Function Need to extend This equation may be viewed as the definition of Memory Function.

20. kinematic shear viscosity General Properties of Memory Function

21. Memory Function The memory function usually decays monotonically. The memory function usually decays more rapidly than the Transverse Current Autocorrelation Function (TCAF).

23. Assumption for Memory Function and obtain τ(k) as a function of Γ and  by fitting the function to the MD simulation data.

24. Normalized Memory Functions

25. ka  =2.0 wavenumber dependence of τ fitting parameters:

26. melting temperature Einstein frequency Scaling of the relaxation time

27. 1.Memory functions for Transverse Current Correlation Functions are calculated. 2.In the strongly coupling regime, non-exponential long time tail was observed in the memory function. 3.The relaxation time in Generalized Hydrodynamics has been estimated in the wide range of parameters. 4.The relaxation time takes the minimum value as a function of the system temperature (or Γ). Summary