Non-perturbative rheological behavior ofa far-from-equilibrium expanding plasma
Syo Kamata

North Carolina State University
Collaboration withA.Behtash, M. Martinez, H. Shi, (NC State U.)
C. N. Cruz-Camacho (Universidad Nacional de Colombia)"Non-perturbative rheological behavior of a far-from-equilibrium expanding plasma", [arXiv 1805.087771]"Dynamical systems and nonlinear transient rheology of the far-from-equilibrium Bjorken flow", in Preparation"Physics of Nonequilibrium systems"
Dec 26th 2018 @YITP

Introduction: motivation• QCD and Heavy-ion collision• Described by quarks and gluon• QGP and early universe --- Thermal process in time evolution• Bjorken flow (Kinetic theory, RTA approximation)• Connection to Hydrodynamics• Renormalized transport coefficients

Introduction: Bjorken flow - RTA approximation [Bhatnagar et al. 54,
Anderso et al. 74]• Collision of Kernel: input from a micro theory• RTA approximation (relativistic massless particle)• Depends on Milne time and momentum• Boost + 2D isometries + parity (z-axis)• Bjorken flow :• Gubser flow :(on(on) [Bjorken 83]) [Gubser 10]

Today'stalk• Integration form
Initial constants

Introduction: BE to Hydrodynamics• Gradient expansion

Hydrodynamics[Chapman, Enskog][Bhalerao et al. 14]

Introduction: Mathematics• Dynamical system (Non-autonomous ODE)
• Analytic method• Transseries analysis (classical asymptotics)
Generalization of asymptotic expansion• Connection to other math tools:• Resurgence theory• Bifurcation theory• Conley index theory• Etc...

• Basic tools for application to other physical system• Non-relativistic system • Non-perturbative RG eq.
• Rheology• Etc...

Contents• Introduction• Transseries analysis for Bjorken flow
• Boltzmann equation to dynamical system• Transseries analysis• Global structure of the dynamical system• Phase portrait• Initial value problems• Conclusion

Transseries analysis
for Bjorken flow

Boltzmann equation (Bjorken flow)

Momentexpansion

Chapman-Enskog
expansion

Dynamical system

Hydrodynamics
EM tensor
Transport coeffs（Non-autonomous system）
Navier-Stokes
limit (?)

Transseries Analysis(Resurgence)Reduction to Dynamical System• Moment expansion

Legendre polynomial
Laguerre polynomial• EM tensor[Grad 49, Romatschke et al. 11][Molnar et al. 16]

Reduction to Dynamical System• Dynamical System (Non-linear ODE)

From E conservaiton law

Reduction to Dynamical System• Dynamical System (Non-linear ODE)Transseries Analysis[Ecalle 81-85, Costin 98]• Generalization of asymptotic expansion• Divergent series (Radius of convergence = 0)
• Factorial growth:(singularities on the positive real axis in the Borel plane)• Signal of existence of higher level trans-monomials• Resurgence relation• Costin's formula•(converge into an IR fixed pt.)
• Transseries ansatz and coefficients are uniquely determined.
(up to normalization of integration consts)• Imaginary ambiguity cancellation works (if you want).

Evolution equation
Transseries ansatz

Substitute Boltzmann eq.

Recursively solveorder by order.

Transasymptotic matching[Basar et al. 15]• RG equation of transport coefficients• Simultaneous PDE

ODE and solvable if L=1, N=0

Comparison with the exact solution
L=1, N=0, O(1/w)

Deviation from the NS limit• NS limit~ 1/w
EM tensor is related only with• However ...

has also the same asymptotics.

• Deviation from the NS hydro should exists in ~ 1/w
due to!!

Deviation from the NS limit• Hydrodynamic limit -> 1/w …> c01• However... c11 ~ 1/w• Energy momentum tensor is not enough todescribe the late time behavior

Global structure of
the dynamical system

Phase portrait• Trivial fiberization• Base space :Skew product• Fiber space :Outstanding problem: How to make"Time dependent
control parameter"??in Bifurcation theory

Phase portrait
Singularity

Source (UV)

Sink (IR)Saddle (UV)Phase portrait
Near = 0

Initial value problem• Integration form:
essentiallyin w coordinate• Transseries: # of =•gives a one dimensional orbit onspace• Invariant subspace of the flow is two dimensions
Outstanding problem: How to make??

Conclusion• Boltzmann equation ⇒ Dynamical system of Bjorken flow
via. moment expansion• Application of Transseries to the dynamical system⇒ Beyond hydrodynamics
Non-hydro mode can be uniquely determined.
• Renormalized transport coefficients• Deviation from the NS hydroFuture work

• (Beyond) Linear response theory• More realistic model ⇒ space dependence ⇒ PDE• Condensed matter, non-relativistic system, ...