A fresh look at hydrodynamics from fluctuation formulas Shin-ichi Sasa and Masato Itami 1

Purpose of my talk 2 1)Basic concepts in non-equilibrium statistical mechanics (20-th century ) - from a viewpoint of large deviation theory 2)Recent developments in non-equilibrium statistical mechanics (for last two decades) - symmetry and variational principle in large deviation theory 3) Presentation of our recent work (2015) - a Surface-Bulk correspondence in non-equilibrium statistical mechanics

3 PART I Thermodynamics and Large deviation

4 Thermodynamics Thermology dynamics heat temperature work pressure Unified theory of heat and work

How is it unified? 5 Entropy

Simple Example Thermally insulating material-2 material-1 Remove the constraints Thermally conducting 6 What is the equilibrium value?

Answer (Thermodynamics) material-2 material-1 Thermally conducting 7 Variational principle (which comes from the second law of thermodynamics)

Question 8 Can you obtain the equilibrium value without thermodynamics (entropy) ?

Quick answer 9 Yes, the equilibrium value is given as the most probable (typical) value almost all microscopic states show

Example : the most probable value (throwing many coins) 10 coins:“head” of i-th coin “tail” of i-th coin score:

Probability “density” of X 11 Law of large numbers

12 Asymptotic form of

Large deviation theory 13 The probability of rare (atypical) values large deviation function In general, non-negative and convex function (with a more precise definition) large deviation property most probable value (coin problem)

Remark: central limiting theorem 14 Gaussian distribution: quadratic function If the dispersion is proportional to 1/N, and If the tail can be ignored

Energy distribution 15 The most probable value The principle of equal-weight determines the large deviation function for a given microscopic model

Large deviation and entropy 16 Frequency of rare fluctuationThermodynamics “fluctuation” (e.g. energy fluctuation) “response” (e.g. heat capacity) (Einstein formula, 1908)

17 PART II Dynamics and Large deviation

A simple example 18 viscous fluid Impulse force at equilibrium with no external forces friction force velocity friction coefficient slow motion

What corresponds to thermodynamics ? 19

Hydrodynamics :Stokes (1851) 20 Solve with the BC at the surface of the ball Calculate the force from :radius of the ball :viscosity Stokes’ law

Question 21 Can you derive Stokes’ law without the hydrodynamic equations ?

Microscopic setup 22 fluid particles CM of the ball short-range interaction radius of the fluid particles radius of the ball as (spherical symmetric ) microscopic mechanical state Force acting on the ball

Basic assumptions 23 solution of the Hamiltonian equation for any Separation of length and time scales and other scale separation conditions that will be explained in later arguments

Large deviation 24 correlation time of the force relaxation time of the ball momentum

Symmetry 25 Fluctuation Theorem

Brief history 26 (pioneers) Evans, Cohen, Morriss, FT in a determinisitic “toy” model (1993) Gallavotti-Cohen, a mathematical proof for the FT (1995) Jarzynski, essentially same identity (work relation) (1997) Kurchan, FT for Langevin systems (1998) Lebowitz-Spohn, Maes, Crooks, FT for Markov stochastic systems (1999) (developments) Kurchan, Tasaki, quantum FT (2000) Hatano-Sasa, steady state thermodynamics (2001) Sagawa-Ueda, information thermodynamics (2010) (experiment) Nakamura et al, quantum coherent conductor (2010)

Response formula 27 Central limiting theorem (non-linear response theory) Fluctuation theorem (linear response theory)

Kirkwood(1946) 28

29 PART III Proof of the fluctuation theorem

Time dependent probability density 30 Probability density just after the impulse time-reversal time evolution (Liouvile’s theorem) reversibility

Identity 31 (Liouvile’s theorem) (energy conservation) (reversibility) (Equation of motion)

Probability density of time-averaged force 32 Reflection symmetry

33 PART IV Problem

Where are you ? 34 hydrodynamic equations Fluctuation of the time averaged force to the ball Stokes (1851) Kirkwood’s formula (1946) How do you calculate the fluctuation intensity ?

Trajectories of particles interacting with the ball 35

Decomposition of the force 36 z-component of the force per unit area on the surface (= average stress)

Random collisions 37 The average stress obeys the central limiting theorem : the dimensional analysis (with some physics) valid for dilute gases Area law

Nontrivial nature of Stokes’ law 38 linear law NO Central Limiting Theorem It indicates the existence of the long-range correlation of the time-averaged stress at the surface !

Trajectories of particles interacting with the ball 39

A key to solve the problem 40 How is the viscosity related to fluctuations at the surface?

Green-Kubo formula (1954) 41 correlation time of the stress fluctuation relaxation time of the momentum density field

Landscape 42 hydrodynamic equations Green-Kubo formula (1954) Stress fluctuation in the bulk Stress fluctuation at the surface Kirkwood’s formula (1946) Stokes (1851)

The heart of the problem 43 hydrodynamic equations Green-Kubo formula (1954) Stress fluctuation in the bulk Stress fluctuation at the surface Kirkwood’s formula (1946) Stokes (1851) Formulate the connection between bulk and surface

44 PART V effective theory in the bulk

Two fluctuation formulas 45 correlation time of the stress fluctuation relaxation time of the momentum density field correlation time of the force The basic assumption: The same can be found in the two formulas relaxation time of the ball momentum

Coarse-grained description 46 correlation length of the stress Coarse-grained and time-averaged stress field The Green-Kubo formula

Macroscopic fluctuation theory 47 for is a ultraviolet cut-off in a macroscopic description (space mesh)

Statistical properties 48 traceless part Scalar part

Statistical property of 49 Fundamental assumption: The fluctuating stress fields are balanced in each region The scalar part is determined from the traceless part !

Probability density of stresses 50 :spherical coordinates Boundary conditions: (spherical symmetric potential between the ball and particles) large deviation functional ( large deviation principle )

51 PART VI Highlight

Stress at the surface 52 z-component of the force per unit area on the surface (= average stress) (continuity of the total stress) macroscopic fluctuation probability density of the average stress

Saddle point estimation 53 (large deviation property) (contraction principle) variational function (Lagragian) Lagrange multiplier

Variational problem 54 boundary conditions We impose (natural boundary condition) Euler-Lagrange equation

Euler-Lagrange equations 55 (equivalent to the Stokes equations !)

Result 56 Stokes’ law for the slip BC in hydrodynamics !

Rough surface of the ball 57 boundary conditions natural boundary conditions so that the E-L is obtained at the surface Stokes’ law for the stick BC in hydrodynamics !

Summary of the result 58 hydrodynamic equations Green-Kubo formula (1954) Stress fluctuation in the bulk Stress fluctuation at the surface Kirkwood’s formula (1946) Stokes (1851) Formulate the connection between bulk and surface Itami-Sasa (2015) Arxiv: We have re-derived Stokes’ law from Kirkwood’s formula and Green-Kubo formula with the aid of large deviation theory.

New prediction 59 at the surface Violation of CLT (NO divergence) Stokes’ law Short-range disorder is exactly cancelled by the long range correlation

Remark 60 (large deviation property) (contraction principle) Additivity principle : A variational principle determining the large deviation of the time averaged current for non-equilibrium lattice gases (Bodinue and Derrida, 2004;Beritini,Sole, Gabrielli, Jona-Lasino,Landim, 2005)

61 PART VII Epilog

Summary of my talk 62 1)Basic concepts in non-equilibrium statistical mechanics (20-th century ) large deviation theory 2)Recent developments in non-equilibrium statistical mechanics (for last two decades) symmetry and variational principle in large deviation theory 3) Presentation of our recent work (2015) a Surface-Bulk correspondence in non-equilibrium statistical mechanics

Last message 63 Non-trivial correlation at the surface can be calculated by the variational principle in the bulk Are there any relations with holography ?