Nonlinear response theory in long-range Hamiltonian systems - PowerPoint PPT Presentation

2014/05/27@The Galileo Galilei Institute for Theoretical Physics Advances in Nonequilibrium Statistical Mechanics: large deviations and long-range correlations, extreme value statistics, anomalous transport and long-range interactions Nonlinear response theory in long-range Hamiltonian systems Yoshiyuki Y. YAMAGUCHI (Kyoto University, JAPAN) In collaboration with Shun Ogawa (Kyoto University)

Main topics We propose a nonlinear response theory for long-range Hamiltonian systems. 1) Reponse to external field → Strange critical exponents and scaling relation 2) Reponse to perturbation → Discussion on limitation of the theory

Response Response Observing the response, we get information of the black-box.

Response Response External mag. field /perturbation Magnetization Ferro mag. body Response

Hamiltonian mean-field model A paradigmatic toy model of a ferro magnetic body Each spin interacts with the other spins attractively All interactions are only through the magnetization (mean-field) � M N N N p 2 2 − 1 j � � � H = cos( q j − q k ) − h cos q j 2 N j =1 j,k =1 j =1 h : external mag. field

Critical phenomena in HMF ( h = 0 ) M T Tc Critical phenomena of mean-field systems are analysed by Landau theory

Landau theory F ( M ) = a 2( T − T c ) M 2 + b 4 M 4 + · · · − hM Free energy: d F d M = a ( T − T c ) M + bM 3 − h = 0 Realized M :

Landau theory d F d M = a ( T − T c ) M + bM 3 − h = 0 Realized M : Critical exponents

Landau theory d F d M = a ( T − T c ) M + bM 3 − h = 0 Realized M : Critical exponents β = 1 M ∝ ( T c − T ) β h = 0 : 2

Landau theory d F d M = a ( T − T c ) M + bM 3 − h = 0 Realized M : Critical exponents β = 1 M ∝ ( T c − T ) β h = 0 : 2 � d M � ∝ ( T − T c ) − γ + h � = 0 : γ + = 1 T > T c � d h � h → 0

Landau theory d F d M = a ( T − T c ) M + bM 3 − h = 0 Realized M : Critical exponents β = 1 M ∝ ( T c − T ) β h = 0 : 2 � d M � ∝ ( T − T c ) − γ + h � = 0 : γ + = 1 T > T c � d h � h → 0 ∝ ( T c − T ) − γ − γ − = 1 T < T c

Landau theory d F d M = a ( T − T c ) M + bM 3 − h = 0 Realized M : Critical exponents β = 1 M ∝ ( T c − T ) β h = 0 : 2 � d M � ∝ ( T − T c ) − γ + h � = 0 : γ + = 1 T > T c � d h � h → 0 ∝ ( T c − T ) − γ − γ − = 1 T < T c M ∝ h 1 /δ T = T c : δ = 3

Landau theory Scaling relation γ ± = β ( δ − 1) Critical exponents β = 1 M ∝ ( T c − T ) β h = 0 : 2 � d M � ∝ ( T − T c ) − γ + h � = 0 : γ + = 1 T > T c � d h � h → 0 ∝ ( T c − T ) − γ − γ − = 1 T < T c M ∝ h 1 /δ T = T c : δ = 3

Question Landau theory gives critical exponents in the context of statistical mechanics. Q. Does dynamics give the same critical exponents ? q j = ∂H p j = − ∂H ˙ , ˙ ∂p j ∂q j For simplicity, we start from themal equilibrium states: → β = 1 / 2 .

Vlasov approach N -body: N N � � p 2 2 − 1 j � � H = cos( q j − q k ) − h cos q j 2 N j =1 k =1 1 -body: H [ f ] = p 2 � cos( q − q ′ ) f ( q ′ , p ′ , t ) dq ′ dp ′ − h cos q 2 − Vlasov equation: ∂f ∂t = ∂ H [ f ] ∂f ∂p − ∂ H [ f ] ∂f ∂q = {H [ f ] , f } ∂q ∂q

Linear response theory - Patelli et al., PRE 85 , 021133 (2012) - Ogawa-YYY, PRE 85 , 061115 (2012) - Ogawa-Patelli-YYY, PRE 89 , 032131 (2014) Critical exponents β = 1 M ∝ ( T c − T ) β h = 0 : 2 ❄ � d M � ∝ ( T − T c ) − γ + h � = 0 : γ + = 1 γ + = 1 � d h � h → 0 γ − = 1 ∝ ( T c − T ) − γ − γ − = 1 4 M ∝ h 1 /δ T = T c : δ = 3

Nonlinear response theory We need a nonlinear response theory for δ . Check the scaling relation γ = β ( δ − 1) . Critical exponents β = 1 M ∝ ( T c − T ) β h = 0 : 2 � d M � ∝ ( T − T c ) − γ + h � = 0 : γ + = 1 γ + = 1 � d h � h → 0 γ − = 1 ∝ ( T c − T ) − γ − γ − = 1 4 M ∝ h 1 /δ T = T c : δ = 3 δ = ?

Idea f ini : Initial stationary state f 0 : Initial state with perturbation ǫg 0 f A : Asymptotic state

Idea Normal decomposition: f = f ini + ǫg H [ f ini ] drives the system (cf. Landau damping)

Idea Our decomposition: f = f A + ǫg T H [ f A ] drives the system

Asymptotic state Contours of f 0 Contours of f A f A = (average of f 0 over iso- H [ f A ] curve)

Asymptotic state Contours of f 0 Contours of f A f A = (average of f 0 over iso- H [ f A ] curve) ⇓ ( θ, J ) : Angle-action associated with H [ f A ] f A = � f 0 � J : Average over θ (iso − J curve)

Idea of re-arrangement itself is not new f A = � f 0 � J : Re-arrangement of f 0 along iso- J curve 1 -level waterbag initial distribution - Leoncini-Van Den Berg-Fanelli, EPL 86 , 20002 (2009) - de Buyl-Mukamel-Ruffo, PRE 84 , 061151 (2011) multi-level waterbag initial distribution - Ribeiro-Teixeira et al., PRE 89 , 022130 (2014)

What’s new Landau like equation for asymptotic M = ⇒ Critical exponents Justification of theory (omitting ǫg T ) by the hypotheses H0. The asymptotic state f A is stationary. H1. f ( t ) is in a O ( ǫ ) neighbourhood of f ini . H2. We may omit O ( ǫ 2 ) . = ⇒ Discussion on limitation of the theory

Self-consistent equation for M � f A = � f 0 � J = ⇒ M = cos θ � f 0 � J dqdp ✻ J depends on M through H [ f A ] H [ f A ] = p 2 2 − ( M + h ) cos q We expand the self-consistent equation for small M .

Expansion of self-consistent equation We focus on homogeneous f ini ( p ) . � M = cos q � f 0 � J dqdp √ ✲ power series of M + h Expansion p √ Separatrix width is of O ( M + h ) q

Initial condition f 0 ( q, p ) = Ae − p 2 / 2 T (1 + ǫ cos q ) � ✒ ✻ � Homogeneous Maxwellian Perturbation After long computations...

Landau like equation − ǫa ( M + h ) 1 / 2 + b ( T − T c )( M + h ) + c ( M + h ) 3 / 2 − h = 0 a, b, c > 0 cf. Landau theory: a ( T − T c ) M + bM 3 − h = 0

Landau like equation − ǫa ( M + h ) 1 / 2 + b ( T − T c )( M + h )+ c ( M + h ) 3 / 2 − h = 0 ǫ = 0 : M ∝ ( T − T c ) − 1 h T > T c : Linear response

Landau like equation − ǫa ( M + h ) 1 / 2 + b ( T − T c )( M + h ) + c ( M + h ) 3 / 2 − h = 0 ǫ = 0 : M ∝ ( T − T c ) − 1 h T > T c : Linear response M ∝ h 2 / 3 T = T c : Nonlinear response δ = 3 / 2

Response to external field (numerical test) 10 0 T c = 0 . 5 T = 0 . 50 T = 0 . 51 T = 0 . 55 10 − 1 T = 0 . 60 T = 0 . 70 10 − 2 Slope= 2 / 3 M 10 − 3 10 − 4 Slope= 1 10 − 5 10 − 6 10 − 6 10 − 5 10 − 4 10 − 3 10 − 2 10 − 1 h Ogawa-YYY, PRE 89 , 052114 (2014) [slightly modified]

Scaling relation in Vlasov dynamics Scaling relation holds even in the Vlasov dynamics ! γ − = β ( δ − 1) Critical exponents β = 1 β = 1 M ∝ ( T c − T ) β h = 0 : 2 2 � d M � ∝ ( T − T c ) − γ + h � = 0 : γ + = 1 γ + = 1 � d h � h → 0 γ − = 1 ∝ ( T c − T ) − γ − γ − = 1 4 δ = 3 M ∝ h 1 /δ T = T c : δ = 3 2

Origin of the strange exponents The Vlasov equation has infinite invariants called Casimirs: � C [ f ] = c ( f ( q, p )) dqdp ∀ c smooth

Response to perturbation − ǫa ( M + h ) 1 / 2 + b ∆ T ( M + h ) + c ( M + h ) 3 / 2 − h = 0 ∆ T = T − T c h = 0 : � 2 � ( b ∆ T ) 2 + 4 ǫac � − b ∆ T + T > T c : M = 2 c

Response to perturbation − ǫa ( M + h ) 1 / 2 + b ∆ T ( M + h ) + c ( M + h ) 3 / 2 − h = 0 ∆ T = T − T c h = 0 : � 2 � ( b ∆ T ) 2 + 4 ǫac � − b ∆ T + T > T c : M = 2 c M = a T = T c : c ǫ

Response to perturbation (numerical test) 0.1 T c = 0 . 5 T = 0 . 50 T = 0 . 51 T = 0 . 55 0.08 T = 0 . 60 T = 0 . 70 0.06 M 0.04 0.02 0 0 0.05 0.1 0.15 0.2 ǫ Ogawa-YYY, PRE 89 , 052114 (2014)

Discrepancy ? We omitted O ( ǫ 2 ) term. ⇓ The transient part ǫg T can be omitted. 0.1 T = 0 . 50 T = 0 . 51 T = 0 . 55 0.08 T = 0 . 60 T = 0 . 70 Omitting transient part ǫg T implies 0.06 M 0.04 omitting the Landau damping. 0.02 0 0 0.05 0.1 0.15 0.2 ǫ T = T c : Damping rate is zero, and the theory works well. T ր : Damping rate grows, and the theory gets worse.

Numerical evidences � f − f 0 � L 1 � f − f ini � L 1 0.14 0.14 0.12 0.12 T = 0 . 5 T = 0 . 7 0.1 0.1 || f − f ini || L 1 || f − f 0 || L 1 0.08 0.08 T = 0 . 6 T = 0 . 6 0.06 0.06 0.04 0.04 T = 0 . 7 (a) 0.02 0.02 T = 0 . 5 (b) 0 0 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 t t Ogawa-YYY, PRE 89 , 052114 (2014) 0.7 0.6 0.5

Summary [Ogawa-YYY, PRE 89, 052114 (2014)] We proposed a nonlinear response theory for long-range Hamiltonian systems. It works not only for thermal eq. but also for QSSs. Response to external field: γ − = β ( δ − 1) Landau theory Response theory M ∝ ( T c − T ) β β = 1 / 2 β = 1 / 2 dM/dh ∝ ( T − T c ) − γ + γ + = 1 γ + = 1 dM/dh ∝ ( T c − T ) − γ − γ − = 1 γ − = 1 / 4 M ∝ h 1 /δ δ = 3 δ = 3 / 2 Response to perturbation: The theory works well at the critical point (no damping).

Thank you for your attention.

Appendix A T-linearization and omitting the transient part ǫg T