New concepts emerging from a linear response theory for - PowerPoint PPT Presentation

New concepts emerging from a linear response theory for nonequilibrium Marco Baiesi � Physics and Astronomy Department “Galileo Galilei”, � University of Padova, and INFN � � in co lm aboration with Christian Maes, Urna Basu, � Eugenio Lippie lm o, Alessandro Sarracino, Bram Wynants, Eliran Boksenbojm GGI Workshop, 26-30 May, 2014

Overview Linear response � Linear response to temperature kicks � Entropy production and something else � Time-symmetric quantities: how many?

A macroscopic FPU? Livia Conti, Lamberto Rondoni, et al: www.rarenoise.lnl.infn.it

Linear response for FPU? How does a Fermi-Pasta-Ulam chain react to a change in one temperature? � Nonequilibrium specific heat variance of the energy � 6 = Compressibility in nonequilibrium?

Fluctuation-Dissipation Th.: Kubo � d h A ( t ) i = 1 d ds h A ( t ) V ( s ) i E → E − h s V dh s T An observable A(t) reacts to the appearance of a potential V(s) � 1 d is the entropy production from -V dsV ( s ) T

Out of equilibrium: many FDT a1) perturb the density of states and evolve (Agarwal, Vulpiani & C, Seifert & Speck, Parrondo &C,…) � a2) “bring back” the observable to the perturbation (Ruelle) � b) probability of paths (Cugliandolo &C, Harada- Sasa, Lippiello &C, Ricci-Tersenghi, Chatelain, Maes, …) � short review: Baiesi & Maes, New J. Phys. (2013)

Path probability (Markov), ω → { x s } for 0 ≤ s ≤ t Overdamped Langevin � p dx s = µ F ( s ) ds + 2 µT dB s � Discrete states C, C’, …with jump rates W ( C → C 0 )

Diffusion Probability of a sequence dx 0 , dx 1 , dx 2 , . . . ⇢ ( dB i ) 2 � dP i = (2 π dt ) − 1 / 2 exp 2 dt ⇢ [ dx i − µF ( i )] 2 � = (4 π µTdt ) − 1 / 2 exp 4 µTdt P ( ω ) = lim dt → 0 Π i dP i

Diffusion + perturbation dx s = µ F ( s ) ds + h s µ ∂ V p ∂ x ( s ) ds + 2 µT dB s Perturbation changes the path probability � Ratio of path probabilities is finite for dt → 0 P h ( ω ) P ( ω )

Susceptibility (h>0 for s>0) Generator ⇢ h Z t P h ( ω ) � 2 T [ V ( t ) − V (0)] − h P ( ω ) = exp LV ( s ) ds 2 T 0  P h ( ω ) ⌧ �� h A ( t ) i h � h A ( t ) i = A ( t ) P ( ω ) � 1 Susceptibility h A ( t ) i h � h A ( t ) i χ AV ( t ) = lim h h → 0

Markov generator ∂ x + µT ∂ 2 L = µF ∂ In this case: ∂ x 2 h LV i = d dt h V i interpret as expected variation: �� ⌧ dV � LV ( x ) = � dt � x

Response function Z t χ AV ( t ) = R AV ( t, s ) ds 0  d � R AV ( t, s ) = 1 ds h V ( s ) A ( t ) i � h LV ( s ) A ( t ) i 2 T 1/2 entropy production � minus 1/2 “expected” entropy production

Response function Z t χ AV ( t ) = R AV ( t, s ) ds 0  d � R AV ( t, s ) = 1 ds h V ( s ) A ( t ) i � h LV ( s ) A ( t ) i 2 T Entropic term Frenetic term Baiesi, Maes, Wynants, PRL (2009) � Lippiello, Corberi, Zannetti, PRE (2005) �

Negative response The sum of the two terms may be <0 �  d � R AV ( t, s ) = 1 ds h V ( s ) A ( t ) i � h LV ( s ) A ( t ) i � 2 T Example: negative mobility for strong forces Baerts, Basu, Maes, Safaverdi, PRE (2013) �

Negative mobility

The structure of R(t,s) is different for inertial systems � Achieved in a standard path-space comparison, with different drift terms (Radon-Nicodym derivative, Girsanov Th.) p dx s = µ F ( s ) ds + 2 µT dB s

The structure of R(t,s) is different for inertial systems � Achieved in a standard path-space comparison, with different drift terms (Radon-Nicodym derivative, Girsanov Th.) p dx s = µ F ( s ) ds + 2 µT dB s What happens if we change the noise term?

Mathematical problem The response to T kicks involves changing the noise term! � P h ( ω ) not well defined for different noises � P ( ω ) However: we are interested in the limit h—> 0

T(1+h), small deviation from T dP h − [ dx t − µF ( t ) dt ] 2 ⇢ � = (2 π T (1 + h ) dt ) − 1 / 2 exp t 4 µT (1 + h ) dt ) /dP t dP t ⇢ h � T + ( dx t ) 2  �� 2 µdt + µ 2 F 2 ( t ) dt + µT ∂ F = exp ∂ x ( t ) dt � F ( t ) � dx t 2 T Dangerous term (mathematically) entropy production, � new terms as before ⇢ h � 2 [ dS − B S dt ] + h = exp 2 [ dM − B M dt ]

Four terms: heat flux / temperature dS ( t ) = dQ ( t ) = � 1 T F ( t ) � dx t Entropy production � T Expected entropy production �  � B S ( t ) dt = − µ F 2 ( t ) + T dF dx ( t ) dt T � [Sekimoto, Stochastic Energetics] �  ( dx t ) 2 � dM ( x t ) ≡ 1 Activity (?) � 2 µdt − T T Expected activity (?) B M ( t ) dt ⌘ h dM i ( x t ) = µ 2 T F 2 ( t ) dt

Four terms: heat flux / temperature dS ( t ) = dQ ( t ) = � 1 T F ( t ) � dx t Entropy production � T Expected entropy production �  � B S ( t ) dt = − µ F 2 ( t ) + T dF dx ( t ) dt T � [Sekimoto, Stochastic Energetics] Time symmetric �  ( dx t ) 2 � dM ( x t ) ≡ 1 Activity (?) � 2 µdt − T T Expected activity (?) B M ( t ) dt ⌘ h dM i ( x t ) = µ 2 T F 2 ( t ) dt

Dynamical activity Very relevant in kinetically constrained models � Count the number of jumps between states � time-symmetric quantity � Lecomte, Appert-Rolland, van Wijland, PRL (2005) � � Merolle, Garrahan, D. Chandler, PNAS (2005) �

Relation with activity The (dx)^2 term should scale as the number of successful jumps in an underlying random walk description

Susceptibility Susceptibility Z t Z t χ A ( t ) = 1 ⌧ i� h A ( t ) S ( t ) − B S ( s ) ds + M ( t ) − B M ( s ) ds 2 T 0 0 antisymmetric � symmetric terms � (entropy produced) (frenetic contributions + activity)

Example: harmonic spring Susceptibility of the internal energy

Inertial dynamics � dx t = v t dt p m dv t = F ( x t ) dt − γ v t dt + 2 γ T d B t Harada-Sasa/stochastic energetics for the entropy production terms T dS ( x t , v t ) = mv t � dv t � F ( x t ) v t dt T B S ( x t , v t ) = γ m [ T − mv 2 t ] Harada-Sasa for transients/jumps: Lippiello, Baiesi, Sarracino, PRL (2014)

Inertial dynamics T dM ( x t , v t ) = ( m dv t ) 2 − T − m γ F ( x t ) dv t 2 γ dt " ◆ 2 # ✓ F ( x t ) T B M ( x t , v t ) = γ v 2 t − 2 γ Susceptibility Z t Z t χ A ( t ) = 1 ⌧ i� h A ( t ) S ( t ) − B S ( s ) ds + M ( t ) − B M ( s ) ds 2 T 0 0 antisymmetric symmetric terms

Conclusions General scheme: probability of trajectories -> physics � Time-symmetric quantities in response formulas � Not only dissipation, but also “activation” � Name(s): dynamical activity, frenesy, traffic, … � Attempt to compare trajectories with different T