New concepts emerging from a linear response theory for

nonequilibriumMarco Baiesi!

Physics and Astronomy Department “Galileo Galilei”, !University of Padova, and INFN!

!in collaboration with Christian Maes, Urna Basu,!

Eugenio Lippiello, 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!

Compressibility in nonequilibrium?

6=

Fluctuation-Dissipation Th.: Kubo!

An observable A(t) reacts to the appearance of a potential V(s)!

is the entropy production from -V

dhA(t)idhs

=1

T

d

dshA(t)V (s)i E ! E � hsV

1

T

d

dsV (s)

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),

Overdamped Langevin!

!

Discrete states C, C’, …with jump rates

! ! {xs} for 0 s t

dxs = µF (s) ds+p

2µT dBs

W (C ! C 0)

Diffusion

Probability of a sequence dx0, dx1, dx2, . . .

dPi = (2⇡dt)�1/2exp

⇢(dBi)

2

2dt

�

= (4⇡µTdt)

�1/2exp

⇢[dxi � µF (i)]

2

4µTdt

�

P (!) = limdt!0

⇧idPi

Diffusion + perturbation

Perturbation changes the path probability!

Ratio of path probabilities is finite for

dxs = µF (s) ds+hsµ@V

@x

(s)ds+p

2µT dBs

dt ! 0

Ph(!)

P (!)

Susceptibility (h>0 for s>0)

Susceptibility

hA(t)ih � hA(t)i =⌧A(t)

Ph(!)

P (!)� 1

��

�AV (t) = limh!0

hA(t)ih � hA(t)ih

Generator

Ph(!)

P (!)= exp

⇢h

2T[V (t)� V (0)]� h

2T

Z t

0LV (s)ds

�

Markov generator

In this case: L = µF

@

@x

+ µT

@

2

@x

2

hLV i = d

dthV i

LV (x) =

⌧dV

dt

�����x

interpret as expected variation:

Response function

1/2 entropy production !

minus 1/2 “expected” entropy production

�AV (t) =

Z t

0RAV (t, s)ds

RAV (t, s) =1

2T

d

dshV (s)A(t)i � hLV (s)A(t)i

�

Response function

�AV (t) =

Z t

0RAV (t, s)ds

RAV (t, s) =1

2T

d

dshV (s)A(t)i � hLV (s)A(t)i

�

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!

!

Example: negative mobility for strong forces

Baerts, Basu, Maes, Safaverdi, PRE (2013)!

RAV (t, s) =1

2T

d

dshV (s)A(t)i � hLV (s)A(t)i

�

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.)

dxs = µF (s) ds+p

2µT dBs

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.)

What happens if we change the noise term?

dxs = µF (s) ds+p

2µT dBs

Mathematical problem

The response to T kicks involves changing the noise term!!

not well defined for different noises!

However: we are interested in the limit h—> 0

Ph(!)

P (!)

T(1+h), small deviation from T

dP

ht

dPt= (2⇡T (1 + h)dt)

�1/2exp

⇢� [dxt � µF (t)dt]

2

4µT (1 + h)dt

)

�/dPt

= exp

⇢h

2T

�T +

(dxt)2

2µdt

+

µ

2

F

2(t)dt+ µT

@F

@x

(t)dt� F (t) � dxt

��

= exp

⇢h

2

[dS �BS dt] +h

2

[dM �BM dt]

�

Dangerous term (mathematically)

entropy production, !as before new terms

Four terms:

Entropy production!

Expected entropy production!

!

!

Activity (?)!

Expected activity (?)

dS(t) =dQ(t)

T

= � 1

T

F (t) � dxt

BS(t)dt = � µ

T

F

2(t) + T

dF

dx

(t)

�dt

dM(xt) ⌘1

T

(dxt)2

2µdt� T

�

BM (t)dt ⌘ hdMi (xt) =µ

2TF

2(t)dt

[Sekimoto, Stochastic Energetics]

heat flux / temperature

Four terms:

Entropy production!

Expected entropy production!

!

!

Activity (?)!

Expected activity (?)

dS(t) =dQ(t)

T

= � 1

T

F (t) � dxt

BS(t)dt = � µ

T

F

2(t) + T

dF

dx

(t)

�dt

dM(xt) ⌘1

T

(dxt)2

2µdt� T

�

BM (t)dt ⌘ hdMi (xt) =µ

2TF

2(t)dt

[Sekimoto, Stochastic Energetics]

heat flux / temperature

Time symmetric

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

�A(t) =1

2T

⌧A(t)

hS(t)�

Z t

0BS(s)ds+M(t)�

Z t

0BM (s)ds

i�

antisymmetric!(entropy produced) symmetric terms!

(frenetic contributions + activity)

Susceptibility

Example: harmonic spring

Susceptibility of the internal energy

Inertial dynamics!

Harada-Sasa/stochastic energetics for the entropy production terms

T dS(xt, vt) = mvt � dvt � F (xt)vtdt

T BS(xt, vt) =�

m

[T �mv

2t ]

dxt = vtdt

mdvt = F (xt)dt� �vtdt+p

2�T dBt

Harada-Sasa for transients/jumps: Lippiello, Baiesi, Sarracino, PRL (2014)

Inertial dynamics

T dM(xt, vt) =(mdvt)2

2�dt� T � m

�

F (xt)dvt

T BM (xt, vt) =�

2

"v

2t �

✓F (xt)

�

◆2#

�A(t) =1

2T

⌧A(t)

hS(t)�

Z t

0BS(s)ds+M(t)�

Z t

0BM (s)ds

i�

antisymmetric symmetric terms

Susceptibility

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